Copyright © James R Meyer 2012 - 2018 https://www.jamesrmeyer.com
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 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.
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.
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 that 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.
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.
Diverse opinions and criticisms are welcome, but messages that are frivolous, irrelevant or devoid of logical basis will be blocked. Difficulties in understanding the site content are usually best addressed by contacting me by e-mail. Note: you will be asked to provide an e-mail address - any address will do, it does not require verification. Your e-mail will only be used to notify you of replies to your comments - it will never be used for any other purpose and will not be displayed. If you cannot see any comments below, see Why isn’t the comment box loading?.
There is now a paper that deals with the matter of language and the diagonal proof, see On Considerations of Language in the Diagonal Proof.
There is now a new page on a contradiction in Lebesgue measure theory.
I found that making, adding or deleting footnotes in the traditional manner proved to be a major pain. So I developed a different system for footnotes which makes inserting or changing footnotes a doddle. You can check it out at Easy Footnotes for Web Pages (Accessibility friendly).
I have now added a new section to my paper on Russell O’Connor’s claim of a computer verified incompleteness proof. This shows that the flaw in the proof arises from a reliance on definitions that include unacceptable assumptions - assumptions that are not actually checked by the computer code. See also the new page Representability.
For convenience, there are now two pages on this site with links to various material relating to Gödel and the Incompleteness Theorem
– a page with general links:
– and a page relating specifically to the Gödel mind-machine debate:
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Copyright © James R Meyer 2012 - 2018