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:
First of all we assume that there actually can be a function matching the natural numbers to all the real numbers; the object of the proof is to prove that this assumption cannot be correct. We call the assumed function List(x). Whenever we replace the x in List(x) by some natural number, the definition of List(x) gives some real number to correspond with this natural number.
Now we go through the natural numbers one by one, starting at zero, until we get two values for the function List(x) that are between the values 0 and 1 (which will be real numbers) – we call these first two natural numbers n1 and n2, so that the first two real number values are List(n1) and List(n2), so that we have a new interval List(n1) - List(n2) which includes all the real numbers between List(n1) and List(n2).
We repeat this procedure and continue to go through the natural numbers one by one until we find the next two values of List(x) (again, these are real numbers) that are within the interval List(n1) - List(n2) For these two numbers, let the smaller be List(n1,1), and the other List(n2,2), so that we have a new interval List(n1,1) - List(n2,2).
We continue this process again and again. As we do so, the interval becomes smaller and smaller. But between any two real numbers there is always another real number, since there is no such thing as a smallest difference between two real numbers. That means that no matter how long we continue the process for, there will still be two real numbers with an interval between them.
The proof now says: Seeing as the interval never vanishes, there must be at least one number (which we will call c) which lies between every interval of every List(n1,1,1, …) - List(n2,2, 2, …), for which there cannot be any corresponding natural number (which we call nc). For if there were such a natural number nc, it would follow then when that number nc was reached in the above process, that the corresponding real number c would have been added as some endpoint of some interval as List(n1,1,1, …) or List(n2,2,2, …).
This is a contradiction, since it must lie between the sequence List(n1,1,1, …) and the sequence List(n2,2,2, …). Since there is a contradiction, then the initial assumption that there can be a function List(x) that maps the natural numbers to the real numbers is wrong, and there can be no such list.
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 function 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.
Rationale: Every logical argument must be defined in some language, and every language has limitations. Attempting to construct a logical argument while ignoring how the limitations of language might affect that argument is a bizarre approach. The correct acknowledgment of the interactions of logic and language explains almost all of the paradoxes, and resolves almost all of the contradictions, conundrums, and contentious issues in modern philosophy and mathematics.
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