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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, "o
by VyseofArcadia 9y ago
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.
- cliffy 9y agoCan you unpack 2 for me? How can there only be finitely many axioms if NBG is 'equivalent' in proving power to ZFC?
- VyseofArcadia 9y agoUnfortunately 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 https://en.wikipedia.org/wiki/Axiom_schema_of_specification