> The first person to notice this difficulty was Jules Richard in 1905, and the manner in which he formulated the problem is now called Richard’s paradox. Here is how it goes.
> Since all possible texts in French (Richard was French) can be listed or enumerated, a first text, a second one, etc., 2 you can diagonalize over all the reals that can be defined or named in French and produce a real number that cannot be defined and is therefore unnameable. However, we’ve just indicated how to define it or name it! In other words, Richard’s paradoxical real differs from every real that is definable in French, but nevertheless can itself be defined in French by specifying in detail how to apply Cantor’s diagonal method to the list of all possible mathematical definitions for individual real numbers in French!
I followed this up to "and produce a real number that cannot be defined". Would this be from some computation on some/all of the diagonalized reals? I can't see how to guarantee that this generated number wasn't already in the set.
> I can't see how to guarantee that this generated number wasn't already in the set.
The trick here is the diagonal argument [1].
The set (S) contains all real numbers that can be described by a valid French sentence (enumerating French sentences in some fixed order). For example we could come up with an enumeration such that the first few elements are:
S_0 = .7139847654
S_1 = .111111111111
S_2 = .93939
S_3 = .313331333133 repeating
...
We can construct a diagonal number r such that the ith digit of r differs from the ith digit of S_i for all S_i in S:
Ah got it. This seems different than the idea of there being 'more' real numbers and is rather a quirk of an unclear specification of set membership. This seems to be covered in 'Richardian Numbers'[0]
I am similarly confused, Is Richard's paradox really a paradox, since the 'diagonalized' mapping of texts to reals is not the same as the implied 'meaning-based' one?
'meaning-based' mapping is not bijective - both texts 'pi' and 'the number pi' map to the same real.
On the other hand the 'diagonalization' mapping doesn't care about the new real: texts which describe how to obtain it are already assigned to an arbitrary real.
> Since all possible texts in French (Richard was French) can be listed or enumerated, a first text, a second one, etc., 2 you can diagonalize over all the reals that can be defined or named in French and produce a real number that cannot be defined and is therefore unnameable. However, we’ve just indicated how to define it or name it! In other words, Richard’s paradoxical real differs from every real that is definable in French, but nevertheless can itself be defined in French by specifying in detail how to apply Cantor’s diagonal method to the list of all possible mathematical definitions for individual real numbers in French!
I followed this up to "and produce a real number that cannot be defined". Would this be from some computation on some/all of the diagonalized reals? I can't see how to guarantee that this generated number wasn't already in the set.