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What's your favourite incredible fact?
The background to this question is as follows: Philipp I went to the mountains yesterday with a visiting Australian mathematician called Geordie. It was one of the best days we've had in the mountains - great conditions (though it may be time to abandon the winter trousers for the season), beautiful views, pleasant company and some fascinating discussions about mathematics*. On the train on the way back, we started discussing the question above, although being mathematicians we phrased it as "what's your favourite incredible theorem?"
I think it's an interesting question, and I'd like to throw it open to the floor. You may like to contribute theorems, but you don't have to: for instance, one of my favourite incredible non-mathematical facts is that traffic congestion can increase if you build more roads (even without an increase in total traffic volume, IIRC). Or how about "The Universe is nearly 15 billion years old, and contains around 10^23 stars"? That's pretty damn incredible, when you think about it.
By the way, our favourite incredible theorems were
Geordie: for n != 4, there is exactly one differentiable structure on R^n. For n = 4, there are uncountably many.
[Or: there exists a surjection from R to R^2 - the famous Peano curve being an example]
Philipp: I can't remember, even though he's just told me again. Something to do with coverings and Baier measure, whatever that is.
Me: there exists a countable model of ZF set theory. In other words, there's a universe of "sets" that satisfies all the usual axioms of set theory, but only has countably many objects.
Incredible, no?
* The only slight downside was that I'd been out for a romantic meal at the Shish Mahal with![[livejournal.com profile]](https://www.dreamwidth.org/img/external/lj-userinfo.gif)
wormwood_pearl the night before, and I'd made the schoolboy error of drinking five pints and then ordering the vegetable paal. For those of you who don't eat Indian food much, paal dishes are typically too hot to feature on menus, and I had to ask for this one specially. It was so hot. As Philipp put it when he tried some later, it was "hot beyond Good and Evil". After a couple of mouthfuls, even breathing became painful, as the air flowing over my palate re-ignited the chilli thereon. I managed about a third of it, and had to ask for the rest to be bagged up.
It was fantastic :-)
Though I have to agree withwormwood_pearl, in that it's better to think of it as a relish for other dishes than a dish in itself. Currently it's in the freezer, frozen into very small portions... anyway, my digestive system coped remarkably well, but I was still feeling a bit iffy the next morning on the train.
The background to this question is as follows: Philipp I went to the mountains yesterday with a visiting Australian mathematician called Geordie. It was one of the best days we've had in the mountains - great conditions (though it may be time to abandon the winter trousers for the season), beautiful views, pleasant company and some fascinating discussions about mathematics*. On the train on the way back, we started discussing the question above, although being mathematicians we phrased it as "what's your favourite incredible theorem?"
I think it's an interesting question, and I'd like to throw it open to the floor. You may like to contribute theorems, but you don't have to: for instance, one of my favourite incredible non-mathematical facts is that traffic congestion can increase if you build more roads (even without an increase in total traffic volume, IIRC). Or how about "The Universe is nearly 15 billion years old, and contains around 10^23 stars"? That's pretty damn incredible, when you think about it.
By the way, our favourite incredible theorems were
Geordie: for n != 4, there is exactly one differentiable structure on R^n. For n = 4, there are uncountably many.
[Or: there exists a surjection from R to R^2 - the famous Peano curve being an example]
Philipp: I can't remember, even though he's just told me again. Something to do with coverings and Baier measure, whatever that is.
Me: there exists a countable model of ZF set theory. In other words, there's a universe of "sets" that satisfies all the usual axioms of set theory, but only has countably many objects.
Incredible, no?
* The only slight downside was that I'd been out for a romantic meal at the Shish Mahal with
wormwood_pearl the night before, and I'd made the schoolboy error of drinking five pints and then ordering the vegetable paal. For those of you who don't eat Indian food much, paal dishes are typically too hot to feature on menus, and I had to ask for this one specially. It was so hot. As Philipp put it when he tried some later, it was "hot beyond Good and Evil". After a couple of mouthfuls, even breathing became painful, as the air flowing over my palate re-ignited the chilli thereon. I managed about a third of it, and had to ask for the rest to be bagged up.It was fantastic :-)
Though I have to agree with
wormwood_pearl, in that it's better to think of it as a relish for other dishes than a dish in itself. Currently it's in the freezer, frozen into very small portions... anyway, my digestive system coped remarkably well, but I was still feeling a bit iffy the next morning on the train.
no subject
Really? How does the power set axiom work?
no subject
The powerset of N is still uncountable, but it's uncountable within the model. Which is to say that while we, with our godlike exterior view, can look at P(N) and see that it's countable, this countability is not witnessed by a bijection N <-> P(N) in the model itself.
This theorem is a consequence of the downward Lowenheim-Skolem theorem (http://en.wikipedia.org/wiki/Lowenheim-Skolem_theorem), by the way, which you should know more about than me...
no subject
Ah yes, having checked, that was indeed on the b1 further logic syllabus. This is the extent of my knowledge of model theory.
So, given that I took the course but had forgotten, how come, having not taken the course, you knew what was on the syllabus??
Am trying to think of an incredible fact of my own, but have nothing better than `projective geometry! Ooh!'
no subject
The reason I know is because I looked at that course quite carefully before deciding I didn't want to take it - this formed part of my rationale for switching from Maths and Philosophy. I've since had a couple of fascinating chats with Jeff Egger (a reformed model theorist), which have convinced me that there's actually some deep and cool content in set/model theory, and I was wrong to dismiss it so lightly as a result of bad experiences being taught it as an undergraduate. See also: representation theory.
no subject
no subject
*woogles*
no subject
no subject
Ok - let's try it like this:
Do you believe the universe is understandable? Isn't that incredible?
no subject
That puts me in mind of a rather lovely proof of the fact that there exist irrational (can't be expressed as a fraction) numbers x, y such that x^y is rational (which is itself pretty surprising):
Recall that sqrt(2) is irrational. Let z = sqrt(2)^sqrt(2). Either z is rational or it's irrational. If it's rational, put x = y = sqrt(2) and we're done. If it's irrational, put x = z and y = sqrt(2). Then x^y = (sqrt(2)^sqrt(2))^sqrt(2) = sqrt(2)^(sqrt(2)*sqrt(2)) = sqrt(2)^2 = 2, which is rational.
At no stage do we need to know if sqrt(2)^sqrt(2) is rational or irrational - I'm not even sure if it's known which it is :-)
Hilbert strikes back
About Baire: A subset of a topological space is of Baire category 1, if it is a countable union of nowhere dense sets. They're called meagre, which seems fair enough. But: The real line can be covered by a set of Baire category 1 and a set of measure 0. Meagre is maybe is the wrong description.
Re: Hilbert strikes back
(PS. The formulae for sin & cos in terms of cot-csc are given by stereographic projection of the circle onto a line).