Timeline for Is it possible to derive the rules of set theory as transfers from the pure finite set world, and can we extend this further?
Current License: CC BY-SA 4.0
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| Dec 28, 2021 at 19:50 | vote | accept | Zuhair Al-Johar | ||
| May 10, 2018 at 17:06 | history | edited | Zuhair Al-Johar | CC BY-SA 4.0 |
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| May 10, 2018 at 16:57 | history | edited | Zuhair Al-Johar | CC BY-SA 4.0 |
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| May 10, 2018 at 14:34 | history | edited | Zuhair Al-Johar | CC BY-SA 4.0 |
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| May 10, 2018 at 14:22 | history | edited | Zuhair Al-Johar | CC BY-SA 4.0 |
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| May 9, 2018 at 21:54 | history | edited | Zuhair Al-Johar | CC BY-SA 4.0 |
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| May 8, 2018 at 12:09 | history | edited | Zuhair Al-Johar | CC BY-SA 4.0 |
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| May 6, 2018 at 11:22 | history | edited | Zuhair Al-Johar | CC BY-SA 4.0 |
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| May 2, 2018 at 21:42 | review | Close votes | |||
| May 15, 2018 at 3:00 | |||||
| S May 2, 2018 at 21:40 | history | bounty ended | Zuhair Al-Johar | ||
| S May 2, 2018 at 21:40 | history | notice removed | Zuhair Al-Johar | ||
| May 2, 2018 at 20:32 | comment | added | Andrej Bauer | @Zuhair: an important part of asking a question is listening to the answer. | |
| May 2, 2018 at 15:49 | history | edited | Zuhair Al-Johar | CC BY-SA 4.0 |
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| May 2, 2018 at 7:01 | comment | added | Harry Altman | Ah, I see, it's OK for the transitive closure to be a class rather than a set in this context. Thanks, I didn't realize that. | |
| May 2, 2018 at 4:36 | comment | added | Zuhair Al-Johar | the ternary principle was already mentioned there, "before the answer", but I've just made it more conspicuous by the further edit, the other edits are either typos or are in response to technical comments of Joel, which I've fixed. so most of these edits are actually fixing minor issues or highlighting something that already had been written. the main subject of the posting is still the same. | |
| May 2, 2018 at 4:30 | comment | added | Zuhair Al-Johar | @ we have $V$ being a transitive superclass of any class in this theory, now the minimal transitive superclass can be proved from class comprehension. | |
| May 1, 2018 at 20:07 | comment | added | Harry Altman | Question: How do you prove a transitive closure exists without using something like the axiom of infinity? | |
| May 1, 2018 at 18:27 | comment | added | user44143 | Given the 23 changes to the question, with some edits explicitly designed to reject a proposed answer, I will wait to see if this survives beyond the bounty period; if so, I will vote to close as unclear what you’re asking. | |
| May 1, 2018 at 17:57 | history | edited | Zuhair Al-Johar | CC BY-SA 3.0 |
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| May 1, 2018 at 11:35 | comment | added | Joel David Hamkins | Yes, any second-order theory can be deemed as a first-order theory, by considering the second-order part to be added explicitly as objects. But to my way of thinking, to do this is to abandon a useful distinction. For this reason, I find the one-sorted approaches to KM to be unhelpful, a distraction. In my experience, the experts who are currently seriously studying the various second-order set theories (and I know them; I myself have several papers analyzing this hierarchy) are careful to observe the first-order/second-order distinction. | |
| May 1, 2018 at 11:22 | history | edited | Zuhair Al-Johar | CC BY-SA 3.0 |
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| May 1, 2018 at 11:16 | history | edited | Zuhair Al-Johar | CC BY-SA 3.0 |
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| May 1, 2018 at 11:11 | history | edited | Zuhair Al-Johar | CC BY-SA 3.0 |
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| May 1, 2018 at 11:02 | history | edited | Zuhair Al-Johar | CC BY-SA 3.0 |
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| May 1, 2018 at 6:56 | comment | added | Zuhair Al-Johar | MK is a "first order axiomatic theory", this is the traditional description about it, MK is an extension of FIRST ORDER LOGIC, but you was right regarding me being wrong when I said "first order language of set theory" since this do have a different connotation, since the language of MK is 'termed' as "second order language of set theory" but this is just a matter of terminology MK is still a "first order axiomatic set theory", see this wikipedia on MK (second line) for example. en.wikipedia.org/wiki/Morse%E2%80%93Kelley_set_theory, anyhow I've corrected this error of mine | |
| Apr 30, 2018 at 20:13 | history | edited | Zuhair Al-Johar | CC BY-SA 3.0 |
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| Apr 30, 2018 at 18:27 | comment | added | Joel David Hamkins | But why not use the well-developed and widely understood theories, which seem fully capable of expressing the ideas you are trying to convey, instead of developing your own idiosyncratic notation and axioms? It seems to me that your questions would find a much better reception if you did so. | |
| Apr 30, 2018 at 15:56 | comment | added | Zuhair Al-Johar | @joel iknow the stuff related to the naming I know it is confusing but still its first order axiomatic theory I know it is called second order language of set theory but I'm not speaking in that sense | |
| Apr 30, 2018 at 15:50 | comment | added | Zuhair Al-Johar | @Matt F.44 no the only primitives are membership and equality and I mean finiteness in this theory | |
| Apr 30, 2018 at 15:23 | comment | added | Joel David Hamkins | By the way, there is a standard convention in second-order set theory (and second-order logic generally) to use lower-case letters to quantify over the first-order objects and upper-case letters to quantify over the second-order objects. Thus, in set theory, $\forall x$ means for all sets $x$ and $\forall X$ means for all classes $X$. | |
| Apr 30, 2018 at 15:22 | answer | added | Joel David Hamkins | timeline score: 9 | |
| Apr 30, 2018 at 14:58 | comment | added | user44143 | “Shortest formula defining finiteness” is sensitive to both the language and the background theory. It looks like your language has a constant (null set), a unary function (singleton), a binary function (union), a unary relation (set), and a binary relation (element) — is that the right signature? And are you looking for the shortest formula defining finiteness in MK or in some other theory? | |
| Apr 30, 2018 at 14:51 | comment | added | Joel David Hamkins | In the established terminology, the "first-order language of set theory" usually (i.e. basically always) refers to the language with only $\in$ and all quantifiers range over sets; and the "second-order language of set theory" refers to the languages where one also allows class quantifiers. For example, ZFC is a theory in the first-order language of set theory, and GBC and KM are stated in the second-order language of set theory. The fact that one may regard models of GBC and KM as first-order (two-sorted) structures is beside the point, since one can do this with any second-order structure. | |
| Apr 30, 2018 at 14:37 | comment | added | Zuhair Al-Johar | Yes, I fixed the matter about the empty set being finite.about the language it is a first order axiomatic theory, the class quantifiers won't make the underlying logic second order logic, there is no quantification over function symbols nor over predicates, i.e. there is not quantifier that quantifies over all subsets of the universe of discourse, so it is formulated in FIRST order logic. The Classes are "elements" of the universe of discourse of this theory, and the quantifiers only quantify over those. Much as Morse-Kelley class theory | |
| Apr 30, 2018 at 14:33 | history | edited | Zuhair Al-Johar | CC BY-SA 3.0 |
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| Apr 30, 2018 at 14:22 | history | edited | Zuhair Al-Johar | CC BY-SA 3.0 |
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| Apr 30, 2018 at 14:21 | comment | added | Joel David Hamkins | Your definition doesn't seem to include the empty set as finite. Also, you say that you think your definition is the shortest way to define "finite" in the first-order language of set theory, but your definition is not in the first-order language of set theory, but rather in the second-order language of set theory, since you are using class quantifiers. | |
| Apr 30, 2018 at 14:19 | history | edited | Zuhair Al-Johar | CC BY-SA 3.0 |
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| Apr 30, 2018 at 14:10 | history | edited | Zuhair Al-Johar | CC BY-SA 3.0 |
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| S Apr 30, 2018 at 9:38 | history | bounty started | Zuhair Al-Johar | ||
| S Apr 30, 2018 at 9:38 | history | notice added | Zuhair Al-Johar | Draw attention | |
| Apr 29, 2018 at 20:29 | history | edited | Zuhair Al-Johar | CC BY-SA 3.0 |
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| Apr 28, 2018 at 15:34 | review | Close votes | |||
| Apr 30, 2018 at 9:38 | |||||
| Apr 28, 2018 at 14:25 | history | edited | Zuhair Al-Johar | CC BY-SA 3.0 |
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| Apr 28, 2018 at 13:26 | history | edited | Zuhair Al-Johar | CC BY-SA 3.0 |
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| Apr 28, 2018 at 13:20 | history | edited | Zuhair Al-Johar | CC BY-SA 3.0 |
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| Apr 28, 2018 at 13:08 | history | edited | Zuhair Al-Johar | CC BY-SA 3.0 |
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| Apr 28, 2018 at 13:00 | history | edited | Zuhair Al-Johar | CC BY-SA 3.0 |
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| Apr 28, 2018 at 12:44 | history | edited | Zuhair Al-Johar | CC BY-SA 3.0 |
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| Apr 28, 2018 at 11:56 | history | edited | Zuhair Al-Johar | CC BY-SA 3.0 |
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| Apr 28, 2018 at 11:46 | history | edited | Zuhair Al-Johar | CC BY-SA 3.0 |
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| Apr 28, 2018 at 11:41 | history | edited | Zuhair Al-Johar | CC BY-SA 3.0 |
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| Apr 28, 2018 at 9:40 | history | edited | Zuhair Al-Johar | CC BY-SA 3.0 |
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| Apr 28, 2018 at 9:35 | history | edited | Zuhair Al-Johar | CC BY-SA 3.0 |
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| Apr 28, 2018 at 9:26 | history | edited | Zuhair Al-Johar | CC BY-SA 3.0 |
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| Apr 28, 2018 at 9:17 | history | asked | Zuhair Al-Johar | CC BY-SA 3.0 |