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You out yourself by using the phrase "Reification of Zero," which is something I've only ever heard from Objectivists. It's funny to hear you say this, since O

by jfarmer 13y ago

You out yourself by using the phrase "Reification of Zero," which is something I've only ever heard from Objectivists. It's funny to hear you say this, since Objectivists are usually so apoplectic about axioms and premises. You can object to ZF(C) as the basis for set theory, but I don't see how one can call it "wrong." If you want to build up a theory without the Axiom of Infinity you're free to do so — many mathematicians have. Either Cantor's arguments follow from ZF(C) or they don't (hint: they do). It's not a philosophical problem at all. One can certainly explore what it would mean to do mathematics without access to "infinity", as many mathematicians in the first half of the 20th century did. IMO this conversation is all air until you propose your own precise set of axioms.

dmfdmf 13y ago
> It's not a philosophical problem at all.... Well, that was my main point. You are free to disagree. > IMO this conversation is all air until you propose your own precise set of axioms. Apoplectically, the three basic axioms are Existence, Identity and Consciousness. Whatever the proper mathematical axioms (and methods) are they have to be consistent with these three. Cantor's method treats infinity as a real thing, which violates the axiom of identity. You could even call it the Reification of Infinity but the error is easier to see in Rand's identification of Reification of Zero by the mystics who attempt to treat nothingness as real.
- jfarmer 13y ago
  The three basic (metaphysical) axioms according to Objectivists are Existence, Identity and Consciousness. Anyhow, as far as I can tell, all you're saying is that either (1) Cantor's argument is invalid within ZF(C) or (2) you find ZF(C) objectionable on the grounds that it conflicts with the basis of your metaphysics. That's fine — I don't care whether it does or not — but mathematics has no opinion on the matter. You're free to choose a set of mathematical axioms which you don't find objectionable and go about doing your math in that universe. If you want to claim Cantor's argument is invalid then you need to tell me what axioms you're taking as the foundation for your mathematics. It is valid in ZF(C) and I'm 99% sure it's valid in plain ZF. Which of the axioms of ZF do you find objectionable and what would you replace them with? Assuming you're trying to engage sincerely, I'd ask you to be precise. This means listing whatever foundations of mathematics you don't find objectionable and talking about how they differ from ZF(C). If you're unfamiliar with it, I'd read up on proof theory: http://en.wikipedia.org/wiki/Proof_theory http://en.wikipedia.org/wiki/Proof_theory

by jfarmer 13y ago

dmfdmf 13y ago

> It's not a philosophical problem at all.... Well, that was my main point. You are free to disagree. > IMO this conversation is all air until you propose your own precise set of axioms. Apoplectically, the three basic axioms are Existence, Identity and Consciousness. Whatever the proper mathematical axioms (and methods) are they have to be consistent with these three. Cantor's method treats infinity as a real thing, which violates the axiom of identity. You could even call it the Reification of Infinity but the error is easier to see in Rand's identification of Reification of Zero by the mystics who attempt to treat nothingness as real.

jfarmer 13y ago
The three basic (metaphysical) axioms according to Objectivists are Existence, Identity and Consciousness. Anyhow, as far as I can tell, all you're saying is that either (1) Cantor's argument is invalid within ZF(C) or (2) you find ZF(C) objectionable on the grounds that it conflicts with the basis of your metaphysics. That's fine — I don't care whether it does or not — but mathematics has no opinion on the matter. You're free to choose a set of mathematical axioms which you don't find objectionable and go about doing your math in that universe. If you want to claim Cantor's argument is invalid then you need to tell me what axioms you're taking as the foundation for your mathematics. It is valid in ZF(C) and I'm 99% sure it's valid in plain ZF. Which of the axioms of ZF do you find objectionable and what would you replace them with? Assuming you're trying to engage sincerely, I'd ask you to be precise. This means listing whatever foundations of mathematics you don't find objectionable and talking about how they differ from ZF(C). If you're unfamiliar with it, I'd read up on proof theory: http://en.wikipedia.org/wiki/Proof_theory http://en.wikipedia.org/wiki/Proof_theory