Page last updated 10 Feb 2022
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 “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. 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 “sizes”:
First of all, we note that we can enumerate all symbol combinations of the form of 0⁄10, 0⁄100, 0⁄1000, … to define a set A, and that we can enumerate all symbol combinations of the form of 1⁄10, 1⁄100, 1⁄1000, … 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. Furthermore, any subset of a denumerable set is also denumerable. This means that every possible set of combinations of symbols of the form of 0⁄10, 0⁄100, 0⁄1000, … and of the form of 1⁄10, 1⁄100, 1⁄1000, … is denumerable. And for a real number, a defined sequence of digits such as 0.101… can be used to represent a sequential sum of fractions such as 1⁄10 + 0⁄100 + 1⁄1000 + …, so every decimal expansion of a number is denumerable, that is, the digits of the decimal expansion of any real number constitute a countable set: there is a first digit, a second digit, and so on, and every digit has an associated natural number that denotes its position in the sequence. This applies regardless of whether there can or cannot be indefinable real numbers.
But 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 function f (n) for any such indefinable number, that could, for any given natural number n, correctly output the nth digit of that indefinable real number - otherwise, if there were such a function, then that would completely define that real number and that number would be definable.
Now we have a contradiction, since we also know that there cannot be real numbers that have “more” digits in their infinite decimal expansion than any other irrational real number that has an infinite decimal expansion - there aren’t digits missing in the infinite decimal 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. This shows that when there is a 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 digits of any hypothetical indefinable number - even though there cannot be any “extra” digits in the decimal expansion of any such indefinable number.
Hence the notion - 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 - 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 “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 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 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 decimal expansion of an indefinable real number, but the enumeration function is itself also indefinable. Or that an enumeration function might “exist” for the union set 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?
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 - for more on this see the page The Axiom of Choice and Well-Ordering.
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. 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.
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