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
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.
It says absolutely nothing about whether there are “more” real numbers than natural 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.
Also 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.”
But, of course, that is not a 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.
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 is an 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.
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.
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.
This is accompanied by the claim that, since by the definition the cardinality of a set is the number of elements in the set, then since different infinite sets can have different cardinalities, then different infinite sets can have different numbers of elements.
What the assumption in the definition conveniently ignores is that every infinite set has limitlessly many elements - and it is never explained how there can be limitlessly large sets that are smaller than other limitlessly large sets, when for both sets, there is no upper limit on how many elements they contain. It is impossible that one infinite set can have “more” elements than another, since there is no upper limit on the quantities of elements of either set.
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.
Some final observations
We may note that for finite sets, a one-to-one correspondence can be given without any reference whatsoever to the properties of the elements of the sets, by an actual list (e.g., 1:A, 2:B, 3:C). But for infinite sets this is not possible, and all known instances of a one-to-one correspondence are cases where there is a definition that defines a correspondence in terms of the properties of the elements of the sets. For example, the function 2x defines a one-to-one correspondence between the set of positive integers and the set of even positive integers, but it cannot define a correspondence for letters of the alphabet.
And we can also observe an instance where an element of a set has a property that prevents the definition of a one-to-one correspondence - the Power set proof relies on the definition of one subset where that definition gives the subset a property such that a one-to-one correspondence cannot be defined.
And we can further observe 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.
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.
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