>> 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.
>> 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.
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.
>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.