The piece is just a bit of fun, a bit of a stunt — TonesInDeepFreeze

A simplification too far IMO, and as evidence, it was presented in this thread as being meaningful. A belief I corrected.

It is meaningful. People who understand that proof is relative, and who are already familiar with incompleteness, might appreciate that a brief spoken word bit doesn't have to provide all the qualifications.

My opinion is supported by the fact that Kurt Gödel was a Platonist, and believed that there was a true fact of the matter for every mathematical proposition. — fishfry

That Godel was a platonist supports your opinion that Boolos's rhetorical lark is wrong and meaningless?

People who understand that proof is relative, and who are already familiar with incompleteness, might appreciate that a brief spoken word bit doesn't have to provide all the qualifications. — TonesInDeepFreeze

Of course. But that's not who the article is aimed at. I respect that you have a different opinion, this is not a hill I'm going to choose to die on.

That Godel was a platonist supports your opinion that Boolos's rhetorical lark is wrong and meaningless? — TonesInDeepFreeze

It's a slow afternoon over here too. But since you asked ... yes. "The whole of math" includes the ultimate truth or falsity of any given proposition, irrespective of its provability in any given axiomatic system. If one is a Platonist, as Gödel was. As "the greatest logician since Aristotle," he'd never have agreed with what Boolos wrote, even in a lighthearted manner. Gödel doesn't strike me as much of a lighthearted person.

I hope you don't mind if I stop responding to this topic. I've said my piece and stand by my opinion.

that's not who the article is aimed at — fishfry

It seems it wasn't an article but something from a lecture. In any case, I would allow him some liberties when the purpose is to have some rhetorical fun rather than purporting to be an academically rigorous article.

"The whole of math" includes the ultimate truth or falsity of any given proposition, irrespective of its provability in any given axiomatic system. — fishfry

Yeah, I don't think that's what's mean by Boolos.

And it wouldn't make sense anyway to take him that way, since there is no known axiomatic theory that upholds such a collection of truths.

rhetorical fun — TonesInDeepFreeze

Would it be fair for me to say that I'm perfectly willing to allow Boolos his rhetorical fun; but that I was perfectly justified, and in fact helpful to the discussion, to point out the error of a poster quoting Boolos as if the out-of-context paragraph were meaningful and true?

It's a little like Hilbert's hotel, which Hilbert mentioned only once in his life in a public lecture, and never mentioned again; versus the legions of philosophers and especially theologians like William Lane Craig, who make a living off misunderstanding it? Hilbert was having fun, but those who misinterpret or misuse the story need to be corrected. Likewise, my complaint isn't with Boolos having fun and grossly simplifying a complicated subject. It's with anyone who takes what he said so literally that they'd use it as a talking point in a discussion on incompleteness.

"The whole of math" includes the ultimate truth or falsity of any given proposition, irrespective of its provability in any given axiomatic system.
— fishfry

Yeah, I don't think that's what's mean by Boolos. — TonesInDeepFreeze

What do you think he meant? I think me meant exactly what the incompleteness theorems say, namely axiomatic systems meeting certain technical conditions. And instead of saying that, he used "the whole of math" in a nontechnical sense. He clearly did not mean for people to be quoting him in conversations about incompleteness.

I think it is eminently helpful to point out to people not familiar with mathematical logic that proof is relative. I mentioned it myself about the Boolos piece. But then I realized I had read it too quickly and that it very much seems to me that he does intend provability to be relative to a theory (whatever theory one takes as encompassing the branches of mathematics) even if unspecified which theory. I allow him that especially since it is very common for people in foundations to say that ZFC axiomatizes the branches of mathematics. So I'm just saying, "Ward, don't you think you're being a little hard on the Beaver?"

Cantor's proof is not a reductio ad absurdum. — TonesInDeepFreeze

What?

Cantor's diagonal argument is a reductio ad absurdum proof. The link to incompleteness results should be obvious.

The diagonal argument was not Cantor's first proof of the uncountability of the real numbers, which appeared in 1874. However, it demonstrates a general technique that has since been used in a wide range of proofs, including the first of Gödel's incompleteness theorems and Turing's answer to the Entscheidungsproblem.

I'm guessing he means constructive in the sense that, given any countable list of real numbers, one can construct a real number not on that list, from that list. This shows that every injection fails to be a surjection. (In the same way, Euclid gave a way to construct a prime not already on a list of prime numbers. )

I think that what you say is the diagonal argument. A REAL NUMBER that is NOT on that list. That's the negative part. And how do we get that real number? From the list itself.

That's the reference part.

Cantor's diagonal argument is a reductio ad absurdum proof. — ssu

Actually not. It can be expressed in positive form. It shows that for every list$^*$ of real numbers, there is some real number that's not on the list. Equivalently, no list of reals enumerates or contains all the reals. No reductio necessary. It's often presented as a reductio, which is why everyone think's that's a necessary feature of the argument.

(*) A list is an ordered set order-isomorphic to the natural numbers in their usual order 0, 1, 2, 3, .... That is, a list is countable by definition; and it's NOT some exotic alternative ordering like 0, 2, 4, 6, ..., 1, 3, 5, 7, ... in which all the evens precede all the odds.

So there is no enumeration of the naturals onto the reals. — TonesInDeepFreeze

This shows that every injection fails to be a surjection. — T H E

What they said.

The link to incompleteness results should be obvious. — ssu

Yes, the Halting problem, the first incompleteness theorem, Cantor's theorem, etc. have all been subsumed into a category-theoretic framework as described here.

https://ncatlab.org/nlab/show/Lawvere's+fixed+point+theorem

There's a somewhat more accessible version of this idea here.

http://math.andrej.com/2007/04/08/on-a-proof-of-cantors-theorem/

An interesting historical note is that Cantor didn't invent the diagonal argument. It was invented by Paul du Bois-Reymond in 1875. He was investigating the growth rates of functions, essentially anticipating modern complexity theory in computer science. Although from the checkmarked answer on that page, maybe he didn't really invent the diagonal argument. It's kind of borderline.

https://hsm.stackexchange.com/questions/3812/did-du-bois-reymond-invent-the-diagonal-argument-before-cantor

Thanks for correcting me, and . Learn bit more every day. Yet of course, in math something important can be said in many ways.

So I'm just saying, "Ward, don't you think you're being a little hard on the Beaver?" — TonesInDeepFreeze

You realize that you have just outed yourself as being of a certain age . . . :razz:

You realize that you have just outed yourself as being of a certain age . . . :razz: — EricH

Is that an archaic movie quote, or something of antiquity?

So I'm just saying, "Ward, don't you think you're being a little hard on the Beaver?"
— TonesInDeepFreeze

You realize that you have just outed yourself as being of a certain age . . . :razz: — EricH

Or perhaps something else.

Quote from a 1960s USA TV show. You'll have to find the rest yourself . . . :nerd:

Aryamoy Mitra Quote from a 1960s USA TV show. You'll have to find the rest yourself . . . :nerd: — EricH

I did - and whilst it's a novelty relating to a statement first imparted more than 4 decades prior to my arrival, it's unsurprising that this particular reference hasn't entirely obsolesced (thematically).