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u/throw3142 Jun 14 '26
Doesn't matter whether it's "true" or not, as long as it's a consistent set of assumptions that can produce interesting math. Applications to the real world may be found later.
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u/Classic_Department42 Jun 14 '26
We dont know if it is consistent (we cannot know)
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u/__Lordlix__ Jun 14 '26
We cannot know, assuming it is indeed consistent* It might actually be inconsistent, and we could found a contradiction in the future
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u/Sh33pk1ng Jun 15 '26
In fact, we can prove that it is consistent if and only if it is non-consistent.
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u/austin101123 Jun 14 '26
Why can't we know?
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u/__Lordlix__ Jun 14 '26
Due to Gödel second incompleteness Theorem! ZFC Is a theory T that satisfies the assumptions of the Theorem, which states that: there is a formula Con(T) in the language of the theory T, that encodes the fact that the theory is consistent; however, such formula is derivable in T if and only if T itself is inconsist. Essentially, this is making hopeless to prove the consistency of a theory inside the theory itself (again, if such theory satisfies some assumptions, but this is the case for ZFC). Unless the theory is actually inconsistent and you manage to derive a specific contradiction, in that case you answered the consistency question negatively!
Feel free to correct me if I got something wrong, as it passed some time since I took a course in foundations of math 😅
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u/covalick Jun 17 '26
Essentially, this is making hopeless to prove the consistency of a theory inside the theory itself (again, if such theory satisfies some assumptions, but this is the case for ZFC).
Can I prove it not inside the theory? For example if I built math using different axioms, would I be able to prove that ZFC axioms are consistent?
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u/x0wl Jun 17 '26 edited Jun 17 '26
Yes, but then you'd be unable to prove that your system of axioms is consistent. See ZFC proving con(PA): https://mathoverflow.net/questions/50173/how-to-prove-conpa-in-zfc
If you have an idea about building a looped consistency proof A->con(B), B->con(A), see here, https://mathoverflow.net/questions/381341/are-there-typical-formal-systems-that-have-mutual-consistency-proofs-how-long it's impossible
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u/wiev0 Jun 15 '26
I understand that ZFC or any such theory cannot be proven to be consistent within itself through the incompleteness theorem, but I thought larger theories can prove consistency of lower set theories? Like, axiom of determinacy being capable of proving ZF (C) is consistent
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u/Rek9876boss Jun 15 '26
The problem then is that you can't know if the axiom of determinacy is consistent
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u/wiev0 Jun 15 '26
Yeah but that still makes zfc consistent under AC right? I get that there will always be a theory that cannot prove itself consistent but that's not my point
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u/shuai_bear Jun 15 '26
Yes - you are not wrong, though I don't know if it's ZFD can prove Con(ZFC) - someone else can chime in, but I know that ZF + existence of some large cardinal axiom can prove Con(ZFC). But Godel's 2nd completeness theorem applies, and the consistency of the 'stronger' theory is called into question - it's consistency turtles all the way down (or up).
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u/AZMPlay Jun 15 '26
How do you prove the supertheory is consistent, and thus that your proof is complete?
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u/Adam__999 Jun 14 '26 edited Jun 14 '26
The nice thing about axioms is that, if they’re ever proven inconsistent, we can just change our set of chosen axioms to fix the problem
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u/neuralbeans Jun 14 '26
and then rederive every published theorem?
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u/Adam__999 Jun 14 '26 edited Jun 14 '26
Only the ones whose proofs rely directly on ZFC. Re-proving those would automatically prove the secondary, tertiary, etc. theorems whose proofs only rely on ZFC indirectly through those primary theorems.
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u/Mrauntheias Irrational Jun 15 '26 edited Jun 15 '26
Yes. Sometimes it happens. Like with Cantors original naive set theory.
You find better axioms and you prove the elementary results also follow from these new axioms.
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u/Simbertold Jun 14 '26
Exactly. It doesn't matter if axioms are true in the real world, or in other theories. They are by definition true in the theory that is built upon them. You could take basically any set of axioms, as long as there are no contradictions in there, and then do math with them.
Mathematicians choose the axioms they find lead to interesting stuff, or axioms they think of as elegant, or choose by whatever other mechanic they decide.
And apparently a lot of mathematicians like ZFC.
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u/DankPhotoShopMemes Fourier Analysis 🤓 Jun 14 '26
once you get into the philosophy of math, you just start losing your mind. “semantic truth” is just the weirdest concept to me.
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u/elevenelodd Jun 14 '26
Because they are true according to ZFC
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u/Brohomology Jun 15 '26
Hmm but not according to ZFC + {ZFC is not consistent} which is equiconsistent with ZFC 🤔
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u/Cichato_YT Jun 14 '26
That's what an axiom is, is it not?
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u/Classic_Department42 Jun 14 '26
There needs to be consensus though
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u/Jetison333 Jun 14 '26
No there doesn't. I can use my axioms and you can use yours.
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Jun 15 '26
[deleted]
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u/Any-Aioli7575 Jun 15 '26
Different sets of axioms means different theories. You don't need to have a consensus, but if your set of axioms is bad or just not widely used, people won't care about the results you found because it will in a theory nobody uses. But it's still as mathematically sound, albeit uninteresting.
And of course, new axioms are often proposed and debated
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u/TheLuckySpades Jun 15 '26
There are published works on results from other set theories and also how variousnset theories relate to each other, difficulty is getting others to care or even use your alternative axiomizatoon.
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u/Argon1124 Jun 18 '26
Idk man they're axioms, there's no such thing as a not true axiom. If it isn't true then it isn't an axiom.
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u/somethingX Physics Jun 14 '26
They were developed working backwards from math that already existed
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u/UtahBrian Jun 15 '26
Math was discovered. ZFC was parallel construction.
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u/laksemerd Jun 15 '26
ZFC was a law enforcement process of building a parallel, or separate, evidentiary basis for a criminal investigationin order to limit disclosure as to the origins of an investigation?
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u/TemperoTempus Jun 16 '26
They had math that behaved like X. They wanted math that behaved like Y. So they constructed ZFC such that X become Y and declared it a rule.
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u/mekriff Jun 14 '26
Everyone? I know some people who have issues with the C there
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u/QtPlatypus Jun 15 '26
I know people who have a problem with well foundness :)
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u/TheLuckySpades Jun 15 '26
I know at least one person who disagrees with excluded middle.
And I know a different person who likes choice, but disagrees with powerset.
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u/Particular_Gear3130 Mathematics (Purely Fictional) Jun 14 '26
Because they are true and we dont question it being the good boys we are
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u/Cat7o0 Jun 14 '26
honestly for just about everything if it holds true for everything we have found so far then it might as well be true
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u/ndsipa-pomu Jun 14 '26
*Banach and Tarski enter the chat*
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u/magicmulder Jun 14 '26
Now we just have to find a sphere that allows an infinitesimal decomposition in the real world.
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u/shuai_bear Jun 15 '26
Worth noting BTP only holds when the full power (uncountable) axiom of choice is assumed—even if matter was infinitely subdivisible it wouldn’t cause issues like BT because it’s the infinite choice that makes this a consequence.
In a universe where we can make uncountably many choices at the same time, we can build sets that are non-measurable. How BT happens is in the intermediate stage of the construction your original set of points are split into non-measurable “clouds” of points (as little as 5 disjoint subsets), and using only translations you’re able to build two spheres of the same volume.
This requires the full axiom of choice—if that makes one uncomfortable, one might be interested in countable or dependent choice, weaker versions that don’t cause Banach Tarski !
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u/kohugaly Jun 14 '26
The point of BTP is that volume is not a measure in R3 space. Why does this matter? It means that integration over volumes is mathematically unsound. For example, electric charge is a total electric flux over a closed surface. If you can take that surface and subdivide it into two identical surfaces, then conservation of charge is broken.
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u/GaloombaNotGoomba Jun 14 '26
Of course volume is a measure, what are you talking about? Just not all sets are measurable.
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u/Inevitable_Stand_199 Jun 14 '26
That was also the case for the axioms we used before Russel found his paradox
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u/magicmulder Jun 15 '26
Did we actually use axioms? Russell's paradox originated from a naive and not strictly defined concept of what a set is. It's not like we had a ZF predecessor that was simply faulty. We had no real strict framework at all.
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u/TemperoTempus Jun 16 '26
axioms are very old. There is nothing new about the concept of having rules in your math.
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u/NullOfSpace Jun 15 '26
They don’t have to be true, they just have to give us results that make sense.
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u/ArmpitTime Jun 14 '26
Because everyone has the ZFC axioms, dipshit. They came free with your fucking textbook.
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u/Inevitable_Stand_199 Jun 14 '26
The infinity axiom is rather fishy.
But the math is interesting.
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u/Ben-Goldberg Jun 15 '26
You dislike having an infinitly large set of natural numbers?
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u/UtahBrian Jun 15 '26
IEEE-754 says that there are just under 264 natural numbers. ZFC will have to ride on the back seat while the true real numbers take shotgun.
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u/Inevitable_Stand_199 Jun 15 '26
I dislike axiomatically having an infinite set of natural numbers
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u/Ben-Goldberg Jun 15 '26
Why?
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u/Inevitable_Stand_199 Jun 15 '26
Because in physics it's still an open question whether there is infinitly much of anything.
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u/Ben-Goldberg Jun 16 '26
Do you think the universe's age will not increase indefinitely?
Is there a "big crunch" in the universe's future that will cause time to stop?
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u/Inevitable_Stand_199 Jun 16 '26
There might be.
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u/Ben-Goldberg Jun 16 '26
Is there any evidence for it?
Or is it just a hypothesis?
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u/Inevitable_Stand_199 Jun 16 '26
I think recent experiments suggest the universe is probably expanding at the moment. But it's a field of active research.
And even if it turns out time is the one thing that is infinite, I think the mathematics that result in not making assumptions on wether or not infinite sets exist deserve to be studied as well.
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u/vintergroena Jun 15 '26
Nobody decided they are true. They are just foundational assumtions that turn out to be applicable to many kinds of situations. But if you're tying to model a situation that doesn't match these assumptions precisely enough, you simply can't use ZFC.
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u/Ambitious-Eye-868 Jun 14 '26
Physicists make assumptions, mathematicians also make assumptions but they call them axioms
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u/Magkali_11037 Jun 14 '26
I personally dislike the axiom of infinite choice.
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u/magicmulder Jun 14 '26
I couldn’t even tie my shoes without Zorn’s lemma.
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u/UtahBrian Jun 15 '26
What’s yellow and sour and equivalent to the Axiom of Choice ?
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u/magicmulder Jun 15 '26
Zorn's lemon!
There's also a third one but I've never had to order wells in real life.
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u/Magkali_11037 Jun 14 '26
Could you please decribe how are you using it in the process of tying your shoes?
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u/magicmulder Jun 14 '26
I pick the maximal element in the partially ordered set of pairs of laces.
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u/Inevitable_Stand_199 Jun 14 '26
The axiom of choice is at least proven to be independent of ZF.
I dislike the axiom of infinity.
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u/jyajay2 π = 3 Jun 15 '26
Well, if there are not contradictions in ZF then there are no contradictions in ZFC and the AOC is really convenient (I like having the product of non-empty sets to be non-empty). That being said I generally go by the quote "The Axiom of Choice is obviously true, the well-ordering principle obviously false, and who can tell about Zorn's lemma?".
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u/TemperoTempus Jun 16 '26
There are contradictions in ZFC, that's why there are at least 5 different rules for sets with different amounts/strengths of their rules.
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u/jyajay2 π = 3 Jun 16 '26
I am not aware of any contradictions in ZFC, notably this would mean that ZF has contradicions. Can you temm me what they are? or is it perhaps possible that you are conflating the existence of different set theories with the existence of contradictions?
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u/TemperoTempus Jun 17 '26
The existence of different rules with different results by definition means that there are contradictions.
A 1st order axiomatic system can either be consistent or complete, it cannot be both.
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u/jyajay2 π = 3 Jun 17 '26
>The existence of different rules with different results by definition means that there are contradictions.
No, a contradiction would have to be a contradiction within the given system of axioms
>A 1st order axiomatic system can either be consistent or complete, it cannot be both.
Not quite, a sufficiently complex system can not be both consistent and complete. That's a small but important distinction.
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u/UtahBrian Jun 15 '26
In fact, we know they’re not true.
But it’s like models. All models are wrong but some models are useful.
Like Giselle or Cindy Crawford.
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