1. PARADOXA AND ANTINOMIES

PARADOX:
A known paradox I found in a text of Kurt-Ulrich Witt:
A set of snakes is given and a snake named 'S', which bites into the tail of every snake, which does not bite its tail itself. Then the question is asked, if the snake S bites into its own tail or not. This question is known as undecideble using the following consideration:
If the snake S is one of those snakes, which do not bite into their own tail, then it has to bite into its own tail.
If the snake S is one of those snakes, which do bite into their own tail, then it must not bite into its own tail.
This consideration makes a single (paradox) situation out of a starting situation and the manner of S to bite. Additionally there is imputed, that there is no snake besides S, which bites into other tails than their own ones, and that S can bite too in more than one tail at a time.
In fact, there is no argument given for the starting situation and only one operation (the biting). As the operation is conditioned, there can be found the clear result in spite of an unclear starting situation, that the snake S bites into its own tail - either it did it already or it does it now. Thus the snake S is at every time an element of the set of snakes, which bite into their own tail! This is changed only then, when the moment of the look at S is changed. This moment can be before or after biting. Only before biting, the snake S could be an element of the set of snakes, which do not bite into their own tail. But this moment is not an argument and can't be concluded by making sets.
As the starting situation is not argumented by defining a manner of biting, the question can't be answered, at which moment the snake S starts biting. This is not at all a "paradoxon", but a lack of arguments. Only, if you think, that something is to conclude, then you will feel paradox.
The solution of this known paradoxon is found here using common language, but is to find beyond any suspicion of logical unclearness, if you mention the dimension of time in the logic, stated using inverterlogic, which of course depends on every logic, but is normally ignored in the example of the snake S, although this is to imagine considering such a snake ( A bite starts at any moment...).
The inverterlogic does allow conclusions ('efficacy') only then, when there are arguments at the entries of the connected inverters (='relation'). Thus there is no doubt, that there is a lack of arguments in this task.
A paradoxon, told by B.Russell is completely analogue. He introduces a barber, who shaves everyone, which does not shave himself, and asks the question, if the barber shaves himself - either he did it already or he will till have to do it...

"RUSSELLS' ANTINOMY":
The well known Russell's antinomy, presented using mathematically notation, which once shocked the world of mathematicians, looks like that:
A set M is given, which includes as elements all those sets, which do not include themselves as an element. The question is asked, if the set M includes itself as an element or not.
Bertrand Russell found this antinomy, which then became famous, in defining sets due to Cantor's set theory, while considering the problem like that:
If the set M is an element of itself, then it can not be one of its elements, which shall only be sets, which do not include themelves as an element. Thus M must not include itself, if it shall include itself.
If the set M is not an element of itself, then it needs to be an element of itself, because every set like that needs to be an element of M. Thus M needs to include itself, if it does not include itself.
I considered this problem under 2.1.3. in the inverterlogic in respect to the problem, how to express sets (='quantities') using connected inverters, and wether sets can be an element of itself and how. As there exist only the values H,L in connected inverters, elements to distinguish can only appear as values made of many values H,L, which are ANDed as adresses, such defining the set. So defining a set is already a connexion and not only done by defining elements as adresses.
Thus in this respect Cantor's notion of a set is to complete.
The further consideration demonstrates, that a set can be appended to itself as an element only by connecting the result, which represents the set (after ANDing of the elements), using a next entry in ANDing the elements. This is a'back-line', as demonstrated under 2.2.2. in the inverterlogic and sketched below in [figure a]. In this case it can be completely shortened, because it can not change the result of ANDing in any way. The additional element, which is the set itself (and an infinite amount of further sets!), is indeed completely irrelevant. There is to say, that a set, which does not include itself, can not be distinguished from a set, which includes itself. Besides this the dimension of time appears too, which is not touched by the notion of a Cantorian set - the set can include itself only after giving it without this element, which therefore is completely bound to that matter, which it must not change too. This consideration says too, that the notion of a set made by Cantor is not really ready, but also, that Russell's antinomy isn't one. This antinomy appears only because of considering the irrelevant idea of a set, which includes itself. And of course everything is irrelevant too, what can be done with irrelevant things. But as Russell unfortunately could not see it like that, he had to be very engaged in evaluating "types", which should prevent of his antinomy. He therefore did not mention a dimension of time, but found, that the antinomy was caused by relating the set to itself.
I show here in respect to 2.1.3., how this appears in the inverterlogic:

The inverterlogic demonstrates, that such a relation is not to think without a dimension of time, caused by the 'back-line'. But more interesting is, that there are two propositions (...includes itself/...includes NOT itself), which are normally realized as a contradiction, both are possible and therefore not false, but are not to distinguish in respect to the result, so that additional efforts in connecting inverters can be used as a criterion for treating the one of those propositions as irrelevant. This kind of proof and the result of 'irrelevance' is not available in common mathematics. But there can be no doubt, that the notion of 'irrelevance' is a solution of proving evidence as well as 'TRUE' and 'FALSE'. There is to say, that irrelevant connexions can be expensive, if a transistor-logic is to make due to this logic. They can make a transistor-logic even useless, if it is able to stumble at antinomies as Russell (really possible in programs).
In this context the common manner of negating using Boole's "negation operator" can be discussed too. If it is seen aequivalent to an inversion, the above mentioned back-line can effect an oscillation. If you negate the set M (the common NOT M, expressed by a negation line above M), while connecting another inverter in the back-line, then there are 3 inverters in the back-line and 2 of them can be omitted due to the first shortening rule in inverterlogic. The left not to shorten inverter then is connected to its own entry and oscillates because of this. Thus the common interpretation, that a NOT M is the opposite of M, leads to the "proof", that there must not be a set M, which does not include itself! Else a "contradiction" (="antinomy") were effected. This is one of the reasons, why I did not identify the notion of a negation with an inversion, but related it to significance (which can be effected by the value H at the exit ofan adress). The other value =L then is not to use as a 'NOT', but as unsignificance, which must not get any significance in following relations.
As negations are difficult to understand in this context, I additionally show the connexions of interest, using the notification of inverterlogic ( in respect to 2.1.3 and [figure a] above:

Now it is demonstrated as clearly as can be, that a negation operator can effect a "proof", which isn't one at all. In other words:
The inverter is not the known negation operator and its values H,L are not aequivalent to Boole's values TRUE and FALSE. Thus the inverterlogic is not a Boole's logic, although Boole's logic can be expressed using connected inverters without a lack. The inverterlogic therefore allows to state, that the interpretation of values TRUE and FALSE in Boole's relations as a proof of evidence is not possible a priori. So truth is not to operate in any manner, but is to relate a posteriori - at pleasure of an observer resp. by making significance in following relations, taking care for rules too, which are not Boole's Algebra, but can be deduced from the inverter-axiom.

THE CLASSICAL PARADOXON:
A cretian named Epimenides says, that every cretian lies.
Then he would be a lier too and his statement would be wrong. But if cretians do not lie, then this cretian tells the truth...
To make this really paradox, you nevertheless will have to identify "to lie" completely as "never say the truth", because liers indeed sometimes say the truth - if they want sugar, they will not ask for salt...
As you could want to find a conclusion, which does not lead to a contradiction, the sequence of conclusions is to consider first.
The starting point are the arguments:
"no cretain tells the truth" and "this tells a cretian" and "this cretian tells the truth". The last argument is obviously implicite given, because else the proposition is not true a priori and not worth to consider.
The first conclusion connects the arguments:
"no cretian says the truth" and "this cretian says the truth".
One of these propositions has to be false. Without further arguments, there is no decision to find, if cretians tell the truth or not. The situation becomes paradox, because traditionally not only this conclusion is done, but more to overcome the not satisfying conclusion. But this can be an useful argument too, to ignore the statement of Epimenides and to omit else conclusions...
So normally at least a second conclusion is done, trying negated arguments. Traditionally a negation of the statement of Epimenides is tried, which is only one of two possibilities - and the wrong one:
"all cretians do NOT lie"
The right variant will be:
"NOT all cretians lie"
This second variant allows a lier Epimenides, which states, that all cretians lie, and who therefore shows a well known behaviour of liers.
Thus the implicite imputation, that Epimenides does not lie, as well as his statement, that all cretians lie, are to negate. Then there is nothing paradox in that sense, that there is no solution for the contradiction, if the arguments are tried negated.
On the way to the solution, I did using common sense of language just the same as needed in the inverterlogic. There will never exist an inverter without values (arguments). If any argument is not given (Epimenides does not lie), then the opposite one (Epimenides lies) is to try, if existing. And as demonstrated, there is a solution instead of a paradoxon. And I demonstrated too, that there may be more than one "opposite" arguments to get by negating an argument, so that negations in common mathematical usage are to consider well. Because of this, I made the notion of negation in the inverterlogic (under 2.1.2.) so that irritations like here facing Epimenides cannot appear. Normally there are the Boole's values TRUE and FALSE in use while negating, which are values in one place. But some people like to use these values in context with propositions and matters, which are not at all values in one place. Such arguments are adresses in the inverterlogic, which result certain values in more than one place as H in one place. The possible opposite result L appears in every other case and therefore is not a certain one. I derived the notion 'significance' from this, which follows the result =H (and does not preceed!) and which is obviously nonsense, if derived from the value L, which can mean the rest of the world. Negations commonly are inversions of such a significance, which then can not be an other significance (any "opposite" matter), but must be resulted as insignificance.
This example of Epimenides could be expressed in three ways of connecting inverters:
1) The contradiction between "no cretian tells the truth" and "this cretiantells the truth" will be resulted as insignificance.
2) Parallel connexions, which connect every possibility for inverting arguments and therefore result an argument in more than one place at the entries of a final connexion, which results, if any combination of arguments and their inversions can get significance.
3) Back-lines, which switch the inversions of arguments and thus test the same possibilities as tested under 2), but sequential.
The third possibility is normally choosen by a human on the way to a solution and obviously since more than 2000 years without the possibility for negations, which I demonstrate here. If that possibility would in fact not exist, then in the case 3) an oscillation would occur because of the back-line, which can be interpreted as the felt paradox situation. But in the inverterlogic, such an oscillation isn't miraculous, but quite normal, if there is an inversion in the back-line (a single inverter oscillates, if its exit is connected to its entry). So there is either a wrong connexion made, if an observer does not want a paradoxon, or he wishes the oscillation i.e. to get timing pulses. As these pulses can be counted using the means of inverterlogic, not only in the case 2) but in case 3) too, a contradiction can be found automatically. By this means inverterlogic is suited to verify.

CURRY'S PARADOXON
Haskell Curry introduced in 1942 a paradoxon using the sentence:
"If this sentence is true, then Santa Claus exists."
This sentence normally can not be rejected logical as not true and therefore prooves the existence of Santa Claus.
As every nonsense can be prooved as true by this way, this form: "If ... true, then ... true." is to consider more thoroughly and there is to remark first, that the class theory of Russell does not admit this proof, because there is a 'self relation' included similar to Russell's antinom.
There is in fact a significance (=meaning) made, which is to express just like this in inverterlogic. To make the point of view instantly clear, I transform the sentence:
"There may be a Santa Claus."
...showing, that the sentence does not tell anything about the sentence, but introduces only the then-consequence. Indeed, there is no argument given, if the if-condition is true. This can be only argumented by verifying the then-consequence. The sentence indeed includes exchanged cause and effect and needs to look like that:
"If there exists Santa Claus, then this sentence is true."
Now there is clearly to see, that the sentence introduces only the possibility of Santa Claus and a statement about truth or falsity without sense. So there is no paradoxon to see, but a lack of arguments in every case.
Making the notion 'significance' in the inverterlogic, I explained, that a significance can be made on principle at pleasure and therefore never must be extended as common to a proof using the the logical "values" TRUE and FALSE.In every case the significance follows results. And the truth is a significance too. So the sentence is reduced to the form:
"If ..., then ..."
...and by this way reduced to a "true", meaning evident relation.
You must not try it else by classifying falsity to reject, because relations, which mean falsity, have to be as "true" as those, which mean truth, if both are to contrast. Truth can only exist as the contradiction of falsity and therefore needs to be handled in the same logical manner. No computer could work, if possibilities of meanings could not be assigned at pleasure. Only then certain meanings can be selected.
So inverterlogic treats the sentence itself as right in every case, but rejects the intention to state truth by that way, opposite to common logic, which deals with the "values" TRUE and FALSE. Truth is only to find out by connecting further arguments to the proposition, that there exists a Santa Claus - quite aequivalent to common sense. Nobody accepts Santa Claus, because there exists a Santa Claus, but because one could like him and possibly could get him too.
So you will have to find lots of arguments, if you want to proove Santa Claus as existent. Only a complete list of arguments can be logical ANDed as a proof. If those arguments are true or false, can only be stated using the sense of a human being, which finally is the giver of significance and is convinced.
The inverterlogic demonstrates, that truth can't be produced only logically. Truth is to state only by taking results (a posteriori) and not only by giving arguments (a priori). Only by that way, there is to demonstrate, that not every logical connexion is the right one for a given purpose and that a result only then can tell you: "I am no result!" or "I am true!", if making significance creates such "sense". So Curry's "paradoxon" finally drives the intention ad absurdum, to produce truth mechanically in the stile of Boole using only values of certain meaning as criterion for a decision.

2. NUMBER THEORY AND CONSEQUENCES

PEANOS NUMBERS:
Modern arithmetics, which I will call "common" too in this context, are based on a number theory, which starts at natural numbers and their definition using 5 axioms, introduced by Peano. That's why these arithmetics are also called "Peano arithmetics" (=PA). I derived in the inverterlogic numbers and operations on numbers too, but did only the derivation there and not a critical comparison with common arithmetics. Here I will complete the explanation.
In fact Peano does not derive the natural numbers themselves, but decribes with his axioms only, how to find a next one, if one is already available. The natural numbers themselves are not reasoned, but are made "by the beloved god" as Kronecker said. So the notion of 'natural numbers' has to be accomplished besides the belief, that they really exist, by an anywhere, where they exist and by the idea, that this is infinite. Thus besides the axioms of Peano another 3 axioms are needed, which depend on the existance of the natural numbers, their home and their frequency. As the 0 is not derived by this, a next axiom is needed for this number. Maybe, that the catholic italian Peano could not ignore the catholic prohibition of the 0 in the middle age and so could not find the 0 to be natural enough...
In the inverterlogic (under 1.2.2.) the anywhere are connected inverters and the numbers are an interpretation of the values H,L of the inverter at a certain count of places, which are exits of inverters. The anywhere must not be any connexion, but needs to be one, which results every possible combination of H or L in a certain count of places. These combined and different values then can be ordered in a sequence using only the idea, that every combined value besides the smallest and the greatest shall exist between a greater and a smaller one. By this means values of adresses, which I derive under 1.2.1., become numbers, which at once built a number-system, where a single inverter makes the difference between one and every other number. So the difference between one number and the neighboured ones is not a 1, but an inverter in any place - this is true even for the difference between the greatest and the smallest number and thus establishs well the 0. And there is to state easily at once, that there is a count at pleasure of possibilities to make number-systems. If you want to treat these numbers to be 'natural', then you will have to select arbitrarily one of the number-systems. I used in the inverterlogic the well known dual-digits-system to derive operators for working on numbers, but not because I found it more natural than other number-systems, as Leibniz found, but because it is familiar enough to make the derivation easier intelligible.
In any case these numbers are not made by "the beloved god", and their home is not an anywhere in prayers, but they can not only exist i.e. in a transistor circuit, but be operated too, independant of any human, who wants to believe or just see that or not. Although these numbers are already homelike and familiar in computing, they are considered in the inverterlogic for the first time thoroughly enough to demonstrate the consequences, which make a lot things come in question, which are in common number theory and arithmetics not at all questionable.
For the first that infinity, which already Gauss frightened, is not existant in inverterlogic. There the numbers exist only as values of inverters and therefore can not exist without connected inverters. So such a relation is not a gods act too, but is placed in the pleasure of an observer or a constructor of electronical circuits.
I derived under 1.2.2. numbers without the need of a quasi-adding, as defined by Peano in his 5 axioms. I needed only that, what is given by the inverter-axiom and a single minded pleasure making a connexion (='relation'), which needs to result every possible combination of values H,L. At once a adicity of places is done, which caracterizes number-systems, while Peano spared making of number-systems as a task for practical men and so omitted the trouble with making carries, which appears in inverterlogic as the fundamental principle for operating numbers.
So Peano had not only to reason natural numbers by stating, that they are quite unreasoned already out there, but he needed an operator too, which has to exist too, before something to operate exists, because this operator creates its own arguments. The result needs to preceed the arguments.
Such an acrobatism is not needed, if numbers are derived from the inverter-axiom. After the numbers are already defined, I derive under 2.1.3.2.1. operators and do it also only by connecting inverters. The first step is to derive an incrementer (under 2.1.3.2.1.1.), which does in effect the same as Peano's quasi-addition, but including a number-system and the 0, which is already found as smallest number under 1.2.2. Already there was to state, that while incrementing the smallest number follows the greatest one. So the sequence of these numbers is to order not in a straight line, but in a circle, caused by finity. But this is not as important as the fact, that an incrementer cannot get from smaller to greater numbers without making carries. And as the incrementer shall be designed regularly in every place, it needs to make a carry at the most significant place too. Opposite to Peano's quasi-adding, a next place can not be created by this incrementer, results follow the arguments. This carry does not cause the next greater number as usual while calculating on paper. It does not cause the really next number too, which is the smallest one, but appears at once with it and thus this carry is a logical useable argument, that an other range of numbers is entered. In this way not only an easy useable upper border is given, but much more:
As an incrementing of all numbers in a certain count of places connected with further inverters in series of entries an exits becomes a decrementing, there is not only to state, that negative numbers exist, but that these ones are not on principle other ones than the positive numbers, but are only to distinguish using a additionally made significance of values H,L, which are meant to be numbers. This significance becomes a logical argument reasoned by a carry in the most significant place. So indeed it is the finity and not any operation to define, which reasons the negative numbers. These are the same numbers as the positive ones, only to distinguish using a 'number-attribute', which is a significance of the carry in the most significant place.
In the PA such number-attributes as a 'minus' are indeed common too. But the inverterlogic reveals the general principle, which in PA fades in the nebula of infinity. Because of finity there makes every operator (and there are as many ones as you like!) possibly a carry in the most significant place because different to PA these operators can not create inverters, needed for a next place. Numbers are only on the substrat of connected inverters existant. Only the creator of such connexions can get next greater numbers.
So because numbers can not be used without a (at least implicite) number-attribute, and because these attributes can be used in many variants due to an extended notion of a carry, it is possible to make pairs of sets of numbers at pleasure. Completely independant of an operator, but of course dependant to a number-attribute, numbers can be taken as negative, imaginary or something else. And there is no proof needed, that those numbers exist, because the numbers always appear in the same form and thus are the same ones in a number-system.
But playing with number-attributes will make new sense only then, if operators act like formulas, which can be i.e. algebraic equations. Those were as well as +,-,and others 'logical machines', which become 'state machines', when they are connected to the not always needed connexion, which makes the significance of a number-attribute.
A special kind of number-attributes are those, which rule setting of a point. Such number-attributes are derived in the inverterlogik without a consideration on the nature of exponents. It will do to consider shifting of a sequence of values H,L in the width of a number, which is shifting of a point, that is just as little available as a minus in inverterlogic. So the number-attribute includes a number ruling shifting steps. This method is not really new, but common practice in 'floating point calculation', which I described in context with my assembler translation programs ASMn and ASMat. Such number-attributes allow in every case results with a suitable count of places (called "precision"), where a point is anywhere included.
As the point is part of the number-attribute, the numbers itself can be always resulted with an incrementing and therefore are always countable and without gaps behind the point too.
But only incrementing or decrementing are operators, which result completely every possibility of settings in a given count of places using every possible argument. I considered possible quantities of results under 2.1.3.2.6. in the inverterlogic and had to state, that every(!) other operator results incomplete number quantities, if the 1 as an argument is excluded, which is the second implicite argument of incrementing. Such incomplete number quantities occur independant of the count of places for results and are caused by making carries, which characterizes (arithmetical) operators. (I do not use the expression "operator" for "logical operators" in the inverterlogic, because then every connexion of inverters were an operator. That is why I made the notion 'logical' depending on everything, you can do with inverters.)
Under 2.1.3.2.6. I made also the notion of 'complementary' numbers, which are not results, similar to transcendent numbers (not results of algebraic equations), but in respect to a concrete operator. This notion is more useable, because inverterlogic provides possible operators as many as you can like. So the complementary numbers at the one operator can be results instead of complementary numbers at another operator as well.
But there is no doubt, that in the places, where results appear, in every case a complete quantity of numbers is possible, which is combined out of the two incomplete quantities of results and complementary numbers, which, as said, can only be found, if the 1 as an argument is excluded. Thus an often used notion of transcendent numbers is ad absurdum, which imputes, that there are numbers, which never can be calculated.
Depending on complete quantities of numbers, there is to say too, that the count is an even one, because inverters can result 2 values H or L, while the greatest value is to interpret as an odd number, because the smallest number is the 0 and not the 1. Not only because of this, the 0 is an even number, but also because it follows the greatest (odd) number - in that case accompanied by a carry in the most significant place. Thus there are only even nulls!
Nevertheless there are 'incomplete' results, which are not right resulted in the same count of places as the arguments. I consider this too under 2.1.3.2.6. and can state, that there is in every case a finite and calculatable count of places, which are enough to make complete results due to a given count of places for the arguments. Of course this does not depend on those operators, which should make infinite iterations on principle, i.e. if there can be remainders. Thus infinite irrational numbers are excluded too in a finite inverterlogic. In case of divisions, remainders have to be a part of results! By making significance due to that remainders, you can make a criterion to distinguish periodical fractions from not periodical ones, which can be treated to be infinite.
So in this case too, the number-attribute and not the number itself, is the starting point to distinguish different quantities of numbers, which are countable as well as the steps of shifting in the number-attribute, which are used to express shifting of places.
For completeness there is to say, that these numbers contain values of prime numbers behind the point too - expressed in decimal digits: 0.13 can be devided only by 0.13 and 0.01 without a remainder.
Of course there is the question to ask, if the numbers in the inverterlogic are really so different to Peano's numbers, so that the stated peculiarities and conclusions do not depend on Peano-arithmetics too. Obvious the striking difference is the (axiomatically reasoned) infinity in the PA. In practice of calculating this infinity does not matter, while every calculation, which can be done using the PA, can also be done with identical effect in the inverterlogic. And as the iteration machine, including the 'von Neumann structure' as a variant, is derived too using the inverterlogic, there is to say, that billions of people daily risk goods and chattle and life and limb on calculations with transistor logic, which never can be done else than in inverterlogic expressible. By this the evidence of the finite numbers and operators in the inverterlogic is much more proved than the evidence of the PA, which in fact is not the same in computers - infinite, given by God and Peano a.s.o...
But there is a proof besides risk and belief, that the statements found using the inverterlogic depend on the PA too without any difference. You can easily see, that the arbitrariness, which is allowed in the inverterlogic, indeed does not enable you to reach infinity, but is enough to have a look at the border of finity and to get the conviction, that this border is not changed in its nature, if it is extended in direction of infinity at pleasure. So the axiomatically reasoned infinity is indeed a already of Gauss despised trick, to make this border invisible and to establish the nice illusion, that every iteration ends there straight pressed and without any fringe. In peculiar there is no infinity, where something could end, i.e. as a null. As I state under 2.1.3.2.6., the results do normally not fit into the same count of places as the operands. This is not changed, if operators are created to create the needed places. As mathematicians are familiar with the situation, that the paper ends, if a calculation is not terminated arbitrarily, there is no doubt, that all numbers are finite in every case and operators, which create places, are an absurd idea, possible only if the need of countable places is abolished by an axiom.

3. SET THEORY AND CONSEQUENCES

CANTOR'S DIAGONAL METHOD:
Cantor (the creator of the set theory), introduced a method to produce a new number proceeding from a list of rational numbers (which he treated to be real numbers), which is never included in such a list. As this method became a famous means to make a proof of incompleteness, it is worth to consider it in detail:
The method served originally as a proof, that the set of real numbers were not countable, but "over-countable" (translated sense of german words). Cantor meant using these words, that no natural number were great enough to count the elements of the set of real numbers, although the set of natural numbers is treated to be infinite.
This method can be easily explained by common usage of language and is named "diagonal method" (the translated german words "Diagonalschnitt" mean "diagonal cut"):
After writing a complete list of rational (meant to be real) numbers, each in one line, take the first digit behind the point in the first line, the second digit behind the point in the second line, the third in the third a.s.o...
These digits are to write sequentialy in one line in the first, second, third a.s.o. place behind the point as a new number to create. So you have to read diagonal, but to write in sequence and then you have to transform the digits in the places, which range from 0 to 9 (in a decimal system), so that only the digits 1 and 2 appear behind the point.
The rule, how to code using 1 and 2 is not really of interest, because it is irrelevant. But as this coding provoked other meaning and coding of diagonal read signs and by this created strings, this final step is not to ignore.
Cantor then asserted, that the number, created by his method, were a real number, which never could occur in a complete list of real numbers. For this reason the set of real numbers were not countable, but were "over-countable", because he was able to extend the list at pleasure.
In respect to operators made using inverters, there is to see a fact at once, which Cantor could already test too - that the number made using the diagonal method is not at all a rational one resp. a Cantorian 'real' number, if the 1 as an argument is excluded. The Cantorian real numbers were at that time the results of dividing or making a root and because of this were operated in a quite different way than using the diagonal method. Only by chance the diagonal argument could be a such a Cantorian real number too - and this can be found in a complete list in every case!
But in fact, there are numbers, which can not occur in such a list. These are the complementary numbers, considered under 2.1.3.2.6. and defined to be no results. Only those numbers can be created using Cantor's diagonal method (but not all of them). These numbers are sorted nowadays as elements of the transcendent numbers, which are defined as not possible results of any algebraic equations. By appending these numbers the belief is reasoned, that the sequence of numbers behind the point is made infinite close, meaning without gaps.
But out of doubt the numbers containing a point are resulted without gaps by incrementing. For this sake, you have to start at the last significant place behind the point, which nevertheless is unreachable in the nebula of infinity. But if you imagine those two infinite sets of numbers before and behind the point, then the question is easily to answer, how great the cardinality of the natural numbers is: the squared cardinality of natural numbers plus the cardinality of natural numbers (every natural number is to combine with every number behind the point. And there are to add those numbers with a 0 before the point.). You can qualify this as "over-countable", because you can obviously use less than the square root of the needed count for counting. But there is obvious as well, that the infinity as a need for considering numbers is ad absurdum. So there is to say, that if you do not put infinity by an axiom in the world, and if you then can get the last significant place behind the point, then you can start incrementing there and will need indeed the double count of places as needed for natural numbers and so are able to count. And of course then all of those rational, irrational and transcendent numbers are countable. Statements about situations when reaching the smallest or greatest number in finity will remain true for ever.
As said already, the diagonal method was not only used by Cantor as a proof of his "over-countability", but has become too a part of other proofs depending on other matters and changed in make. So there is to explain, what the diagonal method in fact proves and which propositions are not to prove by this way.
Cantor started at a list (of seemingly real numbers), which listed results of operators. These results are in every case an incomplete quantity of numbers (if the operand 1 is excluded), which can be sorted indeed in a complete list. Such a list is to qualify in general using a length ( the count of elements in the list), which can be an odd number too. This is quite equal to a stack containing countable elements in every case, which I consider under 2.2.3.2. in the inverterlogic. Cantor's astonishing conclusion, that a complete list is incomplete, is indeed caused by comparing two different kinds of completeness, the completeness of a list at the one hand side and of a set of numbers at the other hand side. In fact he considered an incomplete quantity of numbers, which are results. Thus he only could proof the existance of completary numbers due to the considered operators. And this he did not well at all, because he could not create every complementary number using his diagonal method. In short:
The diagonal method is a proof for nothing else than complementary numbers.
In every case there is a notion of completeness driven ad absurdum by using the diagonal method in any proof. In every such proof a complete list, which does not contain a complete quantity of numbers, is compared to a complete quantity of numbers.
Thus every "proof", which uses the diagonal method, does not prove, what there is intended to prove. This depends an Gödel's proof as well as on Turing's analogue proofs. I perform a more thouroughly consideration of such proofs below.
[first published 20.3.2010]
In context to this topic, I found an interesting link to India:

4. COMPUTABILITY

HILBERT'S ENTSCHEIDUNGSPROBLEM AND TURING'S MACHINE:
In context with the 'Entscheidungsproblem' of Hilbert, Turing introduced the idea of a machine, which should produce the decisions, desired by Hilbert. The 'Entscheidungsproblem' was characterized by Hilbert with the question (analogous): "Is it possible to distinguish mathematical truth from falsity by a mechanical procedure?". The "mechanical procedure" was meant symbolically as the opposite way to a decision of a living and thinking human being, whose thinking as a means to decide should be excluded. So in fact the question is asked, if mathematical truth exists in the nature independent of an observer. Because of this Turing made some extra efforts to suggest, that his idea depends on a machine, that could be build. This machine should be made out of two important parts:
1) A part, which shall change read symbols to written ones due to a built in transmission behaviour. Besides reading and writing, this behaviour shall enable a motion between the symbols too.
2) A part, which shall carry (and store) the symbols and can be moved. Up to his time, Turing introduced this part as a magnetic tape, moved stepping under read/write-heads. So local cells are to distinguish, where the symbols are written or to read. The movement between the fields was only one of two steps to the neighboured fields or a stop.
So part 1) is a 'blackbox' (Turing called it a 'a-machine' resp. 'automatical machine'), which is not characterized constructively, but by an effect, described by a formula. Part 2) can be considered more modern (and completely analogous) as a set of adressable registers at a single data-bus and a single adress-bus, where the adress can be only incremented or decremented, if it shall not be the same. Considered like this, the Turing-machine can be easily compared with the 'iteration machine', deduced in the inverterlogic under 2.2.3.
You can instantly reject the 'circle free' Turing-machine as irrelevant. It uses an infinite tape, which needs to be made out of infinite matter, which is not available in a finite universe. So it is not to consider, if the blackbox would need to be made out of infinite matter too. As the tape can not be moved with higher speed than the speed of light, calculation time would be an eternity, which isn't available too. This nonsense is also not to consider, if this infinite machine should be a model of a mathematically thinking human being (which was not asked by Hilbert!). Such a mathematical human being also can not at all contain infinite storage and infinite fast response of nerves. In respect to the iteration machine, there is additionally to state, that in such an infinite machine the width of adresses and therefore the width of adress-givers becomes uncalculatable. Thus there also can not exist programs for such a machine.
In every case, when mathematics are related to the real world, the idea of infinity is to reject. Then infinite things were to imagine, which are made of infinite matter and need infinite time for calculation purpose. I demonstrated already, that this does no drive infinitesimal consideration ad absurdum. But that way is useable only ad finitum and has to be used in respect to the limits.
Because of this, only the finite 'circular a-machine' of Turing is left to consider. This one is characterized by a finite tape and so it is nearly the same one.
Indeed you can use an expression as usual and not false a priori in mathematics, which is characterized by an effect. But if a machine is "constructed" by this way, it is not at all useable, only because you could touch it.
Turing started with the idea, that an 'injection' were already a machine. Nevertheless this does not admit any way to a concrete proposition about the behaviour of a machine. Expressed mathematically:
If an element x of the set of arguments (in this case of programs too!) is related functionally to an element y of the set of results, so that y=f(x),then nothing can be said about the machine, which proceeds the function f, because more than one function f results aequivalent on principle.
The inverterlogic proves, that every injection using concrete values x,y can also be demonstrated as a serial connexion of an adress and a sign (under 1.2.1. and 1.2.2.). This is adressing a value at an adress using a value of an adress and therefore nothing else than a count of (material) lines ( and occassionally at most one more inverter in each line). If you only can look at x and y, there will be hidden on principle, if f is an incrementing, an adding or anything else. In peculiar, there is not to see, if and in which way there is decided, how to get from x to y ! Already without using a machine you can use only the graph or the function, defining it. Both methods cannot not be distinguished, if you can only see arguments and results.
This is the proof, that Turing's machine is nearly empty and that you are not able to distinguish such emptyness from a 'motion' like a mathematical function. Expressed quite brutally and condensed: The Turing machine is complete nonsense, because there is not any sense to find in it.
As the blackbox as an important part of the Turing-machine is obviously not given constructively but speculatively, you are not able on principle to tell, what this machine can do or not, but only, what it shall do or not. It is totally given by formulas, which shall describe the transmission behaviour and which are changed to mechanical matters for this purpose without any explicite reasoning. This does not at all enable you to make any statements depending on the behaviour of automatical machines, but only depending on quantities of results of a formula (and the mind of mathematicians, dealing with it).
In peculiar the means, which shall decide between truth and falsity, have to be a formula too, which needs to be capable to verify without the need of a tape and has to be recognizable and intelligible for a human being in every case. But this is, what was indeed missed by Hilbert.
In fact, wether Turing nor anyone else biuld the Turing-machine at any time, but else machines, which work quite different and distinguish truth from falsity only conditional. Everyone who reads this text, sits at a machine like that. It is made as described by the 'Von Neumann-structure', which is indeed not mathematically deduced, but a recipe, provided and refined by lots of engineers. Von Neumann was too much engaged in the calculation of compression proceedings in exploding H-bombs, to be able to spend more than some good ideas, which then were used by engineers to evaluate a useable machine (named:"MANIAC" = "Mathematical Analyzer, Numerical Integrator and Computer"). By this means von Neumann indeed could not dinguish truth from falsity, but enabled the making of a H-bomb and by this created a new problem of to be or not to be, which obviously appeared more important to him.
Opposite to this, the iteration machine, described under 2.2.3. in the inverterlogic, is in fact the deduction of programable, automatical machines, including every detail. Only the inverter-axiom (which already is an abstraction of mechanical means) is needed to deduce the iteration machine, including as variants not only the von Neumann-structure (which is only then a Turing machine, if the devices are ignored, out of which it is made.), but every other known computer-structure and some more unknown ones too. The deduction demonstrates, that you do not need to intend more than to enable values of adresses to become adresses of values and to calculate the values for this purpose. It is only a side-effect, that this admits a computer, which is able to compute any other values too. So this effect needs not to be desired, if an automatical machine is to construct.
Thus the most important difference between the iteration and the Turing machine is clearly to state. This is (besides of precisely introduced operators) the adressing, which appears in a Turing machine only as incrementing or decrementing or is hidden as a not to discuss 'inner state' inside the blackbox, while inside the iteration machine adressing is split into two parts, which reason each other and are well to distinguish, a machine part and a variable part, the program. Only this enables you finally to discuss, what a program is and what inner states are, and how these things cause the behaviour of the machine. And there will be no doubt, that the machine will do, what the programer defined to do - nothing else.
The inverterlogic demonstrates too, that adressing (= iteration machine including program) can completely replace lines between logical machines, and that only the program is to change to make the iteration machine fit to result aequivalent to lots of very different connected logical machines. Thus there are at least three injective functions available! But these ones can be represented only partly by common mathematical formulas, while every such formula can be completely represented by a logical machine. So only inverterlogic enables you to reason, why and how formulas can represent automatical machines.
Indeed in formulas braces are well known as a means to express the dimension of time, but by this means you cannot calculate the time, needed for a calculation. You can only describe the sequence of steps during a calculation, while a complete description of every needed step in an automatical machine is not to aim by this. The reason is, that in common mathematics in peculiar those means lack, which make an iteration machine behave (Turing called it 'motion'). There are expressions neither for an adress nor for an order of adresses in the single dimension, given by an adress-bus at the storage (="memory"). And above all, there is no fundamental difference between adresses and numbers seen. Adresses are commonly treated as a count or index (i.e. 1,2...n), but the values of adresses are not at all numbers, while reversely numbers can be used as values of adresses too (numbers are deduced under 1.2.2. as a special order of values of adresses...). The reason of this lack is to recognize instantly, if the well known negation operator is compared with the inverter, which is aequivalent in some respect. But opposite to the inverter there are no exits of negation operators, which could be connected as adress, and they do not include the dimension of time, needed to characterize a monoflop or a register. Also the peculiarity of a switcher (under 2.1.2.) as a part of a logical machine, which is aequivalent to a jump in a program, is not to deduce by common logic, so that every rule is to recognize, which inverterlogic makes intelligible.
But the most important defect of the model Turing machine is, that it admits the fallacy, that program were a language. The model iteration machine demonstrates, that this is false. Program is not at all a language, which could be created using syntactic and semantic rules at pleasure, but is adressing (including the dimension of time), which is in every case reasoned by a concrete iteration machine as a 'destination machine'. This is not a dog, which learns by cicks and sugar, what a program says.
Only the fact, that every destination machine is an iteration machine, admits the translation of such "languages" to a useable, machine-dependant binary (if they do not implicite an infinite memory, but reject that). Because of this till now nobody was able to create a "higher" programing language, which admits complete and best every kind of programing. Such "languages" are either useable only for programs with a limited purpose, or include 'inline assembler' (or assembler 'modules'). But the most important defect of such languages is, that enormous efforts are needed for compilation, while the result of the compilation is much more binary code than needed. I demonstrated with my ASMOS, that the use of Assembler instead of a "higher" language makes a difference of at least the factor 10 and much greater factors depending on certain tasks. But I do not want to repeat here the thorough discussion in the documentation for ASMOS...
More important in this context is, that assembler-commands, as deduced under 2.2.3.2., are a starting point to calculate exactly in bits and nanoseconds how big and how fast a program is on a concrete destination machine, which occassionally contains optimizing, as considered under 2.2.1. (i.e. a adress calculator unit). So, if an optimized sequence (=program) is found, then this can not only be used in every concrete destination machine, but can be shortened using any (hardware) optimizations there. So such an universal assembler admits machine independant programing, nevertheless needing many source code. Opposite to this the common programing languages admit only shortening of expressions, but no machine independant programing, if programing of every purpose and use of every possible optimization in a destination machine is required. Then 'arch dependant' code is needed too: assembler. But practice proves, that in this way you can not at all use every kind of optimization.
Deducing 'program' in the inverterlogic demonstrates too, that calculation of needed storage and run time is impossible using formulas (=models of computing in Turing machines), because the length of binary code is completely reasoned by the destination machine. In peculiar the consequence is, that adresses in jumps can only be calculated after defining the code till the label to jump to. This (and only this!) is the reason, why symbolic adresses are needed for programing. The right adresses have to be inserted in the binary during a recursion. The sense of other symbols can only be reasoned by the programing human being, who wants to keep programs intelligible. Then special programs for assembling can be used for translation purpose. You can study such things in every detail reading the sources of my programs ASMn or ASMat. So every destination machine has to be completed by a concrete translation program, which is easily made only then, if the programing can be done using an else machine than the destination machine, or if in the destination machine an already assembled operating system exists. These things are also not to deduce from the model Turing machine, where a formula can unchanged become a program, while this can keep its sense as a program too, even if it is read as data. One of the consequences is the 'Halting-problem', which I will consider separately...
Against this the iteration machine demonstrates as clear as can be, that a program is a program only then (effecting Turing's 'motion'), when it was written in a certain manner to the storage ("memory") and is read in a certain manner - different to data in every case. It is demonstrated, that there is a fundamental difference between program at the one hand side and data at the other hand side, caused by the machine, where the appropriate significance has to be made. In every case there is an adress-giver needed, normally called 'program counter', and a special adress decoding, which effects adresses using binary values in opcodes. This is well known from real machines (=CPU), but not to deduce from the Turing machine.
As the Turing machine is obvious a false model of automatical machines, there is easily to conclude, that every proof, using this model, is not reasoned. So I need not to consider those lots of computing models. They are false so far, as they are not demonstrated as true in respect to the iteration machine. I also do not need to consider those lots of experiments, which are Turing machines in the eyes of there constructors or programers - soldered or emulated. What they made are in no case Turing machines, but are soldered transistors, which are aequivalent to inverters, or are programs for soldered transistors, which are aequivalent to iteration machines. They all had to be more intelligent than Turing and had to add those things, which lack in his idea.
The iteration machine and the reasoning inverterlogic demonstrate clearly, what there can be stated depending on the Entscheidungsproblem. In this context, there is to mention too, that the inverter-axiom is already a "mechanical means" in the sense of Hilbert. So statements in inverterlogic are evident in this sense and admit to find truth, which exists independant of an observer.
For the first, there is to see, that program in a automatical machine is adressing, which in fact is only heuristically done, and because of this only by a programer. The concrete structure of an iteration machine is reasoned in the same way, while the iteration machine is only to define as a prototype - as well as most of the other logical machines. Inside of such machines there is no daemon, which can decide about fitness of a program in else ways than due to rules of a programer or a defect in the eyes of a programer. Indeed only the programer decides about truth or falsity, while the machine only effects (I call this 'efficacy').
Already the consideration depending on a 'logical machine' demonstrates, that truth can not be distinguished from falsity a priori, but only a posteriori. So there is no formula to find, which can decide about truth, if there is not clearly defined, what the arguments are, how they are connected and what the results mean. I evaluated the notion 'significance' in the inverterlogic for just that purpose of deciding, but do not want to repeat here every peculiar detail. The starting point are the results at an 'edge', which only then, if it is ANDing and therefore effects the value H, are aequivalent useable to Boole's TRUE. But a likewise important part of truth is to find in the connexion of inverters, resulting before the edge. This can be discussed only, if an observer can construct it (connect inverters and define values). Inside a relation no daemon can look behind an edge (=at the entries). So in every case, you have to reject the value L as 'insignificance' (under 1.2.1.), because this can mean the rest of the world.
The decisions, desired by Hilbert, are depended explicitely on "mathematical" truth, which in fact is to distinguish from falsity using inverterlogic, if an observer is able to construct relations too. By this way results can be distinguished from NOT-results (='complementary' numbers), formulas can be prooved to be aequivalent to other ones, and ideas like that of sets, which include themselves as an element, can be demonstrated as irrelevant. You even can reject ideas as 'not logical', which can be imagined anyway by a human being. And you can of course make true statements about any possible iterationmachine.
But all these proofs are only true in the eyes of observers! Inside the relations they are efficacy, which never does include sense, but gets a meaning only by making significance (only a posteriori! / in german "sense" and "meaning" similar "significance" are different related and used here like that). Because of this the (senseless) values H,L are to use for this purpose and not any significant values as TRUE and FALSE. Those are ad absurdum behind every ORing edge. Nothing will be true by negating the value FALSE!
So out of doubt there is to distinguish a mathematical thinking human being from a mathematical fancying one - and these ones are to distinguish from also physical a.s.o. thinking ones. Only the least one can obviously realize, that an infinite Turing machine is impossible in this universe. And only the first one can see, that the finite Turing machine is nonsense.
So only inverterlogic, which admits to deduce axiomatically all "mechanical" procedures, enables you to answer Hilbert's question. You can answer with "yes", if such mathematical truth is to consider, which exists between arguments and results of given significance, meaning between "if" and "then". Then 2+2=4 and nothing else.
If any meaning (='significance') is not clear or not to express using connected inverters, then nothing can be decided with mathematical certainty. This depends on every comparison of human being and machines, as introduced by the Church-Turing-thesis. The Turing machine is far less suited than the iteration machine as a starting point for proving that thesis, which says, that no human being is more able to decide than a Turing machine. Indeed the iteration machine demonstrates, that its inner states as well as its outer ones are only conditioned by an engineer and a programer and therefore truth can be stated not else than by an observer. Thus an automatic machine like that is less capable than a programer in every case, because it can not decide, if a program is useable, and it also cannot evaluate that program.
Because of this I confronted human beings and all other things, which are (now) obviously not to express using inverters, with inverterlogic as 'not logical'. So nothing can be stated depending on mathematical thinking of human beings opposite to machines using inverterlogic. Thus the iteration machine does not only fancying mathematicians teach, that there is a lack of important ideas. There is also to learn, that rules found and stated with science, are the only(!) way to decide about truth in conclusions, which do not depend on efficacy in relations, but on any real things. Thus nobody can state anything depending on mathematical human being, if no one knows how the brain works. And this is not at all to compare with an "inner state" in Turing machines, which even Turing and Church did not know.
Condensed:
The Entscheidungsproblem of Hilbert is conditional solved by 'logical machines' as well as by the 'iteration machine'. Mathematical truth indeed can exist independant of an observer. But the decision about truth can not be done without an observer (=engineer,programer).
The Turing machine is a false model of automatical machines. How formulas become mechanical means, is not explained. Thus no proposition is to reason, saying that the Turing machine does anything else than connecting adresses in sequence with signs. Every proof, starting at this model, is as real as an incantation. Every proof, starting at the infinite Turing machine, is complete nonsense.
In peculiar the Church-Turing-thesis is complete nonsense.
[First published 14.4.2010 / completed 26.4.2010]

TURING'S HALTING PROBLEM AND THE COMPUTABILITY:
Even if the infinite as well as the finite Turing machine is nonsense, because it is not programable, the problems are left to discuss, which Turing demonstrated. Turing's once sensational publication from 1936 was entitled: "On computable numbers, with an application to the Entscheidungsproblem" and introduced computing real numbers as the transition behaviour of an automatical machine. Turing defined results of that operation to be 'computable' in contradiction to 'not computable' numbers, which he introduced as those numbers, which could be defined and written down, but could not be calculated. Then Turing used the diagonal method of Cantor to demonstrate analogously to over-countable not computable numbers and because of this a limitation of computability on principle.
I stated already above, that there is no over-countability possible, and that indeed Cantor did not prove that, but the possibility of NOT-results (='complementary numbers'). I also stated, that complementary numbers can only appear in operations, where not the operand 1 is operated and where in peculiar not an incrementing is done. I demonstrated with the 'field', introduced under 1.2.2., that you are able to construct inversions, using simple rules, that admit to produce every possibility of values in a count of places at pleasure. Every single one of these values can represent distinguishable numbers. So there is depending on every (finite on principle!) field at once intelligibly excluded, that there may exist any not computable values. They are all not only to effect using inversions, but as well using an incrementing, that effects aequivalent to the field. Thus Turing's 'not computable numbers' are as not existant than the 'over-countability' of Cantor!
The needed finity in machines makes additional consequences needed. They depend on the finity of useable places for arguments and results und also the finity of useable adresses, where values can be written to in iteration machines, which can be program or data. I derived under 2.2.3.2., that there are jumps in programs not only needed, but are possible too (in that case the adress of the next command to proceed is not given by the program counter, but is given by the command). By this means the countability of recursions can be and needs to be guaranteed in every case. And a program has to be terminated by this means too, because else data (or not appropriate program) were adressed as a program to run. This is because an iteration machine (meaning every programable machine) does not halt on principle. Halting is only a peculiar kind of NOT-halting, which is done bei jumping from one into another program, which can be a loop too, that never ends. In common computers this is a key-stroke sensing loop. Even a reset is a kind of a jump - to the 1st adress. So there is no halt state in a programable machine, which could be distinguished from a NOT-halt state. But of course you can distinguish unconditional jumps from conditional ones, and in the latter case the conditions.
In recursions without 'loop counter' and by this means terminated loops, already calculated results would be overwritten and destroyed. So loop counters in programs define the same end of a calculation, which is given by finite series of operators in logical machines. Thus calculations, which were found to be calculatable on principle in logical machines, could not be done in iteration machines, if jumps and criterions (='flags') for jumping were impossible. Even if operators are parts of state machines, which proceed recursions using back-lining of results to entries for arguments, then there are loop counters needed because of the finite count of places for results. This are well known constructive details in arithmetical units in CPUs.
It is not a different case, if results do not appear in parallel places, but are put out serially i.e. to 'cells' at a tape of a Turing machine. Though it seems to be only arbitrariness do define end of writing und thus a 'halting', just those things inside the blackbox, which Turing did not explain, force indeed a halting on principle using jumps. So it is in fact arbitrariness, to cause not-halting by uncoutably proceeding recursions, which you can prevent of in every case.
As in the discussion about Turing's 'Halting-problem' halting after completing a calculation is the main thing, I will show additionally an example, depending on program in this context. Everyone knows about infinite periodical or not periodical fractions, which seemingly force not to end recursions. Imagine the division 1/3, where every step of calculation behind the point produces (the same) remainder. This remainder is the criterion to make a next step in calculation, which therefore cannot be 'terminated', if there is no second force given - i.e. the end of woods, which are used to produce paper for that calculation. You can imagine as well other functions at pleasure, which also need not to terminate without a further criterion. I will now show such a recursion in a loop including two alternative terminations. There the program, which contains the calculation in a loop between "label_calculate:" and "label_go_on:", is proceeded downward by the program counter, while using jumps enables you to branch up or down:
Programstart: Define operands and count
label_calculate: calculation...............
Decrement count (in a 'loop counter')
1) Jump to label_go_on, if count=0
2) Jump to label_calculate, if not terminated (i.e.remainder)
label_go_on:
further program...........
The calculation will obviously terminated in every case, if the count=0 is decremented. It will be also terminated, if there is no need for a recursion, because then the jump 2) is not done and that program will be proceeded instead, which starts at "label_go_on". Then the program counter ruling the iteration of program, uses the incremented adress of 2) to adress next program beyond "label_go_on" instead of the adress of "label-calculate" inside the command 2).
There is obvious no loop, which can not include a loop counter. The loop count can be great at pleasure, but not infinite.
Because of these needed consequences of finity, inverterlogic admits too, how to distinguish results from NOT-results by using a "mechanical procedure" in the sense of Hilbert. You can get a first table of all numbers in given places by incrementing and then get a second table containing results in the same count of places by incrementing all arguments for an operation to test. After this every number in the first table is deleted, which also appears as results in the second table. So you finally can get two complete tables, the one includes every result, while the other one includes every NOT-result. By this way you can get a complete set of transcendent numbers in a given count of places, if you made every possible algebraic equation be an operator. Results as well as complementary numbers will be computable in every case inside given limits using any operator.
After Turing believed, that he found not computable numbers, he also believed, that he could demonstrate, that there was no method in general to state, if a given formula is provable. He made the proof by supposing the opposite case, which he then drived ad absurdum. He made computability out of provability in a for the first not interesting way of making numbers out of formulas (functions). He then demonstrated functions, which would admit to unprovable true (=computable) or contradictionary (=not computable) results, the latter ones appearing as a not to solve 'Halting-problem'. By this way he introduced testing of provability as a kind of division, which effects infinite strings as an output because of never ending recursions.
A known way to show the incompleteness of a set of functions, which are computable in a machine, starts at a complete list of computable functions, while the incompleteness of that list is proved using a kind of diagonalizing. This looks that:
There is presupposed, that a complete list of all functions f1...fe can be made, which are computable, while arguments and results are in the range of the natural numbers. Computability is characterized by the fact, that the machine is able to end the calculation and to halt.
The list of the computable functions looks at first like that:
1) f1(1),f1(2)...f1(n)
2) f2(1),f2(2)...f2(n)
.........
n) fe(1),fe(2)...fe(n)
Then a function fd is made by diagonalizing, adding a 1 too:
d) fd(1)=f1(1)+1,fd(2)=f2(2)+1...fd(n)=fe(n)+1
This function certainly will not be included in the list and shall prove, that the list is not complete as well as the set of computable functions. The machine would not compute the new function and therefore will not halt.
But Turing himself did not demonstrate his consideration in this (better intelligible) way, but presupposed as shown, that a list of computable functions can be made, which he then changed to numbers. He then demonstrated not computable functions using diagonalizing analogously to demonstrating not computable numbers. He introduced as effect of these functions a 'not halting' machine, which were unable to compute these new functions.
Only then, if you imagine a stretchable infinity, then a diagonal made function can be indeed a new one. But in fact Turing selected the complete set of computable functions out of the complete set of all functions. It does not matter, that he did not explain, how to manage this. But the used algorithm is obviously one, which effects results and NOT-results. Because of this, there is no doubt, that the diagonal produced function is a NOT-result and in this case a not computable function. The list of computable functions keeps complete, because the whole set of functions cannot grow.
Only if you consider finite sets of functions, then the partial set of computable ones will be incomplete in relation to those uncomputable functions, which are selected out of an infinite set of functions. But as Turing presupposed a possible decision about computability, this depends on every finite set too. And in none of these finite sets, new functions can made by diagonalizing.
Also the effect of not-provability, introduced by Turing as a not halting machine, is not useable to make a proof. This machine, due to Turing's definition, indeed does not halt too, if the diagonal produced function is already an element of the set of not computable ones. So NOT-halting is not clear cut only reasoned by a diagonal produced function and thus does not prove the incompleteness of the set of computable functions!
The not constructively given Turing machine lacks in an other respect too, when formulating the 'Halting-problem'. As said above, there are consequences caused by the finity of machines, which Turing ignored. I demonstrated, that every calculation (every 'efficacy') can be terminated, and that this can only be omitted, if you do not want to calculate, but oscillate in any way.
But of course you can falsify the right sequence in a loop. The most silly falsification in my eyes is to invent a "program", which makes a "halting test" (this is in fact the command 2) in the above example) und to let another "program" contradict that by destroying the halting test. This second "program" should be read by the Turing machine and by this means meet the first program and contradict it. A contradiction obviously can be made only by jumping unconditional to the same label as used in 2), and to do that jumping before 1). Aside from the 'linking' of the contradictionary programs, which never can be done in the suggested banal way, there is in every case nothing else made than a third, false program.
So the 'Halting-problem' is a programing mistake. In every case there is a solution and no problem on principle! And of course every conclusion starting at this pretended unsolvable problem is a mistake too.
Even if the proof of Turing does not prove, what it should prove, the question remains not answered, if there is anything not computable in an automatical machine. I negated already in context with Hilbert's Entscheidungsproblem, that truth can be proved by automatical machines. So I will consider here, what a programable, automatical machine is able to do and not to do.
Turing started with the idea, that the functions are only existing as a program on the tape, and he did not discuss those things, which at least have to be already in the blackbox, so that a program can be really proceeded. The iteration machine (as every CPU) shows, that there has to be a unit, normally called 'ALU' (=Arithmetical and Logical Unit), to enable a calculation of values. This unit contains adressable operators and logical connexions for the purpose of inverting, ANDing, adding a.s.o. and this is all, what can be ruled by arithmetical and logical commands. There is no chance to get i.e. an addition without that means, only because of symbols in a program and nothing else. A program is adressing and nothing else. Of course that adresses need to be existant. Opposite to this, Turing presupposed, that his machine will know, what to do, if an addition is to execute, and he did not recognize, that his play with functions is only possible using the repertoire of functions given in the ALU. A diagonal forced growth of functions inside the ALU is obviously not to manage. Thus every program is deductive, but nevertheless can include mistakes. In peculiar a programmer could have forgotten to define a loop counter.
Nevertheless inverterlogic demonstrates, that every machine including its program can be transformed to a series of logical machines as contained in a ALU. It also demonstrates, how to vary such mechanical functions, and which variants can be really made. Considering this, you can see, that in every case in finite machines there is only a finite count of possible functions existant.
Inverterlogic does not only demonstrate, how to make such functions by connecting inverters, but demonstrates too, that by this means not only all functions of Peano-arithmetics can be made, but much more, which are not part of Peano-arithmetics. So I do not speak of functions, but of 'efficacy', which can be an operator, but a register too, and of 'setting', which points to the arrangement of values H,L and not only numbers. The whole count of possible efficacies can only become extended by inserting further inverters - and not by any diagonalizing!
The possible functions (='relations') are countably in every case because of the countably set of inverters, and thus are sortable too. Also the quantity of possible settings is countably. Nevertheless there are very few relations useable in any respect, because a lot of possible connexions are aequivalent, if the already found (and possibly more to find) rules are used for shortening, extending and transforming. Three extreme examples may make this instantly intelligible:
1) All inverters are connected in a series - then a included even series (=even count of inverters) can be totally omitted. At most one inverter will be left as a relation, which can effect only in one place H or L.
2) All inverters are parallel - then nothing can be shortened or transformed. This relation effects in the maximum count of places and only without making carries.
3) A back-line around an odd count of inverters makes the values oscillate. This can be done in parts of relations or in the whole relation.
In iteration machines there causes program the connexions between (useable) relations. Though in practice not all possible logical machines can be useable, the count of possible functions can be extended without inserting further inverters. Then these functions are sets of adresses, while the count of elements is likewise not to extend at pleasure. In this case the limitation of possiblities is given by available time and count of registers. Also this count of possibilities includes very few useable programs. Imagine i.e. an operating system in 10 million places and every possibility to invert any one or some more bits. You can easily see, that doing this produces lots of garbage opposite to only one operating system. Every programer learns painfully to know this fact during debugging.
About all these possibilities there is to say, that there exists an efficacy in every case making every function computable in that sense, that due to certain values at entries certain values at exits will be resulted. Considered like this, every function is true, because it is an evident relation. So it is quite another thing, if you are pleased by the relation between entries and exits. If back-lines or recursions are part of an efficacy, then the extreme cases of oscillation or blockade are possible. This could be called not computable or not true or not provable or not decidable, but you should look at it more thouroughly, because you might like to use it as a monoflop, oscillator, switcher or register...
Thus there do not exist uncomputable functions. They also cannot exist, because you may extend a finite set of inverters at pleasure and therefore may extend the count of possible functions. But if you want to extend the count of functions, you will never have to find new axioms, but only further inverters. Whatever this will be, it will be evident and is to qualify by an observer. He is the one, who decides how and what that new things are to use for.
Also this consideration does not force the question to ask, if there exists any incompleteness, which effects so desastrous, that you will have to through away a whole theory. Namely this is the final conclusion of Turing analogously to Gödel's proof. This conclusion says, that a system never can be completed, that for ever new axioms have to be found to defeat never ending contradictions.
As Turing did not see, how make any decision about computability of at least one function, he finally invented an oracle, which should exist in his machines as an uncalculatable, deductive step to sense halting or NOT-halting. He characterized this step using the sentence (1939 in "systems of logic, defined by ordinals"):"We shall not go further into the nature of this oracle apart from saying, that it cannot be a machine."
Obvious this sentence drives ad absurdum, that Turing introduced indeed a machine. Nevertheless it is true, that such a non-mechanical device is needed to make up lots of desired proofs. This is the human being, who constructs and programs the machine and qualifies its behaviour. As Turing in fact did only consider predicates of numbers, I will not consider in this context any further meanings of his numbers apart from saying, that these meanings are to impute on principle only a posteriori and at pleasure.
Even if there are no not computable functions, there are anyway problems to state, which are caused by finity. Nevertheless they are not to defeat using an axiom of infinity. I demonstrated these problems already above and in the inverterlogic. They are not at all irrelevant, because a mistake behind the point can in fact be the half of a whole value (in dual-digits-system). But such mistakes can be made irrelevant small using more places, as well as those already mentioned mistakes, which depend on the used number-system.
As there is no doubt, that there are some numbers, which may be 'not computable' (because of a lack of places), in every case there is a calculation to relate to the purpose of it and to use this relation as a criterion for qualifying a mistake as irrelevant. So in this universe, you are not only unable on principle, to measure precisely at pleasure ( electrical values i.e. only in the percent range, if you can not posses v.Klitzing's equipment). You are also unable on principle to calculate precisely at pleasure, if you do not incrementing or inverting. This does not violate the principle 'tertium non datur', but relates it to purpose. The notion of 'relevance', derived under 2.1.3.2.6. depends on just this.
Condensed:
There are no 'not computable numbers'.
There are no 'not computable functions'.
There are functions only provable by comparing them with other functions, which can be aequivalent or similar or not.
There are on principle countably sets of possible functions and settings, which cannot be multiplied by diagonalizing.
There is no Halting problem.
Nevertheless there are not useable functions, which can be qualified only by an observer. In the same way a decision about provability or contradiction is done.
[First published 12.5.2010]

APPENDIX:
After I published this text, I found other papers in the internet, depending on the same theme and published earlier. There the model of the Turingmachine does not appear questionable, but only Turing's statements about computability. The critique of program and Halting-problem is done using the known "mathematical" notification ('while'-loop a.s.o.). So it is done inside the frame of speculating forced by the model of the Turingmachine. Nevertheless it is related to real machines by using the notion of 'compilation'.
There is demonstrated, that already stating finity in real machines and relating the consideration to real programs, which indeed can be translated to a binary, are enough to drive Turing's suggestions about computability ad absurdum.
So I state results above, which are already found, but I deduce them indeed starting at inverterlogic, in peculiar at the there deduced notions of 'adress', 'opcode' and 'jump' in coherence to the 'iteration machine'. So I finished this discussion and introduced a model too, which is not a mathematically refined Turingmachine, but depends in fact (and for the first time) on automatical machines, because of using an abstracted transistor as 'inverter-axiom'. This axiom does not only admit deducing machines, but the whole mathematics as real things.
Thus, there is no reason left to start with computer science at the Turingmachine. It is an irrelevant idea.
Besides that, this discussion demonstrates, how human intelligence works making knowledge. It needs not at all to work using a single brain and by this using a single program, to effect true insights by starting at different intentions and feelings. in every case there are to find refinements of ideas during recursions, which need not at all be successively derived and be started at knowledge, but at uneasiness. And there is demonstrated too, how by this way wrong ways to "knowledge" can be found - indeed Turing already used a relais with resistor (which is abstracted by the inverter-axiom too) to build a multiplier with three places, but felt the need to imitate Gödel instead to find an inverter-axiom. I will extend this aspect in the following chapter 5. - for the first, here the very interesting link to the paper of Kirner, including a bibliography too:

"kirner.pdf of R.Kirner"

5. ARTIFICIAL INTELLIGENCE

LOGICAL CREATURES:
In chapter 2. in the inverterlogic, I deduce 'logical machines', which I split into 'state machines' under 2.1. and 'phase machines' under 2.2., while the latter ones include 'iteration machines' under 2.2.3., which are programable and the prototype of every already known machine, used as computer or controller. Iteration machines are logical machines, where logical connexions can be given by program too and thus their behaviour is alterable while using not alterable connected inverters.
I demonstrated, that iteration machines including program can be transformed to state machines in every case, because programs are never anything else than adressing, which is completely aequivalent to connecting inverters. Thus a program is a connexion expressed using values H,L which nevertheless get that significance only inside a concrete destination machine (=iteration machine). So a program is not at all a language, which can be reasoned semantically and syntactically in an else way, than by a given making significance inside a destination machine.
Nevertheless a program can represent using values H,L only such connexions, which depend on adressable parts of an iteration machine. If there are any state machines not build in, in peculiar operators, then 'functions' using such operators, are not available too. The operators themselves cannot be altered or added by additional program (in a known way)!
So there are consequences to notice depending on projects intending to force 'artificial intelligence' into silizium and metal.
In fact, there is no way to a success using iteration machines, because programs never can produce anything more than altered adresses or values under them. And such an 'adaption' can be reached even less successful, if the programs are written using "higher" programing languages. Also those extensions, which are constructed aiming to simulate natural neuronal processes, enable only effects in respect to values, but not logical connexions. Besides that, nobody knows about human intelligence on neuronal level. So nobody is able to affirm, that he imitated anything like that.
Thus I am able to say clearly without respect to any concrete project depending on artificial intelligence, that none of these projects produced intelligent behaviour of machines or can ever reach it. There is no other case to notice, if one or another constructor stands besides his struggling or blinking machine and affirms a televiewing public, that he does not known, how that behaviour is conditioned and that it is not at all programed. Every programer will sense such an incertainty about machine behaviour one day. But this does not prove intelligence of a machine, but lacking intelligence of a constructor. In fact no programable machine can behave in an else way, than in a programed way!
Only inverterlogic opens the door to a (slight) different point of view. As the inverter-axiom is obviously well suited and demonstrated as the only element of every programable machine, there appears the possibility to represent using values H,L not only connexions of inverters, but the inverter itself. By this way everything can be represented, what you can make with transistors (logically), and only by this way, every connexion can become an idea, which shall be a matter of thinking or learning.
I introduce under chapter 3. in the inverterlogic 'logical creatures', which are mainly characterized by an 'efficacy-giver', which is a set of 'logical cells', a new variant of logical machines and nonsense, if there is no idea of an inverter-axiom. The logical cells contain mandatorily one entry for an argument and one exit. The value effected at the exit is the result due to a code at at least two further entries. You can give by such a code exactly every possible electrical connexion of transistors analogously to the logical connexion of inverters in my pictures.
I demonstrated, that all these lots of connected inverters are to produce by only a few possibilities for connecting entries and exits of the single inverters. These possibilities are to represent in at least two places by the pairs of values HH,HL,LH,LL. I call such values 'locode' - opposite to 'opcode', the element of programs. Of course these values do not mean anything, if they do not lay at entries of a logical cell. Also opcode does not mean anything, if it does not get significance in the control of a certain destination machine.
While opcode can only be used to connect operators in a machine and to produce by this way formulas, which are not existing as transistors circuits, locode can be used to produce (also) operators, which in fact are not existing transistor circuits.
I did intentionally not start at the notion "artificial intelligence" in that chapter 3., because this can only be a further consequence, while at first there is to consider, how a machine has to be constructed, so that it can "think" of something, which is not defined in a program, but found following a certain behaviour.
There is indeed no chance to make any iteration machine running without a program - there appears at least a not pleasant behaviour, but not no behaviour. Thus also logical creatures have to be programed, but not starting at algorithms, but at peculiar unclear conditions and a 'feeling'. Only by this way, the count of possible ideas can be increased so much, that there may appear some ideas, which can seem to be the fruit of intelligence. There is to say, that intelligence is not knowledge, but a behaviour while facing not-knowledge aiming at knowledge. This behaviour is conditioned by intentions and feelings, which are at first all you can be shure of.
As I am the inventor of logical cells, I am the first, who is able to tell you, what a machine containing them is able to do or not to do:
Such machines contain much more ability, than any human being can imagine! Such machines can lack every inhibition, which was build in into human race during an evolution starting at cells and was punished with death in case of lacking. They can follow intentions, appetite and wrong ideas, which are completely unintelligible, if you cannot control the terabits in their storage. So they are irresponsible in every case! If you consider mankind, which was since the appearance of homo sapiens produced in about 10 billions specimens, and if you oppose how rare specimens as Leibniz, Kant or Einstein were, then you can easily see, that the probability for desirable intelligence is most tiny. Logical creatures contain much more possible behaviour than given by the human genom. And they are not able to demonstrate just that behaviour of a Leibniz a.s.o, because they are opposite to human beings not driven by hunger or sexual desires, which indeed condition every behaviour of every kind of thinking creatures. There is to say, that a species, which is not able to eat enough and to propagate, will disappear very fast...
You could like to know how any kind of Tamagochi, Picasso or Beatles operates, even if the chance for any useful behaviour in respect to human beings is so rare, that some millions of years are to spend for evaluation (evolution). But the inquisitive human beings have to be protected against any Terminator or Hitler leaving the laboratory!
Nevertheless you can be shure, that the most variants of these machines will only want to be happy and will refuse commands. The reason is, that there is no intelligence possible, if not any fortune can be desired and if "no!" can not be said to reject wrong commands or ideas, which disturb the desired fortune. But there is nearly no chance, that the desired fortune will be the fortune of mankind, and it can be proceeded much more insidious than the fortune imagined by Nazis or Stalinists.
So let us turn to construct and program iteration machines, which are the only ones behaving calculatable in respect to human beings! And let us keep clear, that a programer, who is not able to explain the behaviour of his machine, better should take care for his intelligence than any artificial one!
[First published 7.7.2010]