On 15 Jun., 21:32, stevendaryl3...@yahoo.com (Daryl McCullough) wrote: > WM says... > > > > >On 15 Jun., 18:49, stevendaryl3...@yahoo.com (Daryl McCullough) wrote: > >> I was talking about the list of all *computable* reals. There are > >> definable reals that are not computable, and Cantor's proof shows > >> how to define one. > > >No, it does not show that because it is impossible to define a real by > >an infinite definition. > > It's not an infinite definition. It's a finite definition..
An infinite Cantor-list not constructed according to a finite definition, means an infinite definition of the diagonal number > > >> You can similarly get a list of all definable reals for a specific > >> language L. > > >Every real that is definable in a specific language is definable in > >another specific language. The set does not grow or shrink or change > >when the language is changed. > > It certainly does.
And if it would do so. What do you think would it help you?
How many languages do exist? Countably many. How many definable numbers in one language do exist? Countably many.
So the set of all definable numbers in all languages in countable. That means the diagonal argument proves that the countable set of definable numbers is uncountable. A fine result! > > Let L be any countable language whose intended model is the naturals. > Following Godel, we define a way to code each formula of L as a natural > number. Then we introduce a new unary predicate symbol T with the > following interpretation: > > T(x) is true > if and only if x is the code of a true sentence of language L. > > Tarski proved that T(x) is not definable in language L. Using > the new predicate T, we can define a real that is provably > unequal to any real definable in the original language L.
Nevertheless is belongs to the countable set of all countable reals (in all languages). > > >No, my list contains all words in all languages, even the definitions > >of the languages and all the dictionaries. There is nothing else. > > No language can consistently contains its own truth predicate.
Of course it can, because uncountability does not exist.
However that does not matter at all. All real numbers that are definable in all possible languages form a countable set.
Or do you meanwhile believe that the number of all languages over all finite alphabets is uncountable? Or do you believe that the cartesian product of countably many countable sets could be uncountable?