I don't want to watch a video right now.

Cantor's diagonal argument says that any list of reals is incomplete. We can prove it directly by showing that any list of reals (not an assumed complete list, just any arbitary list) is necessarily missing the antidiagonal. Therefore there is no list of all the reals. — fishfry

Exactly.

I'll ask again:

But if you start from that there is no bijection, and then prove it by:
If there is a bijection then there is a surjection
There is no surjection.
Therefore, there is no bijection.

Isn't that a proof by contradiction?
— ssu — ssu

I gave you a very detailed answer. I can't do better than what I already wrote. Or, if you like, let me know what you don't understand in my post.

Only countably many interpretations of each sentence. — fishfry

I'm talking about interpretations for languages as discussed in mathematical logic.

There are uncountably many sets, so there are uncountably many universes for interpretations.

Or, another way: Consider just one uncountable universe. Let the language have at least one individual constant. Then there are uncountably interpretations as each one maps the constant to a different member of the universe.

I don't propound the notion that that approach could be adapted for natural languages too, but it doesn't seem unreasonable to me.

a trip to the moon on gossamer wings — fishfry

Seeing just that one phrase from the great song made my night. Such a soul satisfyingly beautiful song by a gigantically great composer.

We're talking about different things. I'm talking about formal theories and interpretations of their languages as discussed in mathematical logic, and such that theories are not interpretations. — TonesInDeepFreeze

enderton page ref please or st*u. second time i'm calling your bluff on references to your magic identity theory.

Exactly. — TonesInDeepFreeze

Stop agreeing with me, that's no fun!

(edit) So you see I do know some logic after all!

I don't propound the notion that that approach could be adapted for natural languages too, but it doesn't seem unreasonable to me. — TonesInDeepFreeze

ok

Seeing just that one phrase from the great song made my night. Such a soul satisfyingly beautiful song by a gigantically great composer. — TonesInDeepFreeze

You're alternately insulting and praising me. Make up your mind!

enderton page ref please or st*u. second time i'm calling your bluff on references to your magic identity theory. — fishfry

Did you mean for that to be in the 'Infinity' thread?

In that thread, you've now seen that I already had given you the Enderton pages yesterday and I gave them to you even though you had not asked for them. There's no bluff and never has been. I've been giving you post after post of correct corrections, information and explanations. It's not my fault that you regard that as inimical.

I know you're kidding. But underneath there lies an actual point for me, which is that I don't think you know how insulting you are in certain threads when you read (if it can be called 'reading') roughshod over my posts, receiving them merely as impressions as to what I've said, so that you so often end up completely confusing what I've said and then projecting your own confusions onto me.

But I do appreciate that you quoted Cole Porter's so charming and magical lyric. And there was another special musical moment for me today, so my evening was graced.

Thank you, @fishfry

It seems that from you I get extremely good answers. Yes, Lawvere's fixed point theorem was exactly the kind of result that I was looking for. It's just typical that when the collories are discussed themselves, no mention of this. I'll then have to read what Lawvere has written about this.

And that not necessary is important for me. This is what @TonesInDeepFreeze was pointing out to me also. I'll correct my wording on this.

It seems that from you I get extremely good answers. — ssu

Thank you.

Yes, Lawvere's fixed point theorem was exactly the kind of result that I was looking for. It's just typical that when the collories are discussed themselves, no mention of this. I'll then have to read what Lawvere has written about this. — ssu

If you're interested in this stuff, do you know the nLab Cafe? It's a category theory wiki. Here's their page on the theorem

It's all very categorical. Like a new paradigm for thinking about math.

And that not necessary is important for me. This is what TonesInDeepFreeze was pointing out to me also. I'll correct my wording on this. — ssu

I'm not sure how the subject came up. It's interesting to know that all these diagonal type proofs can be abstracted to a common structure. They are all saying the same thing.

I know you're kidding. But underneath there lies an actual point for me, which is that I don't think you know how insulting you are in certain threads when you read (if it can be called 'reading') roughshod over my posts, receiving them merely as impressions as to what I've said, so that you so often end up completely confusing what I've said and then projecting your own confusions onto me. — TonesInDeepFreeze

If I crossed any lines, I apologize. But I think you are equivocating the word "insult." If I tell you, Tones, you are a low down rotten varmint who cheats at cribbage!" that's an insult.

But if I don't happen to dwell on every word you write; and if I often find your expository prose convoluted and unclear, especially when you lay out long strings of symbols without any context; my eyes do glaze over, and I do skip things.

That is not an insult. It's just me being me, reacting to whatever you wrote that made my eyes glaze. The fault is all mine, But that's who I am and how I am. I am not insulting you.

Can you see the difference between:

(a) Me actively and directly insulting you; and

(b) Me just being my highly imperfect self, doing something that annoys you.

Surely you can see the difference.

But I do appreciate that you quoted Cole Porter's so charming and magical lyric. And there was another special musical moment for me today, so my evening was graced. — TonesInDeepFreeze

Well that's good, so let's go with the grace.

I'm not sure how the subject came up. — fishfry

From the OP at least I made the connection.

It's interesting to know that all these diagonal type proofs can be abstracted to a common structure. They are all saying the same thing. — fishfry

That's what really intrigues me. Especially when you look at how famous and still puzzling these proofs are...or the paradoxes. Just look at what is given as corollaries to Lawvere's fixed point theorem:

Cantor's theorem
Cantor's diagonal argument
Diagonal lemma
Russell's paradox
Gödel's first incompleteness theorem
Tarski's undefinability theorem
Turing's proof
Löb's paradox
Roger's fixed-point theorem
Rice's theorem

Of course in mathematics a lot theorems have corollaries, but I would just point out to what these theorems are about: limitations in proving, limitations in computation and a paradox, that basically ruined naive set theory and spurred the creation of ZF-logic. All coming from a rather simple thing.

Going back to the OP and the article given there, perhaps in the future it will be totally natural (or perhaps it is already) to start a foundation of mathematics or a introduction to mathematics -course with a Venn diagram that Yanofsky has page 4 has. Then give that 5 to 15 minutes of philosophical attention to it and then move to obvious section of mathematics, the computable and provable part.