On Monday, April 14, 2014 9:17:07 AM UTC-7, Jim Burns wrote: > On 4/14/2014 11:16 AM, email@example.com wrote:
> > Actual infinity implies a contradiction.
> This is the sort of claim that I have been looking for.
So even though you deny being a Platonist, you act like one.
> > A set of numbers > > is "proven" uncountable by constructing or defining another number > > that, together with the former, belongs to a countable set. > > The proof that all constructable and definable numbers belong > > to a countable set is circumvented by asserting the existence > > of undefinable numbers.
> I should give you credit for knowing that this is wrong, > I suppose. That means that you are blatantly lying.
> That spoils the fun for me, [...]
I too have trouble deciphering WM's argument. So let me present an argument that I *think* is close to the argument the he is trying to present.
In the language I am using here, the only things that "exist" are things that are provably well defined. Let's say that a real number is provably well defined iff we have a proof that every digit of the number can actually be computed. And let's say that a list of real numbers is provably well defined iff we have a proof that for all n and m, the n'th digit of the m'th real number in the list can actually be computed. Then the diagonal argument gives us a way to construct a provably well defined real number not on the list. However, that does not imply that it's meaningful to say that there exists a collection of all real numbers, and especially, it does not imply that there "exist" uncountable collections of real numbers.