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Comment by zaptheimpaler | Hacker News Reader

This is crazy. How about just doing what maths does anyways - state your assumptions?

Prove stuff with V=ultimate L? Great, slap a "V = ultimate L => ..." on it.

Prove stuff with forcing axiom? Cool, "forcing axiom implies ..."

The entire premise of asking "whats the one true axiom?" makes no sense. Its not like the real world places any restrictions on math - its a self-contained universe. So obviously you are free to pick whatever imaginary universe you would like to study. If we ever get to a point where we can directly test which of the 2 axioms leads to more useful theories in the real world, great, but that seems pretty unlikely given they come up when talking about infinity. The obvious truth seems to be that infinity could have a few subtly different meanings depending on how you define it - and thats totally fine because the real world does not contain infinities, its a purely mental construct.

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The article frames it as some sort of debate in some sort of mathematical ruling body which doesn't exist. There is no authority that's going to decide that, "ok, V = ultimate L is the axiom we're going with, everyone has to accept it now."

If you're assuming ZFC, you typically don't say anything (although I personally like to explicitly point out if I need the axiom of choice.) There was no ruling body that decided, "ok, we're all using ZFC now." Earlier generations of mathematicians widely adopted it, and now it's just the default.

Fun fact! Some recently popular mathematics leaking into the realm of CS, category theory, is usually not phrased in the language of ZFC. Instead, category theory is easier to formulate in NBG set theory, which is equivalent to ZFC in the sense that a ZFC statement is provable in NBG if and only if it is provable in ZFC.

The advantages afforded by NBG are:

1. the ontology is bigger. NBG has classes and sets, ZFC just has sets, and

2. it is finitely axiomatizable!

Unlike what the linked article says, ZFC does NOT have finitely many axioms! ZFC has 7 axioms and 2 axiom schemas, which are sort of meta-axioms that each generate countably infinite axioms.

On the other hand, ZFC is a bit more explicit and hands on. It's easier to roll up your sleeves and just get elbows deep in sets, which is why it's favored more by actual set theorists and set-theoretic topologists.

Can you unpack 2 for me? How can there only be finitely many axioms if NBG is 'equivalent' in proving power to ZFC?

Unfortunately I'm not a set theorist, so I haven't done any deep dives into the NBG/ZFC equivalence. My area of expertise (Lie theory) doesn't really require detailed knowledge along those lines. I'm only really aware of it because I'm a category theory enthusiast.

That said, my intuition is that NBG is finitely axiomatizeable with equivalent strength precisely because of #1: it has a bigger ontology.

From what I remember of grad school set theory, working with proper classes in ZFC amounts to clunky manipulations of formulas in the language of set theory, because there is no such thing as a class in ZFC. A proper class is a well-formed formula that fails to describe a set. It can be described in the language, but cannot be constructed by the axioms. Axiom schemas give one axiom for all formulas in the language of ZFC.

Meanwhile, classes are an actual object in the language of NBG, so an axiom of NBG can just say "for all classes x..."

There's a wikipedia article[1] that actually goes into a good bit of detail in the case of the axiom schema of specification.

[1] https://en.wikipedia.org/wiki/Axiom_schema_of_specification

In most fields of mathematics, it is not necessary to state that ZFC is assumed. Axiom of choice is singled out sometimes, but the other axioms are considered so self-evident that it's not even worth mentioning. Yet from a logical standpoint, there's nothing special about ZFC to be so universally adopted. We could've had a completely different set of axioms and completely different mathematical concepts as a result. The reason why ZFC is chosen is because it results in concepts that are applicable in the real world. If a new independent axiom could give a rise to some useful concept applicable across many different mathematical fields, eventually ZFC+X might be adopted as the universal axiom system and it wouldn't be necessary to write "assuming X" in every proof.

> other axioms are considered so self-evident that it's not even worth mentioning.

Maybe "considered so common" would be a more precise wording.

I'd also argue that the design space within finite math is reasonably limited. I have never seen an interesting alternative axiomization. For infinite math there is of course much more room.

Yes, you're right, this entire fight is only about which avenues of research are fashionable. For the vast majority of mathematicians, this does not affect their work, and for the mathematicians whose work is directly affected, "answering" this question is like biologists answering the question, "Lions or tigers?" Some people's research will become more fashionable and other people's research will become less fashionable. No answer to this question will be of any more consequence than the philosophical conclusions that a number cannot represent zero quantity, negative quantities are nonsensical, parallel lines never meet, etc.

“Woodin showed that if you can just reach the level of the supercompacts, then there’s an overflow and your inner model picks up all the bigger large cardinals as well,” Koellner explained. “That was a sort of landscape shift. It provided this new hope that this approach can work. All you have to do is hit one supercompact and then you’ve got it all.”

This is the mathematical equivalent of "640K ought to be enough for anybody." Somebody's going to want something more, or something different, and they're going to do what they want. In the long run, mathematicians only care what assumptions generate neat and/or useful mathematics. In the short run this might affect young researchers' employment prospects and might influence them to choose different problems for that reason.

> Its not like the real world places any restrictions on math

There has been an ongoing debate about whether that statement is true for the last couple of millennia. Many people are arguing that mathematics were not developed out of the blue, but to describe the physical world around us.

I would agree that math an the real world are related, but not that the physical world should influence math. If you discover an interesting and self consistent system, than it seems that eventually there is a good chance you will find a connection with physics. But you don't need to let perceptions about the physical world influence the systems you look for.

> and thats totally fine because the real world does not contain infinities

Anyone who positively claims that infinities do or do not exist in nature don’t know what they’re talking about.

Physics is full of infinities. They show up all the time in relativity (gravitational singularities) and quantum physics (renormalization, rigged Hilbert spaces with integration over infinite distributions).

For some of these infinities (like the ones we have to deal with in quantum mechanics), it’s unclear (to me, at least) whether or not they have any sort of “real” equivalent. For things like gravitational singularities, our current theories predict the physical existence of infinite quantities (density, gravitational field, etc.).

Infinities show up in currently used models to describe physical processes. Is there an actual infinite quantity in nature? I’m a mathematician and not a physicist but it always seemed to me that the answer is no.

Wouldn't you need infinite precision to test empirically?

But it's also very obvious that general relativity can only be an effective theory that fails for very strong fields. No one I know actually thinks there are singularities, those just point to places where GR starts to significantly deviate from reality. Navier–Stokes equations are very accurate on macroscopic scale, but regardless of their mathematical properties, there are no actual blow ups in real world, because ultimately, the liquid is composed from atoms. Likewise, the fields in GR must be ultimately quantized in some way, which will almost certainly break the singularities.

The infinities in QFT are really just a hack. I mean, all those calculations are perturbational in the first place, so they can be hardly considered a "true" picture of reality.

Precisely stating your assumptions happens in a limited way in everyday mathematics, enough so that the project of finding the minimal set of assumptions for mathematical theories is a niche project: https://en.wikipedia.org/wiki/Reverse_mathematics. By everyday mathematics, I mean mathematics outside of set theory/logic/ foundations of mathematics.

Rather, what everyday mathematicians do is have a set of axioms that are "safe", and do not need discussion, along with other axioms that require explicit note. They then prove things relative to those axioms, make a limited effort to only use the "unsafe" axioms, and make no effort to restrict their use of safe axioms.

Mathematicians will typically describe their philosophy of mathematics as formalist, but it's arguable that everyday mathematicians act much more like Platonists with respect to ZFC.

Amen. Not a mathematician, but this perfectly articulates my thoughts on the subject.

>the real world does not contain infinities

I have no idea what "real world" even means, but I'd say that infinity does not exist within anything that can be conceptualized.

> infinity does not exist within anything that can be conceptualized.

Yet math is consistently being conceptualized?

Enabling clear thinking about concepts outside of the real world is the while point of math.

> math is consistently being conceptualized

What does this mean?

>Yet math is consistently being conceptualized?

Sure, how else could we capture it or convey it to others? The thing is that I don't see it as that different than conceptualizing things in the "real world". In the latter, conceptualization is done on input of the senses, while in math conceptualization is done on inputs from a higher level of abstraction. In both cases, something is defined and then manipulated for something useful. It's just that I believe that whatever is defined automatically becomes a finite concept otherwise it could not be grasped and manipulated, and a finite infinity seems problematic...

When your hand moves, does it pass through an infinite or a finite number of positions at a finite or infinite points in time? Infinity need not be big.

Could one argue that the Planck length/time means the number of points in space/time in your example is finite?

Why does the continuum need to be made of points? Can you really show me a particular point in space?

Quantum mechanics has several facets which correspond to path integrals, infinite-dimensional operators, and infinite series, sums, and products.

True - you don't even need to venture into quantum mechanics, the concept of infinity has been useful in practical science far before anyone dreamed of quantum mechanics - but there's a difference between a definition of what happens to a formula when a limit approaches infinity and claiming to somehow understand infinity as a concept. I mean when you deal with infinite-dimensional operators, infinite-series etc. does it matter what ordinal of infinity is used?

If you are going to play with the "real world", you are going to have to deal with continuous math; i.e. the real numbers at least. At which point you get to deal with infinity somehow.

I believe that potential infinity (i.e. the idea of intervals becoming ever smaller and smaller) will be sufficient to deal with the "real world". But the infinities described in the article are of a different kind.

I don't think there will ever be a situation like: "According to the axiom of choice, this photon should go to the left, but according to the axiom of something else, this photon should go to the right." Or: "If there is such a cardinal X that mumbo-jumbo(X) is true, then the photon should go to the left, otherwise it should go to the right."

Therefore, this part of math is just playing with words. An interesting game, especially if you can get paid for it, and it can provide indirect benefits such as greater insight into how to correctly play with words, but ultimately just a game with no correspondence to the "real world".

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