The famous diagonal argument goes like this. Suppose you had a countable enumeration of some real numbers, then you could use the diagonal algorithm to produce a real number not in your enumeration. So far so good.

The fallacy occurs when you then claim that this somehow shows that the set of reals is a bigger infinite than the set of naturals. Clearly, the diagonal argument made in the paragraph above contained no statement about comparing infinities, so it is logically impossible to deduce a statement comparing infinities from the diagonal argument in the above paragraph.

You may complain "But not being able to find a countable enumeration is by definition the same as being an uncountable infinity". Well, you can make that definition if you want to but then you would be confusing epistemology for ontology. Just because a countable enumeration of all the reals can't feature in your proofs does not mean the reals are "uncountably large" (whatever that term is really supposed to mean) or that a countable enumeration does not "exist".

I'm posting this because I too once believed in uncountable infinities like most simple-minded modern mathematicians, but the realization of this basic flaw in the argument destroyed my faith in set theory and now I think the entire field is built on a fallacy. This might help liberate others out there who are struggling with the unreasonable demands made by the set theory religion. Thanks.