bio | website | noldorin.com |
---|---|---|
location | London, United Kingdom | |
age | 24 | |
visits | member for | 4 years, 10 months |
seen | 2 days ago | |
stats | profile views | 245 |
entrepreneur; graduate in mathematics / theoretical computer science / theoretical physics; polymath-in-training
based in London, United Kingdom
May 25 |
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Why can't mathematics be formalised in terms of classes rather than sets?
@ToddTrimble: Sounds fair. I don't want it getting out of hand either. Best nip this in the bud while it's still pretty minor! |
May 25 |
accepted | Why can't mathematics be formalised in terms of classes rather than sets? |
May 25 |
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Why can't mathematics be formalised in terms of classes rather than sets?
@TheMaskedAvenger: Thank you for the advice. That sounds fair enough, so I'll try to target the chat room in the future! |
May 25 |
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Why can't mathematics be formalised in terms of classes rather than sets?
@AsafKaragila: Oh I see, fair point. Yes, you are probably right there: $\frak c$ classes can be constructed in 2nd-order surely, but above that... my intuition says not. Yes, sorry for the flooding Nik! |
May 25 |
awarded | Yearling |
May 25 |
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Why can't mathematics be formalised in terms of classes rather than sets?
@FrançoisG.Dorais: Sounds nasty, yes! |
May 25 |
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Why can't mathematics be formalised in terms of classes rather than sets?
@AsafKaragila: I'm afraid I'm not familiar with the notation $\frak c$. But if you mean "sets" of cardinality greater than $\aleph_1$, you be right in that that may require higher than 2nd-order. |
May 25 |
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Why can't mathematics be formalised in terms of classes rather than sets?
@AsafKaragila: Well, you can surely operate on the level of classes in such a system, since it's second-order. After all, real numbers must ultimately be constructed as hereditary sets, within ZFC. |
May 25 |
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Why can't mathematics be formalised in terms of classes rather than sets?
Okay, then do please tell me how model theory is useful beyond this perspective? As far as I'm concerned, it has bootstrapping properties otherwise. |
May 25 |
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Why can't mathematics be formalised in terms of classes rather than sets?
@AsafKaragila: Well, I think in practice it actually makes little difference, since certain classes would take the place of sets. For me, model theory just happens to be the traditional way of dealing with the semantics of formal mathematics; there are plenty of other equally good ones (and better from certain points of view). I mean to say, model theory is essentially just an algebraic way of understanding logic and foundational mathematics. Algebraists and most mathematicians love it for that reason, I feel. |
May 25 |
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Why can't mathematics be formalised in terms of classes rather than sets?
Does anyone who made a close vote here want to actually justify it? Clearly some people have understood the question just fine... |
May 25 |
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Why can't mathematics be formalised in terms of classes rather than sets?
Thanks for clarifying that. I've up-voted both your and Nik Weaver's answer, because they have both proven enlightening and relevant, though I'll wait a bit longer before deciding which to accept. |
May 25 |
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Why can't mathematics be formalised in terms of classes rather than sets?
Fascinating. So ACA is an impredicative second-order system over the universe of naturals, and it suffices for virtually all mathematics? Why on Earth is it not more mainstream then, I wonder? |
May 25 |
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Why can't mathematics be formalised in terms of classes rather than sets?
I see. I would presume the axiomatic system of GBC resembles that of ZFC in many ways then? My own inclinations are finitary, so either a classically or strictly finitary system would be of more interest to me here, not that this isn't! |
May 25 |
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Why can't mathematics be formalised in terms of classes rather than sets?
Aha. Now, it's equivalent to ZF in expressive power? |
May 25 |
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Why can't mathematics be formalised in terms of classes rather than sets?
Yep, this is very much what I'm interested in. Thanks Nik. |
May 25 |
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Why can't mathematics be formalised in terms of classes rather than sets?
Is this the same system referred to be en.wikipedia.org/wiki/…? If so, it seems to be discussing the two-sorted version. But apparently there's a logically equivalent one-sorted version that only deals with classes? |
May 25 |
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Why can't mathematics be formalised in terms of classes rather than sets?
Intriguing! Thank you for this answer. This is very much what I was hoping for. Do these formalisations avoid set theory altogether then, I take it? |
May 25 |
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Why can't mathematics be formalised in terms of classes rather than sets?
You misunderstand though, @JohannesHahn. My point is: can we eliminate the seemingly arbitrary ontology of sets altogether? |
May 25 |
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Why can't mathematics be formalised in terms of classes rather than sets?
@AndreasBlass: Well, I left this aspect open somewhat on purpose. But let's be more specific, for the sake of argument. First, we could envisage a universe of discourse being something like the naturals (recursively defined), with only first-order classes. Alternatively, we could work in some higher-order logic, thus allowing classes to be members of other classes, in a stratified manner. |