Timeline for Is Replacement motivated by ranked iterative conception of sets?
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| Jun 15, 2020 at 7:27 | history | edited | CommunityBot |
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| Nov 25, 2018 at 18:55 | comment | added | Zuhair Al-Johar | Yes I agree that neither Extensionality nor the class comprehension schema are motivated by the cumulative hierarchy. The point is that no one said that about those, but about Replacement you hear a lot of talk in that direction, that's why this point is raised with the motivation for adding Replacement on top of Z. | |
| Nov 25, 2018 at 15:41 | comment | added | Timothy Chow | I agree with your basic point that the concept of the cumulative hierarchy doesn't directly suggest Replacement---that's (part of) the reason why Z (as opposed to ZF) is a viable axiomatic system. I suspect Avron would ask, "Why not Replacement?" since he thinks that Replacement is a basic feature of our conception of sets. Not every axiom has to be justified directly from the cumulative hierarchy; e.g., we don't justify Extensionality by arguing that Extensionality is motivated by the cumulative hierarchy. | |
| Nov 24, 2018 at 8:23 | history | edited | Zuhair Al-Johar | CC BY-SA 4.0 |
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| Nov 24, 2018 at 8:20 | comment | added | Zuhair Al-Johar | In nutshell replacement doesn't come from building sets by iterative powering from below. [note that what is meant by replacement here is more precisely the addition of replacement schema to the rest of axioms of ZF] | |
| Nov 24, 2018 at 8:16 | comment | added | Zuhair Al-Johar | .. [continuation], now lets ask ourselves a simple question, what determines the height of each stage in the hierarchy? it ought to be a "set" I believe, I mean an ordinal that is already formed within the levels of the cumulative hierarchy, since we agreed that sets are objects that are created therein! BUT if we hold that, which is in my opinion the natural way to build a hierarchy from below, now as I said if we hold that then we don't get replacement, we'll only get our hierarchy indexed by ordinals less than the first fixed point on the omega function, much weaker than replacement. | |
| Nov 24, 2018 at 8:03 | comment | added | Zuhair Al-Johar | @TimothyChow from what I've read actually Avron is advocating replacement, but he is against the combination of it with power, and he thinks the later is the culprite. Anyhow I speaking here about the cumulative hierarchy and power is a theorem here, indeed we can have a hierarchy of stages that are not power stages, we can have a hierarchy of constructive powers like in $L$, but this is not the point raised here, I like the head post of the page you've referred in your comment. Look at his question which he puts it in a naive (and the right way I think) manner, to be continued.... | |
| Nov 24, 2018 at 5:10 | comment | added | Timothy Chow | Related: mathoverflow.net/questions/228168/… especially the remarks by Arnon Avron that I quoted. I think Avron would say that Replacement is not motivated by the cumulative hierarchy per se, and I think his argument is a pretty good one. | |
| Nov 23, 2018 at 21:58 | history | edited | Zuhair Al-Johar | CC BY-SA 4.0 |
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| Nov 23, 2018 at 21:51 | history | edited | Zuhair Al-Johar | CC BY-SA 4.0 |
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| Nov 23, 2018 at 21:35 | vote | accept | Zuhair Al-Johar | ||
| Nov 23, 2018 at 21:20 | history | edited | Zuhair Al-Johar | CC BY-SA 4.0 |
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| Nov 23, 2018 at 21:10 | history | edited | Zuhair Al-Johar | CC BY-SA 4.0 |
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| Nov 23, 2018 at 17:57 | answer | added | Noah Schweber | timeline score: 3 | |
| Nov 23, 2018 at 14:28 | history | edited | Zuhair Al-Johar | CC BY-SA 4.0 |
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| Nov 23, 2018 at 11:01 | history | asked | Zuhair Al-Johar | CC BY-SA 4.0 |