Timeline for When does collection imply replacement?
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16 events
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| Jan 8, 2021 at 11:40 | comment | added | Elliot Glazer | That being said, the inductive argument you mention in the end seems fine, so I think the theorem still holds. | |
| Jan 8, 2021 at 11:34 | comment | added | Elliot Glazer | The proof of your last theorem doesn't seem right. Even $\text{ZFC}^-$ doesn't prove the reflection principle you described. | |
| Jan 22, 2010 at 5:05 | comment | added | Joel David Hamkins | I see. That's interesting. I don't really have any feeling for what happens without classical logic. But what you say makes sense. | |
| Jan 22, 2010 at 5:02 | vote | accept | Mike Shulman | ||
| Jan 22, 2010 at 5:01 | comment | added | Mike Shulman | BTW, the reason I ask is that I'm thinking about constructive set theory such as CZF, in which separation stronger that Delta0 is rejected as impredicative, and the version of collection adopted is the one I called "strong" (so does Wikipedia: en.wikipedia.org/wiki/…). The real question in my mind was: if you instead use what I called "weak" collection in axiomatizing CZF, and then you make the logic classical, do you recover ZF? Your argument seems to suggest the answer is yes. | |
| Jan 22, 2010 at 4:54 | comment | added | Joel David Hamkins | About KP. I stated it as having Sigma_1 Separation, but it is equivalent to have only Delta_0 Separation. The Delta_0 collection axioms allows you get Sigma_1 collection for free, since one can essentially absorb another quantifier in the collection process. So perhaps the KP result I mentioned is just what you want. (I agree that the examples with the reals is just a curiosity.) | |
| Jan 22, 2010 at 4:52 | comment | added | Joel David Hamkins | I think that the usual axioms people consider are Sigma_n Separation, Sigma_n Collection and Sigma_n Replacement. None of these is exactly your Strong Collection, and perhaps people would think of that as a combination of Collection and Separation. | |
| Jan 22, 2010 at 4:46 | comment | added | Mike Shulman | What is the standard name for what I called "strong" collection? Or does it not have a standard terminology because (by your proof) it is equivalent to ordinary (what I called "weak") collection? | |
| Jan 22, 2010 at 4:44 | comment | added | Mike Shulman | Thanks for the detailed answer! The example of the reals is interesting, but not really what I'm looking for, because I definitely do want to assume Delta0-separation. I'll have to think a bit about your proof in KP but it looks like that may be what I want. | |
| Jan 22, 2010 at 4:07 | comment | added | François G. Dorais | Yes, of course, H(|a|+) is Sigma_1-elementary in V. That could work... | |
| Jan 22, 2010 at 3:30 | comment | added | Joel David Hamkins | But I think a version of your trick works, where you find the witnesses in H(|a|+). That is, if there is a witness, then take a Skolem hull and collapse things down so that the witness is small. And H(|a|+) is something very like P(a), as you said. For example, every element in H(|a|+) is coded in P(a). | |
| Jan 22, 2010 at 3:25 | comment | added | François G. Dorais | @Joel: Ah, right! I should be more careful when switching contexts like that. I was using phi^a(x) when I used this trick before. | |
| Jan 22, 2010 at 3:13 | comment | added | Joel David Hamkins | @Dorais: Is that right? What if phi(x) says "x has a member, which has a member that has a member that is y", where y is the parameter. This seems to push you up several levels beyond the parameters. | |
| Jan 22, 2010 at 3:06 | comment | added | François G. Dorais | You can get around reflection using the powerset (assuming Delta0 separation). If φ(x) is Delta0 with parameters in a transitive set a, then if there is an x which satisfies φ(x) then there is such an x in the powerset of a. This iterates, adding one more powerset for each quantifier depth. You have to modify the formula a little so the quantifiers range over sets of the appropriate powerset level. | |
| Jan 22, 2010 at 2:57 | history | edited | Joel David Hamkins | CC BY-SA 2.5 |
Combined answers
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| Jan 22, 2010 at 2:14 | history | answered | Joel David Hamkins | CC BY-SA 2.5 |