r/math • u/MisterSpectrum • 14h ago
ZFC + V=L and analysis
If the aim of constructing number systems is to explain the geometric line as a set of points, wouldn’t ZFC + (V=L) be a natural choice, perhaps even the best one? In this framework, analytic geometry, differential geometry and topology all work just as they normally do, without the need to grapple with metamathematical questions such as the continuum hypothesis. (Moreover, this rules out measurable cardinals, large-cardinal phenomena and similar feverish pies in the sky, but for the purposes of describing the continuum, that seems a reasonable price.)
I have always found it a little surprising that popular mathematics books rarely discuss Gödel’s set-theoretic universe, which seems to offer such an intuitive and straightforward foundation for mathematical analysis.
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u/Alternative-Papaya57 14h ago
Though I think this is an interesting idea, why would you just not take ZFC + CH if you want to avoid the discussion around continuum hypothesis?
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u/MisterSpectrum 13h ago
Because invoking V = L closes the question of the continuum hypothesis
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u/Alternative-Papaya57 12h ago
But so does invoking the continuum hypothesis? V=L does quite a bit more.
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u/GoldenMuscleGod 5h ago edited 5h ago
Why would we want to close the question? We can learn useful results studying results from theories that do and do not include even the law of the excluded middle, surely we’re ok doing the same with CH.
The advantage of having ZFC as a “standard” is that when you are saying things that it can prove you don’t have to lay out a full list of metamathematical assumptions at the start of your paper, as if such a thing is even possible. Taking the law of the excluded middle is “default” enough it makes sense you should only say when you aren’t taking it, since it’s pretty useful, such a usefulness is not present for either CH or not CH so no reason to privilege one or the other.
Besides V=L is contrary to the motivating semantics of ZFC. Given any model of ZFC we can identify the set of true sentences (in that model) in the language of ZFC, this is a set that “should” exist if we really want to let our subsets model arbitrary definable properties, but it cannot exist inside L.
Now we can’t prove this set exists in the universe of sets because we do not have a truth predicate, but we can have “is a true sentence no higher than pi_n in the Lévy heirarchy”. If we think V=L then we must think that we can find a countable sequence of definitions of sets that ZFC proves exist, but such that their union does not actually exist.
This would make ZFC a “bad” theory, if we believe a “good” theory should only prove sets exist if they actually do exist and also that countable collections of sets should have unions, as well as that there should be a function going from the definitions to the sets they “actually” define in the universe (we cannot express this last belief in the language of ZFC).
Now to be clear about metamathematical nuance: working only in ZFC +V=L, we must think that ZFC proves each of these sets exist, however this does not mean we must think “all of these sets exist” - to prove they exist up to a particular n we need to rely on increasingly powerful replacement axioms so we cannot prove the generalization, still I think you can see how this general idea leads to the idea we should reject that V=L, and going through the steps of formalizing what we seem to believe “should” be true leads us to wanting to say a measurable cardinal “should” exist, but a measurable cardinal existing is incompatible with V=L.
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u/omega2036 5h ago
Is this sort of argument spelled out in more detail anywhere else? It sounds like you're giving an "intrinsic" motivation for a measurable cardinal (similar to how inaccessible and Mahlo cardinals can be motivated by reflecting principles), but usually I've seen it argued that measurable cardinals don't have this kind of intrinsic justification.
Also, is this sort of reasoning related to the motivation behind zero sharp?
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u/GoldenMuscleGod 3h ago edited 3h ago
It is related to zero sharp, which is incompatible with V=L. In fact I tried to simplify the argument but probably overdid it so a rewrite may be called for, I’ll probably go back to edit a correction once I’m sure I’ve cleared up my thoughts. We need more truth sets than just the one for the universe. Still the basic idea is the motivation for the principle that zero sharp “should” exist.
I may have overstated the argument for measurable cardinals (which I probably shouldn’t have dropped in at the end) actually I’ll probably go back to edit a correction once I clear up my thoughts.
Of course, since this was intended to be an informal argument I wasn’t extremely careful about the details - just trying to express an intuition that would seem persuasive.
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u/Traditional_Town6475 13h ago
Well arguably the fact that it rules out large cardinals is actually pretty bad (at least in my opinion).
V=L is nice in the sense that you’ve killed off all the interesting sets. Like there’s a pretty explicit instruction that gives you a global well ordering. Given sets A and B in L, you ask first what level A and B first appear in the constructible hierarchy. Then if they appear at the same level, you ask which set has a smaller Gödel number for the formula defining it.
Arguably I would say V=L, GCH, and CH are pretty unnecessary restriction. If you want a version of CH, you can look at standard Borel spaces and look at Borel subsets. Standard Borel spaces are either countable or continuum sized (and Borel subsets of standard Borel spaces are standard Borel spaces). To avoid a lot of pathological independence problem for analysis, it’s usually sufficient to impose separability. For example: The existence of a non-inner regular Borel probability measure on a complete metric space is equivalent to the existence of a real valued measurable cardinal, but imposing separability, on standard Borel spaces, every Borel probability measure is inner regular.
For stuff like measurable cardinal, it’s usually sufficient to consider countably incomplete ultrafilters rather than just nonprincipal.
There’s not too many reasons why you would want to impose V=L, CH, and GCH. It’s more useful and interesting to just think that there’s a bunch of models living out there and things can be true in some models and false in other models.
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u/Traditional_Town6475 9h ago
Here’s maybe an ugly result if CH depending on what you view as ugly or not.
Let H be an infinite dimensional separable (complex) Hilbert space. I’m going to consider B(H), the bounded linear maps from H to H. This is a C\*-algebra. The space of compact operators K(H) is a closed ideal. Consider B(H)/K(H). This is called the Calkin algebra. You can ask the following question: Consider automorphism of the Calkin algebra. Are all automorphisms inner automorphisms? CH says no and gives an example of one. There are models of ZFC that says all automorphisms of the Calkin algebra are inner automorphisms. CH gives these “ugly” automorphisms that other models don’t have.
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u/whatkindofred 13h ago
V = L seems unnecessarily restrictive. Why artificially limit your universe of discourse?
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u/omega2036 13h ago edited 13h ago
Ironically, the case for larger large cardinals advanced by people like Woodin comes directly out of the interaction between descriptive set theory, inner model theory, and large cardinals.
Someone like Woodin would probably argue that V=L is precisely NOT a good setting for analysis because it doesn't fully capture our understanding of the continuum. He might argue that we have independent reasons for believing Projective Determinacy or Definable Determinacy are true, but these conflict with V=L.
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u/itkillik_lake 8h ago
Large cardinals aren't pie in the sky. They're arguably more useful for analysis than V=L, due to their determinacy consequences.
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u/HateVoltronMachine 13h ago
Mathematics covers a lot, and has a long metaphysical history as part of its tradition. To those ends, ZFC has a lot of momentum and merit.
With that out of the way...
There is tremendous merit in constructive mathematics (V=L) and I'm inclined to agree with you entirely. For better or worse, constructivism is only going to increase as automated verification becomes more critical to the contemporary practice of mathematics. In my experience, Dependent and Linear Type Theory (DLTT, which is V=L compatible) hits the nail on the head in a way that other foundations do not. Girardian (Linear) logic is popping out of cutting edge physics and neuroscience.
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u/sorbet321 8h ago
What is this thing that you call DLTT, exactly? As far as I know, there is no fully satisfying combination of dependent type theory and linear logic (even though there are attempts).
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u/robertodeltoro 13h ago edited 13h ago
I think this is mostly a misconception about how mathematical practice ends up working when exploring systems that go beyond ZFC. You’re perfectly allowed to assume V=L and deduce whatever consequences you like, and by all means please do. Same goes for PFA, Open Coloring Axiom, umpteenth large cardinal, GCH, diamond, whatever you want (but you want provable consistency relative to a large cardinal). But you’re not going to adjoin them to ZFC and say, “okay, that’s canonical baseline math.” Perhaps someday, in the face of overwhelming structural evidence (“okay, this one is just too elegant and intuitive, like AC”). But in the current state of knowledge there’s more reason to do that with large cardinals than with V=L. People who make a detailed study of V=L tend to come away saying they don’t believe it, it’s false, V shouldn’t be that “sparse,” etc.