I guarantee that a naive presentation doesn't actually include the axioms, and doesn't address the philosophical questions dividing formalism from constructivism.
Uncountable need not mean more. It can mean that there are things that you can't figure out whether to count, because they are undecidable.
> I guarantee that a naive presentation doesn't actually include the axioms
But you said "modern math courses". Are you now talking about a casual conversation? I mean the OP's story is that his wife just liked listening to him talk about his passions.
> Uncountable need not mean more.
Sure. But that doesn't mean that there aren't differing categories. However you slice it, we can operate on these things in different ways. Real or not the logic isn't consistent between these things but they do fall out into differing categories.
If you're trying to find mistakes in the logic does it not make sense to push it at its bounds? Look at the Banach-Tarski Paradox. Sure, normal people hear about it and go "oh wow, cool." But when it was presented in my math course it was used as a discussion of why we might want to question the Axiom of Choice, but that removing it creates new concerns. Really the "paradox" was explored to push the bounds of the axiom of choice in the first place. They asked "can this axiom be abused?" And the answer is yes. Now the question is "does this matter, since infinity is non-physical? Or does it despite infinity being non-physics?"
You seem to think mathematicians, physicists, and scientists in general believe infinities are physical. As one of those people, I'm not sure why you think that. We don't. I mean math is a language. A language used because it is pedantic and precise. Much the same way we use programming languages. I'm not so sure why you're upset that people are trying to push the bounds of the language and find out what works and doesn't work. Or are you upset that non-professionals misunderstand the nuances of a field? Well... that's a whole other conversation, isn't it...
Vitali's counter-example for the existence of a non-trivial measure requires the Axiom of Choice. But not assuming the Axiom of Choice then prohibits choosing and actually using that measure. There is more to it than just Banach-Tarski.
Your guesses at what I seem to think are completely off base and insulting.
When I say "modern math courses", I mean like the standard courses that most future mathematicians take on their way to various degrees. For all that we mumble ZFC, it is darned easy to get a PhD in mathematics without actually learning the axioms of ZFC. And without learning anything about the historical debates in the foundations of mathematics.
Honestly it's difficult to understand exactly what you're arguing. Because I understand laymen not understanding your argument about infinities not being real (and even many HN users don't understand code is math bit a CS degree doesn't take you far in math. Some calc and maybe lin alg) but are we concerned about laymen? I too am frustrated by nonexperts having strong opinions and having difficulties updating them, but that's not a culture problem. We're on HN and we know the CS stereotypes, right?
If instead you're talking about experts then I learned about what you're talking about in my Linear 2 course in a physics undergrad and have seen the topic appear many times since even outside my own reading of set theory. The axiom of choice seems to have even entered more main stream nerd knowledge. It's very hard to learn why AoC is a problem without learning about how infinities can be abused. But honestly I don't know any person that's even an amateur mathematician that thinks infinities are physical
The fact that you think I'm talking about the axiom of choice, demonstrates that you didn't understand what I'm talking about. I would also be willing to bet a reasonable sum of money that this topic did not come up in your Linear 2 course in physics undergrad.
The arguments between the different schools of philosophy in math are something that most professional mathematicians are unaware of. Those who know about them, generally learned them while learning about either the history of math, or the philosophy of math. I personally only became aware of them while reading https://www.amazon.com/Mathematical-Experience-Phillip-J-Dav.... I didn't learn more about the topic until I was in grad school, and that was from personal conversations. It was never covered in any course that I took on, either in undergraduate or graduate schools.
Now I'm curious. Was there anything that I said that should have been said more clearly? Or was it hard to understand because you were trying to fit what I said into what you know about an entirely unrelated debate about the axiom of choice?
> The fact that you think I'm talking about the axiom of choice, demonstrates that you didn't understand what I'm talking about.
Dude... just a minute ago you were complaining about ZFC... Sure, I brought up AoC but your time to protest was then.
The reason I brought up AoC is because it is a common way to learn about the abuse of infinity and where axioms need be discussed. Both things you brought up. I think you are reading further into this than I intended.
> Now I'm curious. Was there anything that I said that should have been said more clearly?
Is this a joke?
When someone says
>> Honestly it's difficult to understand exactly what you're arguing.
That's your chance to explain. It is someone explicitly saying... I'm trying to understand but you are not communicating efficiently.
This is even more frustrating as you keep pointing out that this is not common knowledge. So why are you also communicating like it is?! If it is something so few know about then be fucking clear. Don't make anyone guess. Don't link a book, use your own words and link a book if you want to suggest further reading, but not "this is the entire concept I'm talking about". Otherwise we just have to guess and you getting pissed off that we guess wrong is just down right your own fault.
So stop shooting yourself in the foot and blaming others. If people aren't understanding you, try assuming they can't read your mind and don't have the exact same knowledge you do. Talk about fundamental principles...
That point being that what we mean by "exists" is fundamentally a philosophical question. And our conclusions about what mathematical things exist will depend on how we answer that question. And very specifically, there are well-studied mathematical philosophies in which uncountable sets do not have larger cardinalities than countable ones.
If none of those explanations wind up being clear for you, then I'm going to need feedback from you to have a chance to explain this to you. Because you haven't told me enough for me to make any reasonable guess what the sticking point is between you and understanding. And without that, I have no chance of guessing what would clarify this for you.
The "philosophical questions" dividing formalism from constructivism are greatly overstated. The point of having those degrees of undecidability or uncountability is precisely to be able to say things like "even if you happen to be operating under strong additional assumptions that let you decide/count X, that still doesn't let you decide/count Y in general." That's what formalism is: a handy way of making statements about what you can't do constructively in the general case.
To be fair, constructivists tend to prefer talk about different "universes" as opposed to different "sizes" of sets, but that's all it is: little more than a mere difference in terminology! You can show equiconsistency statements across these different points of view.
Yes, you can show such equiconsistency statements. As Gödel proved, for any set of classical axioms, there is a corresponding set of intuitionistic axioms. And if the classical axioms are inconsistent, then so is the intuitionistic equivalent. (Given that intuitionistic reasoning is classically valid, an inconsistency in the intuitionistic axioms trivially gives you one in the classical axioms.)
So the care that intuitionists take does not lead to any improvement in consistency.
However the two approaches lead to very different notions of what it means for something to mathematically exist. Despite the formal correspondences, they lead to very different concepts of mathematics.
I'm firmly of the belief that constructivism leads to concepts of existence that better fit the lay public than formalism does.
Uncountable need not mean more. It can mean that there are things that you can't figure out whether to count, because they are undecidable.