Cantor's "diagonal argument". My Objection.

INTRODUCTION
I don't make any argument for or against Cantor's methodology. I argue instead that it does not tackle what it was intended to tackle.

DISCUSSION
Cantor's theorem goes something like "there are infinite sets which cannot be put into one-to-one correspondence with the infinite set of natural numbers".

If we helpfully translate that term 'sets' into something more substantive, Cantor is saying that, for example, the real numbers (those between 0 and 1, such as 0.2361...) are uncountable while the natural numbers (any integer, such as 2361...) are countable. Another way of saying this is that there is a different "cardinality" between the real numbers (for example, 0.2361...) and the natural numbers (for example, 2361...). If Cantor is right, then these two types of number cannot be put into a one to one correspondence with each other: there are more real's than natural's.

Cantor's proof employs a made to measure 'square'. The natural numbers are employed as an index or list where each number identifies the position of each real number. Cantor's proof, simply put, amounts to the idea that when we add or subtract 1 to all the digits of the real numbers, then that new real number can't be found on the list given to us by the natural numbers.

MY OBJECTION
My objection is simple: Cantor makes no substantive distinction between the real numbers and the natural numbers. Either can be used to list the other.

PROOF
No distinction is made, for example, between "0.2361..." and "2361...". Either one can be used in a list to identify the position of the other. The inclusion of the decimal point is a redundant pictorial artifice, a non-mathematical glyph.

An assumption in Cantor's 'proof' is that the real's can't be used as a list and that the natural's can be used as a list. We like to think that decimals can't be used as a list because they start at the infinite end, whereas the natural numbers start at the finite end and can be used as a list. Yet there is no practical shortcoming for translating "decimals" into the elements of an index or list. The decimal point, contributing nothing to the list, can be dropped.

CONCLUSION
A list erases the distinction between the real's and the natural's. Either can be used to list the other. Cardinality can be imposed equally on the real's and the natural's, once we get over losing the empty psychological significance of what is now the redundant decimal glyph. Whether or not Cantor's methodology is sound, his proof fails to tackle what he wants it to tackle and that, consequently, it shows no more than that what goes for the real's goes for the naturals.
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Relevant Pages

  • Re: INFINITY Revisited
    ... There is no ball that is added at noon. ... > you said that aleph_0 was the name of infinity for the naturals. ... > decimals each with no last digit printed on them (perhaps this is only ... > with no last digit must be either a repeating rational or an irrational ...
    (sci.math)
  • Re: Cantors "diagonal argument". My Objection.
    ... I don't make any argument for or against Cantor's methodology. ... Yet there is no practical shortcoming for translating "decimals" ... that what goes for the real's goes for the naturals. ... "Understanding Godel isn't about following his formal proof. ...
    (sci.logic)
  • Re: INFINITY Revisited
    ... > At 2 minutes to noon the vase is empty. ... > This corresponds to all of the single digit decimals between 0 and 1. ... and I believe all balls must have necessarily been removed. ... > naturals or else the vase would not be empty. ...
    (sci.math)
  • Re: Cantors "diagonal argument". My Objection.
    ... If Cantor is right, then these two types of number cannot be ... Yet there is no practical shortcoming for translating "decimals" ... that what goes for the real's goes for the naturals. ... The main proof infact starts with the definition of which reals are ...
    (sci.logic)
  • Re: Cantors "diagonal argument". My Objection.
    ... there is no infinite natural number. ... I thought this was what Cantor was trying to show? ... What the naturals are; ... he has attempted to prove the truth of his unorthodox interpre- ...
    (sci.logic)