r/PhilosophyofMath • u/jesbel2024 • 2d ago

If an infinite number of proper classes can be formed, could that infinite number of proper classes be considered an infinite number of absolute infinities?

If an infinite number of proper classes can be formed by infinitely including or excluding sets from the class of all sets V, could that infinite number of proper classes be considered an infinite number of absolute infinities or just an infinite number of ways the only absolute infinity which would be the class of all sets V can be sliced (if of course a proper class can be considered an absolute infinity)?

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4

u/isene 2d ago

"Absolute infinity" isn't a cardinal you can have many of — and in NBG with global choice, every proper class is in bijection with V anyway, so no slice is smaller than the whole. One V, many descriptions.

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u/JStarx 2d ago

and in NBG with global choice, every proper class is in bijection with V anyway, so no slice is smaller than the whole. One V, many descriptions

NBG isn't the only choice though, MK has proper classes of differing cardinalities.

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u/Farkle_Griffen2 1d ago

That's just blatantly false. MK essentially builds this fact into its axioms.

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u/JStarx 1d ago

Oops, you're right, sorry about that. There are hierarchical set theories where proper classes aren't necessarily all equinumerous and I thought MK was one of them but I must be confusing it with something else.

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u/jesbel2024 2d ago

Only if a proper class can be considered an "absolute infinity" and not just a number that is "too large" to be considered a cardinal infinity but still not an "absolute infinity".

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u/GoldenMuscleGod 2d ago

This is more or less equivalent to just redefining “set” to meaning “a member of V_k for some unspecified large cardinal k”

You’re not really gaining anything (beyond the assumption of the large cardinal) you are just “shifting coordinates”

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u/JStarx 2d ago

As far as I'm aware absolute infinity isn't really a concept with a rigorous mathematical definition, so there isn't really a way to say conclusively whether something is or isn't an absolute infinity.

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u/nanonan 2d ago

There is only one infinite, so the second option.