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What's your least favourite axiom? Mine's powerset.

Choice, Countable Choice, LEM, K

@j powerset is pretty bad though too

@axiom K?

@kwarrtz Ah, it sounded like a type theory thing. I haven't had the time I'd like to look into HoTT. Any recommendations for resources?

I'm uncomfortable enough with ZF(C) as a basis for modern maths, that I'm not sure I could handle yet another system right now.

@j Really the only introductory text available right now is the HoTT book by UFP (https://homotopytypetheory.org/book/) It's quite good, though obviously pretty dense. I'm not an expert by any means, but I'll also be happy to try to answer and questions.

Being comfortable with set theory or formal logic isn't necessarily a prerequisite for learning HoTT. In fact, I'd say if you've ever done any (especially functional) programming, type theory may actually be more intuitive than ZFC.

@kwarrtz ) closing bracket for the lispers and those with mild ocd.

@j I feel kinda bad that I didn't even notice...

@kwarrtz Luckily, while forcing will be covered during the class, it won't be on the exam.

@j Ah, that's a relief 😅

@kwarrtz I've one of the stranger course loads in the maths department at the moment:

"Axiomatic Set Theory", "Group Representations", "Quantum Filed Theory", "General Relativity", and "the Standard Model" are my modules this time around.

Bizarrely enough "Set Theory" and "General Relativity" are my favourite modules right now.

It's still a far cry from last semester where I kinda took an introductory haskell module that I could've done in my sleep to drag my grades back up.

@kwarrtz I was actually going to be taking either algebraic topology or geometric this semester, but I switched to set theory just before christmas, after a bad experience with one lecturer. I'm not in the US, so no gen-ed requirements. Up until this year, I was in a theoretical physics degree, which required me to do some lab work.

I finally switched to a maths degree program this year, which means I no longer have to deal with the terrible lectures of the physics department. (No more labs!!)

@kwarrtz Where I am, there was a huge difference in the quality of lectures given by the maths department and physics department.

I'm lucky to be a student of a majority ex-USSR maths department. [The stories are the only thing that makes 3 hours of QFT on a tuesday afternoon bearable]

You might enjoy this essay by Arnol'd: (Mathematics is the part of physics where experiments are cheap.)

@kwarrtz Arnol'd is an idol of mine (please, regardless of how this conversation pans out check out "Mathematical Methods of Classical Mechanics"), so I love any disagreement with this essay in particular.

I believe that mathematics is nothing more than an interpretation of those models that you describe as being built by science.

(Formalism+finitisim ftw!)

Mathematics is *useful*, but it's truth-hood is open to debate.

@j All of them. Use an axiomless deduction system

@j

The infinity axiom seems a lot weirder than the power set axiom. Although I wouldn't say I feel any strong dislike against it.

The infinity axiom seems a lot weirder than the power set axiom. Although I wouldn't say I feel any strong dislike against it.

@abs See, I can at least conceptualize a countably infinite set. The idea that a process can be repeated indefinitely is amenable to even the most ardent finitists. Powerset involves accepting the powerset of a countable set, along with the powerset of that set, along with the powerset of that set, etc. An unimaginably large collection of sets, that I find hard to justify philosophically.

(I've other reasons to be uncomfortable with powerset, but this is the easiest to express in a toot)

@j

I agree that a potentially infinite process (always generating the next natural number) is unproblematic finitistically. Concluding that all of its intermediate results should therefore exist, bundled up into a set isn't, though. That's the point where Hilbert, for example, would draw the line.

Avoiding higher cardinalities seems hard once you have countably infinite sets because even just the set of functions from the natural numbers into the two-set is uncountable. You would have to strongly identify functions with some kind of countable "programming language" to get around that.

On the other hand, if you were to remove the axiom of infinity, the power set axiom becomes pretty unproblematic, I think. This is not the case for the infinity axiom if you remove power sets, as you still have to justify why an actually infinite object should exist.

I agree that a potentially infinite process (always generating the next natural number) is unproblematic finitistically. Concluding that all of its intermediate results should therefore exist, bundled up into a set isn't, though. That's the point where Hilbert, for example, would draw the line.

Avoiding higher cardinalities seems hard once you have countably infinite sets because even just the set of functions from the natural numbers into the two-set is uncountable. You would have to strongly identify functions with some kind of countable "programming language" to get around that.

On the other hand, if you were to remove the axiom of infinity, the power set axiom becomes pretty unproblematic, I think. This is not the case for the infinity axiom if you remove power sets, as you still have to justify why an actually infinite object should exist.

@kwarrtz Yes I think that would be totally acceptable finitistically. (You’re referring to this, correct?)

There is a good paper by Tait that argues this point really well.

@j

@abs That’s exactly what I was referring to, yes.

I’ll have to take a look at that paper. I didn’t realize anyone else had thought about this before (it was just a random thought that struck me reading your post) but I’m not surprised

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