[joe_the_user]

It depends how you conceptualize things.

If one takes natural language as a means to definite sets, you quickly get a plethora of paradoxes, for example Russel's paradox("Takes the sets of all sets that don't contain themselves. Does that contain itself?"). So if one takes "natural language" as one's system of defining set, one has to assume it's inconsistent and any statement is provably true and false. Thus "Richard's Paradox" is in the same boat as all statements in our "system of natural language".

That said, I think the proof in the text falls apart in another way - it neglects the distinction made in Skolem's Paradox. The Löwenheim–Skolem [2] theorem show that any system definable with a finite alphabet has a countable model. This is only an apparent paradox because this model is only countable when "viewed" from outside the model, within the model, it is possible to have ostensibly uncountable sets. So ones could certainly haves a countable model of the real numbers while the real numbers themselves remained uncountable as "countable" and "uncountable" were defined in the countable model.

[1] https://en.wikipedia.org/wiki/Skolem%27s_paradox [2] The https://en.wikipedia.org/wiki/L%C3%B6wenheim%E2%80%93Skolem_...

[gerdesj]

"If one takes natural language as a means to definite sets, you quickly get a plethora of paradoxes"

Which is why I think if you resort to natural language to refute the consequences of a more rigorous treatment of the concept of number then there will be trouble. The whole paper attempts to refute things like Cantor through a weird recourse to French.

However I think it is possible to reduce real numbers as being a sort of subset of French purely through the same construct that Mr Cantor describes because that's the way descriptions work. If you define a real in some way in some form of symbolic language - I recall from GED that SSS might embody "three" and so does "trois". So I don't see why French can't encompass reals SSS can be considered exactly equivalent to trois.

I suspect I need to know and understand the formal, rigorous definition of "real" before I really give it some.