Timeline for Circular, or missing, definition in set theory?
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4 events
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| May 24, 2018 at 19:45 | comment | added | Timothy Chow | @FrankQuinn : The standard way to handle classes in ZFC is to define them to be formulas with one free variable. But you're changing the question. The original question was about circularity. If your real question is about the usability of an implementation, then of course the answer is going to be different. | |
| May 24, 2018 at 17:09 | comment | added | Carl Mummert | It's trivial to use class-based theories, such as Morse-Kelley set theory, where there are both sets and proper classes - the collection of all "sets" in a model of MK is itself an object in the domain of that model. One thing that the last 100+ years of set theory have taught us is that there are many concrete questions about the "collection of all sets" that are not resolved by any generally accepted axioms - for example the existence of various kinds of large cardinals, the continuum hypothesis, etc. Arguably, this shows we do not have a completely concrete notion of "set" to refer to. | |
| May 24, 2018 at 15:33 | comment | added | Frank Quinn | I'm a working mathematician, so am concerned with usable implementations rather than the metatheory. You say "everything is a set". In an implementation the collection of all sets is not a set, but it should be a thing of some kind. Just because it doesn't have the properties required of sets doesn't mean it can't make sense. Or to put it another way, if we cannot make concrete sense of "the collection of all sets" then we don't have a practically useful implementation. | |
| May 22, 2018 at 2:46 | history | answered | Timothy Chow | CC BY-SA 4.0 |