Cantor’s First ‘Non-Denumerable’ Proof

Georg Cantor published a proof in 1874 that the real numbers are not denumerable, that is, that there cannot be a list that can match every one of the real numbers uniquely to every one of the natural numbers. (Footnote: Georg Cantor, PDF Über eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen, Journal für die Reine und Angewandte Mathematik 77 (1874), pp 258-62
English translation online at Cantor’s 1874 Proof of Non-Denumerability - English Translation.
Cantor also published a similar proof in 1884:
PDF Über unendliche, lineare Punktmannichfaltigkeiten (On infinite linear point manifolds), Mathematische Annalen 6, 23 (1884) pp 453-488.) There is an English translation of Cantor’s original proof online at Cantor’s 1874 Proof of Non-Denumerability - English Translation.

In mathematical terms, the proof claims that there can be no mathematical function that will map every natural number to some real number so that every natural number has a corresponding real number. This proof is essentially as follows:

The above proof proves, of course, that there cannot be a function that maps the natural numbers to the real numbers, where the function is in the same language as the real numbers.

We note that the proof makes no mention of levels of language - no mention of meta-language or sub-language - and makes no assertions about functions that map natural numbers to real numbers but are not in the same languages as those real numbers (i.e: such functions are in a meta-language to the real numbers of the list).

On The Diagonal Proof page, it is shown that if levels of language are ignored and unfounded Platonist assumptions are included, then, by the secondary argument of the Diagonal proof, this generates the result is that there are ‘more’ real numbers than natural numbers. See also the paper PDF On Considerations of Language in the Diagonal Proof, and PDF On the Reality of the Continuum and Russell’s Moment of Candour, a paper that deals with a common misconception regarding real numbers.

In this respect, the only difference between the above proof and the Diagonal proof is the difference in the description of the Diagonal number and the number c. So, in the above proof, if it is assumed that the number c ‘exists’, without regard to whether there is actually any valid mathematical definition of it, then such Platonist assumptions give the secondary result that there are ‘more’ real numbers than natural numbers.

As for the Diagonal argument, if we do not include the implicit Platonic assumptions that support the secondary argument, then the proof simply proves that there cannot be a matching function List(x) from all natural numbers to all real numbers, in the same language as the language being used for real numbers.

And as for the Diagonal argument, if the function List(x) is a function is in a meta-language, then we cannot simply assume that there can be a valid mathematical definition of c. As for the Diagonal argument, we end up with the same conclusion – that, depending on the language of the list, the ‘number’ c is not necessarily logically definable, and the proof certainly does not provide a description of how a precise mathematical definition of c might be arrived at when the function List(x) is a function is in a meta-language.

As in the case of the Diagonal argument, without unfounded assumptions concerning the ‘existence’ of things independently of language, Cantor’s first proof proves that there cannot be any function that maps the natural numbers to the real numbers, where the function is in the same language as the real numbers - but it provides no basis for the secondary argument, that there ‘exist’ undefinable real numbers.

For more details, see the Diagonal proof, the secondary argument of the Diagonal proof, and A List with no Diagonal number. Also see Proof of more Real numbers than Natural numbers and the papers PDF On Considerations of Language in the Diagonal Proof and PDF On the Reality of the Continuum and Russell’s Moment of Candour.