Timeline for What gets to be called a "proper class?"
Current License: CC BY-SA 4.0
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| when toggle format | what | by | license | comment | |
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| Nov 30 at 12:07 | comment | added | user3840170 | Another example to consider: Nelson’s Internal Set Theory. $\{x \in \mathbb R: \forall \varepsilon: \operatorname{st}(\varepsilon) \land \varepsilon > 0 \to |x| < \varepsilon\}$ is not a set by the axiom schema of comprehension, because the predicate used is not a ZFC formula (it contains the symbol $\operatorname{st}$ original to IST) and IST assumes comprehension only for pure ZFC formulæ. Yet it’s clearly not “too large” to be a set, since it’s contained in $\mathbb R$, which is a set. As I remember though, treatments of IST usually call it an “external set” rather than a “proper class”. | |
| Jun 26 at 18:27 | history | became hot network question | |||
| Jun 26 at 13:43 | history | edited | Daniel Asimov | CC BY-SA 4.0 |
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| Jun 26 at 11:27 | vote | accept | Mike Battaglia | ||
| Jun 26 at 11:23 | comment | added | Mike Battaglia | Thanks, I misread that. Still curious about global choice functions though, given the Wikipedia page. | |
| Jun 26 at 11:10 | answer | added | Joel David Hamkins | timeline score: 16 | |
| Jun 26 at 11:07 | comment | added | Emil Jeřábek | Definable with parameters. So all sets are trivially definable. | |
| Jun 26 at 11:02 | comment | added | Mike Battaglia | And then there are things like global choice functions, which Wikipedia refers to as proper classes (if they exist); such things are not definable unless $V = HOD$, I think... | |
| Jun 26 at 11:00 | comment | added | Mike Battaglia | Do people really use the term this way? So there are sets that aren't definable, but all classes are definable? So undefinable sets would not be classes then? | |
| Jun 26 at 10:53 | comment | added | Wojowu | At least in the ZF(C) context, the term "class" is commonly used for a collection of sets which is definable (with parameters). A proper class is any such definable collection which is itself not a set. Separation then asserts that any subclass of a set is itself a set. | |
| Jun 26 at 10:27 | history | asked | Mike Battaglia | CC BY-SA 4.0 |