Biedermann
Here is my comment to Cantors cracy Diagonal Argument
Summary: Several arguments are presented which seem to justify serious doubts about the validity of Cantor’s Diagonal Argument.
1) Cantor’s claim to be able to conceive (within his small finite brain!) of the INFINITE ( = never ending!) list of all NEVER ENDING decimal sequences as of a COMPLETE (completed!) list is in itself a contradiction! Contradictory results may be expected therefrom! Cantor appears to have been fascinated by the possibility to define ever higher natural numbers N, yet he must have forgotten that in that process he could never achieve anything as compared with the Infinite, that any such number divided by ‘infinite’ would always yield ZERO ! . 2) Intuitively, serious doubts against Cantor’s infinitely many more reals than rationals arise from our knowledge of the density of the rationals themselves. No closest neighboring rationals can be identified, in whose interspace uncountably many reals could be accommodated.
3) The so convincing demonstration of the diagonal argument on a small finite list of finite decimal sequences cannot honestly be extrapolated to an INFINITE list of infinite decimal sequences. No irrational number, no infinite decimal sequence, can be ‘given’ other than by an algorithm that allows to determine as many of its decimals as any one desires. That is what ‘infinite’ means! But these algorithms may lexicographically be ordered, so they are countable! Cantorians now claim all the uncountably many reals to be indefinable. Certainly, we cannot even differentiate between two undefined numbers, much less count infinitely many of them!
4) Every mathematician is familiar with the definition of real numbers as ‘Dedekind cuts’ on the set of rationals. This definition implies the smallest possible difference between two real numbers @’ and @" to be one rational number @ that lies to the left of @" but to the right of @’. So there can impossibly exist more different Dedekind cuts as well as real numbers than there are rational ones! And certainly no @ can be ‘given’ unless by an algorithm out of the above mentioned lexicographically ordered list.
5) The so convincing demonstration of the diagonal argument on a finite list of decimal sequences, (in fact on square matrix!), looses its validity as soon as in that matrix the number of the horizontal lines exceeds that of the vertical rows. Yet on an infinite matrix we have no means to assure this requirement, especially on such an unspecified list as is used to demonstrate the diagonal argument! It is, however, quite easy to conceive of a never ending process that yields all possible different decimal sequences up to any desired length: starting with the list of all decimal sequences of length 1, we proceed to the list of all sequences of length 2 and so on! That list grows in both directions endlessly in to the infinite, yet much faster in the perpendicular than in the horizontal direction. At length 7 we have 10 million lines, at length 100 the number of lines by far exceeds the number of elementary particles in this universe, and never can an Antidiagonal be derived from this list that would not be contained in it as a horizontal line. If we consider, with Cantor’s imagination, this infinite process to be completed, it yields a list that may be interpreted as an enumeration of all natural numbers, as a list of all rationals in the 0-1 interval (by placing a 0. to the left of these sequences), as well as a list of all possible infinite decimal sequences, including all of Cantor’s ‘indefinables’! Whoever is missing here the definable reals (they would be generated with the completion of the list in the INFINTE!) may feel free to insert their generating algorithms alternatively between the initial decimal sequences.
Conclusion: In spite of the general acceptance of Cantor’s Diagonal Argument among today’s university mathematicians the here presented arguments cast some doubt on the justification of Cantor’s imagination of completed infinities and the existence of his by Hilbert once so enthusiastically welcomed ‘Paradise of ever higher Infinities’.
Eginhart Biedermann Biedermann 10:55, 11 October 2006 (UTC)
P.S. It’s just a pity Cantor was not around at the moment of the Big Bang to start counting, one number per second. Today he would be working his way through those of length 17 ( in the decimal System) and it is questionable whether he would reach those of length 100 before the end of this universe. That experience might give him a lesson about the real meaning of INFINITE ! Biedermann 16:35, 4 January 2007 (UTC)
Hello, here my comment to Gödels Incompleteness
== Counterarguments ==
1) In the long sequence of the some 40 definitions which Gödel introduced for the construction of his famous Unprovable Formula, we find under numbers 16. - 17. the definitiion of Z(n) as the Gödel-number of the presentation of any whole number n which is given in Gödels Formal System through n-fold positioning of the symbol f in front of the symbol 0, so Z(4) is the Gödel-number of ‘ffff0’. So far so good. Further on however Gödel introduces, ( I follow here Nagel-Newman’s way of writing the formulae to get them on a single line of typing) after having defined the Gödel-number of the symbol y to be 19, the formula (1): (x)~Dem(x,sub(y,19,Z(y))) , of which he claims to be able to determine the Gödel-number n, by means of which he then proceeds, by substitution of n (i.e. the representation of this number n in the formal System S ) for the symbol y, to his famous formula (G): (x)~Dem(x,sub(n,19,Z(n)))
with its self-fullfilling interpretation of its own non-derivability.
However: what is Z(y) in formula (1)? From what we read above, it should be the Gödel-number of the symbol-sequence that results from the y-fold positioning of f in front of the symbol 0. But nobody, not even Gödel, is capable of putting the symbol f y-times on a sheet of paper, not even in thought. Equally, we all are incapable of determining the Gödel-number of a non-existing symbol-sequence. Consequently no Gödel-number of formula (1) can exist, and formula (G) operates with a non-existing number n. So: this so nice looking formula (G) simply does not exist as a sequence of symbols of System S! That appears to be
:::: GÖDELS INCOMPLETENESS or THE GAP IN THE PROOF
in more detail: Somebody diagnosed my total lack of understanding. This certainly couldn’t help me to some better understanding. So, let me try to explain in more detail what I think to be my understanding with respect to what Gödel’s term Z(y) could be in the symbols of Gödel’s System. Without such knowledge the Gödel number n of formula (x)~Dem(x,sub(y,19, Z(y))) cannot be determined. But this is essential for the progress of Gödel’s argumentation; just to name it ‘n’ , does not serve the purpose, n not being a symbol of the System. From Gödel’s definitions 16 an17, together with 8 and 9, we know Z(n) to be (to BE, as Gödel says expressis verbis, not to Designate, as some people seem to believe!) the Gödel-number of the number n in the symbols of the System. With the claim that sub(n, 19, Z(n)) be obtained from sub(y, 19, Z(y)) through substitution of 19 (i.e. fffffffffffffffffff0 ) by Z(n), of necessity Z(y) would have to BE the number 19, and the term sub(y, 19, Z(Y)) would in fact be sub(y, 19, 19)), from which the term sub(n, 19, Z(n)) certainly does not result through simple Substitution of n for y ! To my poor understanding, Gödel’s argumentation depends on the assumption that Z( ) could be a function sign expressed in the symbols of the System, in which you may freely exchange the arguments n for y etc., and which could simultaneously yield the mathematical function ‘Gödel number of n’ on the argument n, and the deliberately chosen non-mathematical relation 19 for the argument y ! To my poor understanding it seems evident that such function cannot exist. Would you please be so kind as to explain me what you think to be your better understanding! With kind regards Biedermann Biedermann 12:18, 22 January 2007 (UTC)
2) Another questionable aspect of Gödels reasoning shows up in the word-for-word definition of what the abreviation sub(y,19,Z(y)) is standing for: it reads (due to the definition of ‘19’ to be the Gödel-number for the symbol y !):
‘’The Gödel-number of the term that results from the term with Gödel-NUMBER y via substitution of the VARIABLE y by the representation of the NUMBER y ‘’!!!
The symbol y appears here in two clearly different identities, however, to obtain unumbiguous and consistent results in mathematics one should not start with such ambiguous definitions! Yet this ambiguity is crucial to Gödels selfreferent construct ! No wonder this construct
C A N N O T B E D E R I V E D F R O M T H E A x i o m s !!!!
Biedermann 10:26, 1 October 2005 (UTC)
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Godel's incompleteness theorem
editYesterday, I moved your question about Godels incompleteness theorem to Talk:Gödel's incompleteness theorems/Arguments, where I gave a brief answer. Please don't vandalize article or talk pages by inserting your comments at inappropriate places. CMummert 12:48, 7 January 2007 (UTC)
|Javalenok]] 14:44, 14 January 2007 (UTC) Is that PHILOSOPHY ? I am physicist, my world is build from atoms and other elementary particles, but certainly not from mental concepts! Biedermann 11:50, 15 January 2007 (UTC) , I repeat here my first concern with Gödel: With Gödel’s definitions 16 and 17 it is evident that Z(4) yields the Gödel-number of ffff0, but with y instead of 4 as input in these definitions, Z(y) can impossibly yield the symbol sequence fffffffffffffffffff0, which it would have to yield when we assume the Gödel-number of y to be 19 and as it is introduced in sub(y,19,Z(y)) ! Please explain how you want to get around this impossibility. My second concern is that in this term the symbol y appears in two different identities in violation of Russell - Whitehead’s requirement on page 1 of Chapter 1 of Principia Mathematica. Why do you accept such violation? With kind regards Biedermann 11:44, 12 January 2007 (UTC) Biedermann 11:48, 15 January 2007 (UTC) Biedermann
New version on Z(y) Biedermann 10:25, 2 March 2007 (UTC) However: what is Z(y)? Whereas Gödel takes the pain to tell us expressis verbis that Z(n) IS the Gödel number of the numeral for n, he does not make any remark on what he wants Z(y) to BE! It certainly is NOT the Gödel number for “y”, the number “19”, as would be required for the self referent interpretation of formula (G). And obviously it can certainly not be the “Gödel-number of the (not existing) symbol-sequence that results from “putting the sign “f” y- times in front of 0”. Unquestionably, Z(y) cannot be anything else but that entire recursive definition 16 + 17 : 16. 0 N x = x, and (n + 1) N x = R(3)*n N x ( with R(3) = 2exp3) n N x corresponds to the operation of “putting the sign “f” n times in front of x” 17. Z(n) = n N [ R (1)] , Z(n) is the Numeral (the Gödel number for the numeral) denoting the number n.
(rewritten in the symbols of the system) with all occurrences of “n” being replaced by “y”. But then the formula (G), with Z(n) BEING the Gödel number for “n”, certainly cannot be claimed to be derived from formula (1) through “substitution of “n” for “y”! So, the selfreferent interpretation of formula (G) is lost! No better result is obtained from assuming Z(n) NOT to BE the Gödel number for “n”, but, in parallel to the definition of Z(y), to be the “definition 16 + 17” with argument “n”. Now formula (G) evidently results from (1) through “substitution of n for y”, but (G) no more says of itself to be obtained from (1) through “substitution of n for y” ! Again, no selfreferent interpretation is obtained! That is, what I would like to call:
GÖDEL’S INCONSISTENCY
and it's enthusiastic acceptance in the entire mathematical community over so many decades
Biedermann 10:25, 2 March 2007 (UTC)
is just the biggest blunder that ever happend in science !