Proof of more Real numbers than Natural numbers

Page last updated 29 Aug 2023

If you have come to this page looking for the proof that there are more real numbers than natural numbers, then you have come to the right place. Reading this page will save you a lot of time and effort searching for something that does not exist - because this page reveals that there is not actually any rigorous proof that there are more real numbers than natural numbers. When asked for such a proof, mathematicians and logicians resort instead to various pleas, which are described below. (Footnote: For an example see Why we need formality in mathematics where the author writes an article proclaiming the importance of formal rigor in mathematics, particularly in relation to the question of whether there are more real numbers than natural numbers. However, when challenged in the subsequent comments, he completely fails to provide any rigorous defence of his claim that there are more real numbers than natural numbers.)

Cantor’s Diagonal proof

Georg Cantor’s Diagonal proof is probably the most commonly cited claim of a proof that there are “more” real numbers than natural numbers.

However, the simple fact is that Cantor’s Diagonal argument only establishes that, within any given mathematical system, there cannot exist any one-to-one correspondence between the set of all real numbers and the set of all natural numbers.

That’s all.

It says absolutely nothing about whether there are “more” real numbers than natural numbers.

It is crucially important to realize that there is a fundamental difference between finite sets and infinite sets with regard to one-to-one correspondences. Given two finite sets that have the same number of elements, one can always have a one-to-one correspondence that does not require any reference to the properties of the elements of the sets by simply creating an actual list (e.g. 1:A, 2:B, 3:C). Changing the order in the list or replacing any element by some other element (not already in the set) cannot change the fact that there is a one-to-one correspondence. On the other hand, it is impossible to create an actual list of a one-to-one correspondence between two infinite sets, since such a list would be infinitely long. No, for two infinite sets a one-to-one correspondence can only be given by defining a function that defines that one-to-one correspondence in terms of the properties of the elements of the sets. (Footnote: For example, the function 2x defines a one-to-one correspondence between the set of positive integers and the set of even positive integers.)

By acknowledging this crucial difference, we see that the impossibility of defining a one-to-one correspondence between two infinite sets simply implies that there is a difference in the properties of the elements of the sets, rather than some difference in a property of the sets that is independent of the actual details of the elements of the sets. This is discussed in more detail on the page Cardinal Numbers

It can also be noted that the size of a set is independent of the properties of its elements, that is, one can replace any element by some other element not already in the set and the size of the set is not affected; for more on this see The Origins of Comparisons of Cardinal Numbers.

The diagonal proof is a proof of what is called non-denumerability. Note that there are other proofs of non-denumerability, but, as for the diagonal proof, these proofs also only prove the absence of a one-to-one correspondence within a mathematical system and they do not prove that there are “more” real numbers than natural numbers. (Footnote: Note that there are some fully formal proofs that also prove non-denumerability, but again, they only prove the absence of a one-to-one correspondence within a mathematical system and do not prove that there are “more” real numbers than natural numbers.)

Intuition steps in

When the above is pointed out, an intuitive plea is frequently advanced which is some version of:

“I can’t see how two sets could not be different sizes if there cannot be any one-to-one correspondence between them.”

This is an intuitive assumption that has no logical basis whatsoever, and results in a direct contradiction; the notion that one set with limitlessly many elements can have fewer elements than another set with limitlessly many elements is a blatant contradiction. For an example of how this assumption is slipped in as though it were a proven theorem see On David Hilbert’s On the Infinite, and see The Origins of Transfinite Numbers for Cantor’s justification of this contradictory assumption. Such assumptions do not in any way constitute mathematical proof, and you are entitled to summarily reject the plea on that account. A mathematician should not accept any argument that you present that is not a rigorous proof, and similarly you shouldn’t accept any argument presented to you that is not a rigorous proof.

Indefinable numbers

Another favorite plea is the argument that there exist infinitely many “indefinable” real numbers and that these “indefinable” numbers account for the difference in size of the set of real numbers and the set of natural numbers. This argument arises from a secondary argument that is commonly tacked onto the diagonal argument - the details of the argument and the rather obvious flaw in that argument are given in The Secondary Argument. The absurdity of the notion of “indefinable” real numbers is analyzed at: The contradictions of “indefinable” numbers.

Specific “indefinable” numbers

When faced with the refutations of the above-mentioned pleas, some people proceed to a claim that they can show that there are actually real numbers that can be precisely described but which are “indefinable” - by which they mean that no-one can ever determine that exact number. (Footnote: The claim is that there is at least one digit in any expansion of that real number that can never be calculated.) An example of this is Chaitin’s Omega number. But the flaw in such arguments is easily demonstrated, see Chaitin’s Constant Error.

Cardinality

Another favorite trick is to play the definition of cardinality card. This trick doesn’t even begin to provide a proof - instead it relies on your gullible acceptance of a definition of cardinality which includes an assumption that guarantees the notion that some infinite sets can have “more” elements than some other infinite sets. See The Conventional Definition of Cardinality for an analysis of the definition and why it includes an unacceptable assumption.

Given that definition, the claim necessarily follows:

If there is no one-to-one correspondence between two sets, then the cardinality of one set must be greater than that of the other.

For finite sets that have the same number of elements, a one-to-one correspondence can always be given without any reference to the properties of the elements of the sets, by simply creating an actual list (e.g. 1:A, 2:B, 3:C). Replacing any element by some other element (not already in the set) cannot change the cardinal number of the set, so that the cardinal number of any finite set is a property that is independent of the properties of its elements.

For infinite sets it is a completely different matter, since it is not possible to create an actual list (since such a list would be infinitely long). A one-to-one correspondence between two infinite sets can only be given by a definition of a function that defines that one-to-one correspondence in terms of the properties of the elements of the sets, and the absence of a one-to-one correspondence between the elements of two infinite sets results from differences in the properties of their elements, not from a difference in their quantity.Footnote: For example, the function 2x defines a one-to-one correspondence between the set of positive integers and the set of even positive integers.)

Hence, although the impossibility of defining a one-to-one correspondence between two finite sets implies that there is a difference in a property of the sets, where that property is independent of the actual details of the elements of the sets, the impossibility of defining a one-to-one correspondence between two infinite sets implies that there is a difference in the properties of the elements of the sets, rather than a difference in a property of the sets that is independent of the actual details of the elements of the sets.⁠ (Footnote: The same applies to the impossibility of a surjection between two infinite sets, this is discussed in more detail on the page Surjections and Cardinal Numbers.)

We can note that the Diagonal proof and the Power set proof are cases where the absence of a one-to-one correspondence is a result of the properties of the elements of the sets, rather than any property of the set that is independent of the particular details of their elements.

The entire field of transfinite numbers depends on the uncritical usage of this unfounded assumption. It remains nothing more than an illogical assumption - no proof of this assumption has ever been discovered, see Proof of more real numbers than natural numbers? and Cardinal Numbers and The Origins of Transfinite Numbers. It is hardly surprising that no logically valid proof of it has ever been found since the notion of one limitlessly large set being smaller than another limitlessly large set is inherently contradictory.

There is no reason for you or anyone else to accept an assumption that directly produces such an absurd result. Ask for a rigorous proof, and don’t accept absurd assumptions.

The last resort - the ad hominem attack

When you have rejected all the pleas as not fulfilling the requirement of being a rigorous mathematical proof, it’s not uncommon to be the object of the old ad hominem trick of branding anyone with a dissenting voice as a “crank”. These people fail to see the irony that it is the supposed “crank” who is requesting a rigorous argument.

Why do people continue to defend the indefensible?

It is an intriguing question - why do people prefer to continue to believe something that is blatantly contradictory? And why do they continue to believe it when, despite the claim being prevalent for well over a hundred years, there has never been anything even approaching a rigorous proof of the claim? Why is there an immediate knee-jerk reaction against any suggestion that one might be able to remove this contradiction by a full logical analysis of everything involved? See Why do people believe weird things? for some of the reasons for this strange attitude.