Timeline for Critical points and the Foundation Axiom
Current License: CC BY-SA 3.0
11 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jun 30, 2016 at 16:59 | history | edited | Thomas Benjamin | CC BY-SA 3.0 |
corrected spelling
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| Jun 30, 2016 at 7:48 | comment | added | Walter Bruce Sinclair | No need for apology. "Satisfied" is passive, "generated" is active. Shouldn't a more active role for a set of axioms be invited? Questioned, surely. Dismissed, possibly. I look at the Greek root for "generated" and I look at a universe and I am brought to cusp. If that universe is more than combinatorial, then I am forced to ask: how did those extra propositions arise? | |
| Jun 30, 2016 at 1:00 | vote | accept | Thomas Benjamin | ||
| Jun 29, 2016 at 19:11 | answer | added | Noah Schweber | timeline score: 4 | |
| Jun 29, 2016 at 16:41 | comment | added | Thomas Benjamin | @AndreasBlass: I really should have said 'satisfied by $V$'. By 'generated', I meant 'generated by the axioms and theorems in 'the' cumulative hierarchy'. Since one can use the term 'model' as 'universe', the axioms 'generate' a proper class of similar, but distinct models that satisfy a given first-order theory. Apologies. | |
| Jun 29, 2016 at 15:56 | comment | added | Andreas Blass | I think Asaf and Noah have answered your question: The usual proofs of these equivalences don't use choice. But your last comment raises another question: What do you mean when you say a universe is "generated" by some axioms? One possibility is that you mean a universe that is assumed to satisfy those axioms but is not further specified. Your use of the definite article ("the universe generated by ...") suggests that you have something else in mind, but I can't imagine what. | |
| Jun 29, 2016 at 15:01 | comment | added | Thomas Benjamin | @AsafKaragila: I wished to see if (4), (5) , and $FA$ could be proved equivalent in $ZF$ alone--I am not presuming Choice is needed (as Noah was kind enough to point out in his comment). I presume, then, that (5) can be proved equivalent to (4) and $FA$ in $ZF$ and in (5), $V$=$V_{ZF}$, $V_{ZF}$ being the universe generated by the axioms of $ZF$ (as opposed to the universe $V_{ZFC}$, which is the universe generated by the axioms of $ZFC$)? | |
| Jun 29, 2016 at 13:35 | comment | added | Noah Schweber | What is $V_{ZF}$? (I also second Asaf's question - I don't see where choice is being used in the standard argument that $(4)$ is equivalent to Foundation). | |
| Jun 29, 2016 at 13:13 | review | Close votes | |||
| Jul 3, 2016 at 14:20 | |||||
| Jun 29, 2016 at 12:54 | comment | added | Asaf Karagila♦ | Where do you think the axiom of choice is being used in the proof of equivalence between Foundations and the two statements you wrote? | |
| Jun 29, 2016 at 11:26 | history | asked | Thomas Benjamin | CC BY-SA 3.0 |