Timeline for Which classes are sets?
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9 events
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| Aug 6, 2010 at 19:15 | comment | added | Peter LeFanu Lumsdaine | Regarding your query on NF: surely it is indeed "only if", in a rather trivial way, since for any set $x$, the formula $y \in x$ is itself stratified! (And while this is an almost stupidly trivial point, I can't think of a way of restricting the allowed definitions that would rule this sort of thing out while remaining plausibly sufficient; in particular, adding "…without parameters" makes it certainly insufficient, for cardinality reasons.) | |
| Aug 6, 2010 at 19:02 | comment | added | Andreas Blass |
@Harry: I won't try to explain Bourbaki's notion of set, partly because I don't remember those axioms precisely enough, and partly because Bourbaki's attitude seems to be far more formalistic than what is presupposed in the question. I believe, though, that Bourbaki's set theory can be viewed as describing a notion of set similar to that of ZF but with (here's the part I don't remember well enough) some restriction on the height of the cumulative hierarchy (perhaps just height $\omega+\omega$).
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| Aug 6, 2010 at 18:59 | comment | added | Andreas Blass |
It's true that neither NF nor ZF involves classes, but the original question presupposed them. In both cases, to make sense of the question, one should interpret classes as "virtual" in Quine's sense, i.e., as informal collections $\{x:\phi(x)\}$ of sets satisfying some property $\phi$. And in both cases, the axioms are usually formulated as saying that certain sets exist rather than that certain classes are sets. In both cases, there are extensions incorporating classes explicitly into the theory. Ackermann's theory, in contrast, uses classes from the start.
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| Aug 6, 2010 at 16:29 | comment | added | Jeremy Shipley | Maybe in ML Quine introduces proper classes corresponding to non-stratifiable formulas. | |
| Aug 6, 2010 at 16:27 | comment | added | Jeremy Shipley | I.e., the stratifiability condition in NF isn't a condition on whether a class is a set, but on whether a set exists corresponding to a given wff in the language of NF. | |
| Aug 6, 2010 at 16:23 | comment | added | Jeremy Shipley | I wouldn't say that Quine's approach in NF is "similar", though there's ambiguity in that term. Using a direct restriction on the comprehension principle to dodge the paradox is a fundamentally different approach from the limitation of size approach. Also, I hadn't thought that NF even relies on a set/class distinction in the first place. Does it? | |
| Aug 6, 2010 at 16:10 | comment | added | Jason Dyer | While I appreciate this answer (incidentally, Smullyan and Fitting's book uses NBG) the above quote suggests some think there is an overarching principle that connects all possible definitions. Even if it's wrong as posited, I'm curious what that principle might be. | |
| Aug 6, 2010 at 15:47 | comment | added | Harry Gindi | How does Bourbaki's definition of a set using coll_x and the tau operator compare? | |
| Aug 6, 2010 at 15:37 | history | answered | Andreas Blass | CC BY-SA 2.5 |