User talk:Biedermann
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Here is my comment to Cantors cracy Diagonal Argument Dr.Eginhart Biedermann................................11.10.2006 Rua Baden-Powell R 34 ................................Tel +351-291-935558 9125-036 Caniço de Baixo .............................e-mail: post@e.biedermann24.de Madeira/Portugal
I am perfectly aware of the general acceptance of Cantor’s Diagonal Argument among today’s university mathematicians and their reluctance even to discuss any arguments that could cast doubts on it. And yet, I feel, today’s students should be given a chance to hear at least of the existence of such doubts. So I offer you the following text for publication in your journal. With kind regards Eginhart Biedermann
\documentclass [ journal] \title {CANTOR’S DIAGONAL ARGUMENT} \subtitle {COUNTERARGUMENTS} \author {Eginhart Biedermann } \email {biedermann@clix.pt} \keywords {Cantor -- 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 !
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
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)
From Eginhart Biedermann, e-mail: biedermann@clix.pt 15.9.2005
In fact, the "gap" you have found is really an instance of how Godel's cleverness was able to circumvent the difficulties in making such a proof. Making a self-referential statement is not so easy!
In the statement (x)~Dem(x,sub(y,19,Z(y))), y is a free variable, which has its own value in the numbering system. Z is a recursive function, and so has a value as well. This is totally different from (x)~Dem(x,sub(n,19,Z(n))), where n is not a free variable, but rather a specific integer. Thus the value of Z(y) does not require knowing y; while the value of Z(n) is a function of the number n. So it's perfectly legit to talk about Z(n) where n = Godel number of Z(y). Hope this helps. --Luke Gustafson 11:27, 31 December 2005 (UTC)
Thanks for your comment, but I can't see, how it could help! Just tell me, what you think the Gödelnumber o F(y) to be. To my understanding it not defined, since F( ) is running over the natural numbers and nothing else. 195.23.174.229 16:26, 21 April 2006 (UTC)biedermann195.23.174.229 16:26, 21 April 2006 (UTC)
Welcome!
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