[gerdesj]

Richard's Paradox seems a bit shaky to me (p4): "Since all possible texts in French can be listed or enumerated"

Unless I have completely missed the point then he has simply stated a way to generate another member of the set of French texts which of course is part of that set and so on.

You can easily squint hard enough to generalize to all texts in all languages, now, earlier and possible then allow that grammar, spelling and so can be pretty slack. Now translate that lot into numbers in some way (a bunch of IT bods should be able to manage that!) To be honest French on it's own is probably more than enough.

"How very embarrassing! Here is a real number that is simultaneously nameable yet at the same time it cannot be named using any text in French."

The very act of naming the number (in French) constructs the French text that adds to the set of possible French texts.

I think that the set of possible French texts is exactly as large as the set of reals. So is the set of all language texts and that the "paradox" is merely trying to use the Cantor argument backwards.

[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.

[bo1024]

> he has simply stated a way to generate another member of the set of French texts which of course is part of that set and so on.

Nope, because he used the diagonalization technique to make sure the new text was not in the original set, which supposedly contained all texts.

[gerdesj]

I don't know what the rigorous definition of a real number is nor even what the rigorous definition of French (texts) is but I'm pretty sure it would possible to define "reals" in terms of "French texts" in such a way that all the results hold.

Funnily enough: 0,1,2,3,4,5,6,7,8,9 are French words along with . and ,

So, how do we proceed to define reals in French?

[surement]

> I think that the set of possible French texts is exactly as large as the set of reals.

How? The set of French texts is countable and the set of reals is uncountable.

[gerdesj]

0,1,2 etc are French words. All reals are French words 8)

[jawarner]

This is not true, no French word has an infinite length. Any set of finite strings is countable.

[Retra]

Not all French texts need be finished either. Nobody his picked a date boundary to constrain all French text. Nor has anyone formalized which texts are French, since the French language is evolving.

[porges]

The difference is that a French text must be finite.