Timeline for ZF plus class-choice?
Current License: CC BY-SA 4.0
15 events
| when toggle format | what | by | license | comment | |
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| Oct 24, 2021 at 20:17 | comment | added | Frode Alfson Bjørdal | @Zuhair Al-Johar Yes, what I stated was slightly inaccurate. The best is to take all objects to be classes, and sets are those classes which are members in other classes: i. e. Set(x) <-> Ey(x\in y) | |
| Oct 24, 2021 at 17:20 | comment | added | Zuhair Al-Johar | @FrodeAlfsonBjørdal, you said that Cls(x) is short for "not-Set(x)". But this is NOT the standard definition, the standard is to take every object as a class, and only Proper classes as "non-Sets" | |
| Oct 23, 2021 at 20:54 | comment | added | Frode Alfson Bjørdal | Zuhair Al-Johar No, the definitions are as I stated. My terminology is standard, for class-extensions. If I remember correctly, Gödel uses an equivalent notation in his Choice/Continuum paper. | |
| Oct 23, 2021 at 17:33 | comment | added | Zuhair Al-Johar | @FrodeAlfsonBjørdal, if ML is consistent then it has a countable model, and clearly this is externally well orderable, and so external (Class) choice holds, but of course the set world of ML is NF where choice is negated (Specker). | |
| Oct 23, 2021 at 16:49 | comment | added | Zuhair Al-Johar | I thought Cls(x) is short for x=x; and PropCls is short for "not-Set(x)" | |
| Oct 23, 2021 at 13:09 | comment | added | Frode Alfson Bjørdal | @Zuhair “N otice that there is no known inconsistency with Quines ML + Class choice over sets, but of course it's inconsistent with Set choice.” May you point to literature? | |
| Oct 23, 2021 at 13:06 | comment | added | Frode Alfson Bjørdal | @Zuhair My exposition perhaps simplifed too much. As is usual, Set(x) is short for Ey( x\in y). Cls(x) is short for not-Set(x). The axiom (x)(Cls(x)) is presupposed. | |
| Oct 23, 2021 at 8:45 | comment | added | Zuhair Al-Johar | I think (not sure) that Randall Holmes's purported proof of Con(NF) uses a trick where choice is in some sense external. Notice that there is no known inconsistency with Quines ML + Class choice over sets, but of course it's inconsistent with Set choice | |
| Oct 23, 2021 at 7:51 | comment | added | Zuhair Al-Johar | @FarmerS, the exposition is incomplete, for example Extensinaity is not an axiom and also all axioms of ZF are relativized to the predicate $Set$, I think by using class and set predicates he actuay means a mono-sorted FIRST order language extended with two primitive one place predicate symbols $Cls; Set$ denoting "is a class" and "is a set" respectively. The question actually needs to be re-written by the author to clarify his intentions. | |
| Oct 23, 2021 at 5:49 | comment | added | Farmer S | I presume you mean class variables, not class predicates, since you quantify over $Y$. Isn't the theory just equivalent to ZFC? (You haven't really spelled out the axioms of your theory regarding classes. But I expect the axioms to be arranged so that you can use Separation to separate a choice set from the choice class for $s$, and in the other direction, any choice set for $s$ should give you a choice class for $s$.) | |
| Oct 22, 2021 at 19:08 | history | edited | Frode Alfson Bjørdal | CC BY-SA 4.0 |
added 12 characters in body
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| Oct 22, 2021 at 19:08 | comment | added | Frode Alfson Bjørdal | @ZuhairAl-Johar You are right. Thanks! | |
| Oct 21, 2021 at 16:49 | history | edited | YCor | CC BY-SA 4.0 |
formatting
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| Oct 21, 2021 at 15:34 | history | edited | Frode Alfson Bjørdal | CC BY-SA 4.0 |
edited title
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| Oct 21, 2021 at 15:20 | history | asked | Frode Alfson Bjørdal | CC BY-SA 4.0 |