Page last updated 06 Jun 2023
Cantor’s Diagonal argument proves that there “exist ” real numbers that are indefinable.
There has never been any rigorous logical proof of that fallacious claim.
As shown on the page The Diagonal Argument the fallacious claim that there are more elements in some limitlessly large sets than in some other limitlessly large sets is given by an additional argument that is tacked onto the diagonal argument (see The secondary argument and also the page Proof of More Real numbers than Natural Numbers). This secondary argument claims that if it is impossible to define a one-to-one correspondence between two limitlessly large sets, then one limitlessly large set must have a different Cardinal number than the other limitlessly large set, and have “more” elements than the other limitlessly large set. Along with this claim there is the concomitant claim that there must be real numbers that cannot have any finite definition - that there must “exist” what are called “indefinable” or “inaccessible” real numbers. This notion had its origins in a 1905 paper by Julius König, you can see an English translation online at On the foundations of set theory and the continuum problem. A logical analysis shows that the assumption of the “existence” of such indefinable numbers inevitably leads to irremediable contradictions.
The contradictions of indefinable numbers
Here we show how the assumption of the “existence” of indefinable numbers gives rise to a contradiction that is rather troubling to those who try to promote the concept of infinite sets of differing Cardinal numbers and which have different “sizes”:
First of all, we note that we can enumerate all symbol combinations of the form of 0⁄2 , 0⁄4 , 0⁄8 , … 0⁄2n , … to define a set A, and that we can enumerate all symbol combinations of the form of 1⁄2 , 1⁄4 , 1⁄8 , … 1⁄2n , … to define a set B. So both of the sets A and B are denumerable. By the conventional rules it can easily be shown that the union of the two sets A and B is also denumerable, and furthermore any subset of a denumerable set is also denumerable. This means that every possible set of combinations of fractions of the form of 0⁄2 , 0⁄4 , 0⁄8 , … 0⁄2n , … and of the form of 1⁄2 , 1⁄4 , 1⁄8 , … 1⁄2n , … is denumerable. And for a real number, in the binary base, a defined sequence of digits such as 0.101… can be used to represent the corresponding sequence of fractions - here expressed in decimal notation as: 1⁄2 , 0⁄4 , 1⁄8 , … an ⁄2n , … , where an = 0 or 1; in binary notation, this would be 1⁄10 , 0⁄100 , 1⁄1000 , … an ⁄10n , … ). And so, for any number, the set of all such fractions an ⁄2n where an = 1 defines that number. This applies regardless of whether there can or cannot be indefinable real numbers.
However, for any postulated “indefinable” number, that set of fractions must also be indefinable, otherwise that definition would define that “indefinable” real number. But, beyond that, it must also be the case that the subset of natural numbers n where an = 1 for any such postulated “indefinable” number must also be an indefinable set - and furthermore, it must also be the case that for any such postulated “indefinable” real number, that associated subset of natural numbers cannot be put in a one-to-one correspondence with the natural numbers - otherwise the “indefinable” real number would be definable by that one-to-one correspondence.
Hence if there “exist” indefinable real numbers, which is the conventional claim that follows on from The secondary argument that is tacked onto the Diagonal proof - then it follows that it must be impossible to define an enumeration for any such associated set of natural numbers - otherwise, if there were such a defined enumeration, then that would completely define that real number and that number would be definable.
And this means that there must be subsets of the natural numbers which have a Cardinal number either smaller or greater than that of the natural numbers. In addition, we know that there cannot be real numbers that have “more” digits in their infinite binary expansion than any other irrational real number that has an infinite binary expansion - there aren’t digits missing in the infinite binary expansion of any irrational real numbers, nor do some real numbers have additional digits - no real number has “more” digits than some other irrational real number.
Hence the notion of indefinable real numbers results in absurdities and contradictions. And even if one were to accept the claim that there must be indefinable or “inaccessible” real numbers, then it must also follow from that claim, that simply because it is impossible to define a one-to-one correspondence between two sets, that does not imply that one set must have “more” elements than the other.
This follows since we have just demonstrated an example that shows that this is not the case - we have just shown that it is impossible to define a one-to-one correspondence between the natural numbers and the subset of natural numbers that could define the digits of any hypothetical indefinable number - even though there cannot be any “extra” digits in the binary expansion of any such indefinable number.
Hence the notion - that the impossibility of defining a one-to-one correspondence between two infinite sets implies that there must be “more” elements in one infinite set than the other - cannot apply in at least one case - the case of the digits of a supposedly indefinable real number just discussed.
And that means that the assumption that the impossibility of a one-to-one correspondence between two infinite sets implies that there must be “more” elements in one infinite set than the other is logically untenable, since that assumption itself leads directly to the conclusion that very same assumption cannot be correct.
Hence we have the case where the assumption that the impossibility of a definable one-to-one correspondence between two sets necessarily implies that one set has “more” elements than the other, actually leads to the conclusion that the impossibility of a definable one-to-one correspondence between two sets does not in fact necessarily imply that one set has “more” elements than the other - which directly contradicts the original assumption - which was: if it is impossible to define a one-to-one correspondence between two infinite sets, then one set must have a different cardinal number than the other, and must have “more” elements than the other.
This provides a demonstration of the folly of the naive assumption that the impossibility of defining a one-to-one correspondence between natural numbers and real numbers somehow implies that there are “more” real numbers than natural numbers. There aren’t.
The absurdity of the naive assumption can also be demonstrated by another contradiction, as follows:
Consider any enumerable infinite set of definable real numbers, such as the square roots of all natural numbers. Now consider the union of this set and a set containing one indefinable number. Note that while such a set cannot be defined, since it is claimed that indefinable numbers “exist”, then such sets must also “exist”. But there cannot be a definable enumeration function for this set, since then there would be some n for which the value of the function would be that indefinable number, even though, by the Hilbert’s hotel argument, there clearly could be an association between every natural number and every element in the union set (given the assumption that such indefinable numbers “exist”).
In any case, it is patently obvious that the impossibility of defining such an enumeration function for this union set cannot imply that there are “more” elements in this union set than there are in the set of definable real numbers.
The only rational conclusion is that the impossibility of defining a one-to-one correspondence between natural numbers and real numbers does not imply that there are some real numbers that cannot be defined, and it does not imply not that there are “more” real numbers than natural numbers. The use of abstruse terminology in attempts to whitewash over the inherent contradictions serve to fool only the simple-minded.
Definitions of specific indefinable numbers?
Note that there are claims that there are definitions that reference specific real numbers that cannot be defined - supposedly this means that one specific real number is indubitably referred to by the definition, but in terms such that one cannot actually determine all of its decimal or binary expansion from that definition, so that there will be some digits of that expansion that one cannot determine. However, it is very simple to demonstrate that such claims are fundamentally flawed, see Chaitin’s Omega number and Turing’s uncomputable number.
Language, Logic and indefinable numbers
The problems and contradictions that arise from the unproven assumption of the existence of mathematical entities which are also indefinable all disappear when a logical analysis is applied and which takes full consideration of different levels of language, and when the arguments are divested of any unfounded assumptions that numbers ‘exist’ as ‘actual’ things independently of language. The conclusion is simply that there cannot be a matching function from all natural numbers to all real numbers in the same language as the language for those real numbers.
It is the failure to recognize that the naive arguments advanced in favor of indefinables rely on an implicit invocation of different levels of language, but where there is a concomitant confusion between the different levels of language involved which inevitably leads to contradictions.
An analysis of this confusion of levels of language, and how the elimination of that confusion resolves the paradoxical contradictions is demonstrated at The secondary argument of the Diagonal proof, see also The secondary argument of the Diagonal proof, see also Cantor’s 1874 non-denumerability proof. You can also read more about the notion that numbers ‘exist’ as ‘actual’ things independently of language in The Platonist assumptions in the Secondary Argument.
In fact, one can enumerate every possible sequence of symbols of a well-defined mathematical language - but not in that mathematical language itself. Any such enumeration must be in a higher language, a language that is a meta-language to that well-defined mathematical language. That also means that the meta-language cannot define a diagonal number in terms of any such enumeration, as is demonstrated in A List of Real numbers with no Diagonal Number.
The slippery slope of indefinables
Some people (including a professor of mathematics) have suggested to me that one can circumvent the crippling contradictions of indefinable numbers by the claim that although there can be no definable function that can define a one-to-one correspondence between natural numbers and the digits of any hypothetical indefinable number, nevertheless indefinable functions might also “exist” completely independently of any human or machine - for example, that an enumeration function might “exist” for the digits of the binary or decimal expansion of an indefinable real number, but the enumeration function is itself also indefinable. Or that an enumeration function might “exist” for a subset of natural numbers such as those mentioned above, but that the enumeration function is itself also indefinable (See Indefinable Lists that exist).
Such claims simply beg the question:
If one is to simply assume that there exist enumeration functions that are indefinable, then what would be the logic of the assertion that one cannot also assume that there “exists” an enumeration of the real numbers but it is indefinable? After all, if there were such an indefinable enumeration, then one could not define a diagonal number from that enumeration, and so the diagonal proof fails.
One cannot simply throw logic aside and cherry-pick those sets that one wants to be enumerable and those that one wants to be non-denumerable.
“Well-ordered” sets and the “axiom of choice”
One might imagine that the plethora of difficulties engendered by the unfounded assumption of the “existence” of certain indefinable things would have served to dissuade mathematicians from delving further down that route, and the that the above-mentioned professor is not representative of mathematicians in general. But that is not the case, as evidenced by the problems that arose from Cantor’s claim that every set could be what he called “well-ordered”. Cantor had no proof whatsoever of this notion which was a crucial aspect of his philosophy; that fact is not at all surprising since it relies on an assumption that elements of mathematical sets somehow have a magical property that is completely independent of the definitions of a mathematical system. Cantor’s claims in this respect were little more than claims that there exist some sort of indefinable relationships that are enumerations in disguise. And it is not surprising that such claims also result in contradictions, in a similar fashion to those noted above; for more on this see the page The Axiom of Choice and Well-Ordering.
Seeking a proof that indefinables “exist”
For an example of how someone who claims that indefinable numbers “exist”, but when pressed to provide a proof of this claim, resorts to various ploys to avoid admitting the fact that there is no such proof, see the page The duplicity of Mark Chu-Carroll.
How beliefs change over time
Today we are bemused by the fact that Issac Newton had various strange beliefs regarding alchemy and religion, beliefs that we now consider to be completely irrational and we consider it odd that the man who was so rational in his mathematics and physics could have been so irrational in other ways. (Footnote: To be fair to Newton, he seems that he knew that he was unable to support his beliefs in these matters with any solid arguments, and so he did not make any public claims about these notions.) But humans have always been like this, because humans are good at devising arguments that may seem plausible on superficial inspection, but which under detailed rational analysis can be shown to be fallacious.
Future generations will look back on the mathematicians of our time, with their weird beliefs of different “sizes” of limitlessly large sets and their quasi-mystical invocation of indefinable entities, with the same bemusement as we now have when we consider Newton’s strange outdated beliefs.