Timeline for What's the exact consistency strength of this axiom system for classes and sets?
Current License: CC BY-SA 4.0
47 events
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| Dec 28, 2021 at 19:46 | vote | accept | Zuhair Al-Johar | ||
| May 13, 2021 at 19:03 | comment | added | Pace Nielsen | Ah, I see now. Thanks. | |
| May 13, 2021 at 17:18 | comment | added | Zuhair Al-Johar | @PaceNielsen, by my last line I mean we can also take the formula "$x$ is a stage of the cumulative hierarchy that has a successor stage", and the same argument would prove that $W \in V$. | |
| May 13, 2021 at 17:01 | comment | added | Zuhair Al-Johar | @PaceNielsen, I can prove this indpendent of the question of $W \in V$. This theory proves the reflection axiom of Ackermann's (over $W$), so any formula in the language of set theory from parameters in $W$ that only hold of elements of $W$ would define an element of $W$. Take the formula $x \text{ is an ordinal that have a successor }$, now if $W=V$, then the class of all ordinals that have successors would be an element of $W$, a contradiction. Also we can get to prove $W \in V$ also, I think $W=\bigcup_\alpha W_{\alpha \in W}$ can be proved from foundation, and thus we'll have $W \in V$. | |
| May 13, 2021 at 16:42 | comment | added | Pace Nielsen | @ZuhairAl-Johar How does one prove that $W$ is an element (of any class, let alone $V$)? I'm not seeing it. | |
| May 13, 2021 at 16:14 | comment | added | Zuhair Al-Johar | @PaceNielsen, No! $W=V$ is inconsistent, because $W \in V$ is a theorem, so this leads to $W \in W$ and this is paradoxical. | |
| May 13, 2021 at 16:07 | comment | added | Pace Nielsen | Class comprehension gives a class $V$ of all elements. If we add a sixth axiom $W=V$, does this calibrate your axiom system with respect to NBG and KM? | |
| May 12, 2021 at 21:52 | history | edited | LSpice | CC BY-SA 4.0 |
Proofreading. The editorialising in the body of the post seems inappropriate, but 'elegant' *definitely* doesn't belong in the title, so I removed it in the process.
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| May 12, 2021 at 21:35 | history | edited | Zuhair Al-Johar | CC BY-SA 4.0 |
added 47 characters in body
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| May 12, 2021 at 20:52 | comment | added | Zuhair Al-Johar | @JoelDavidHamkins, Yes! It's a typo. Corrected, thanks. | |
| May 12, 2021 at 20:52 | history | edited | Zuhair Al-Johar | CC BY-SA 4.0 |
edited body
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| May 12, 2021 at 20:22 | comment | added | Joel David Hamkins | Do you have a typo in class comprenhesion? Don't you intend $\phi(y)$ rather than $\phi(x)$? | |
| May 12, 2021 at 19:40 | history | edited | Zuhair Al-Johar | CC BY-SA 4.0 |
added 3 characters in body
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| Nov 9, 2020 at 20:10 | comment | added | Zuhair Al-Johar | @user76284, sorry for late reply, this is it: cs.nyu.edu/pipermail/fom/2019-March/021437.html | |
| Nov 9, 2020 at 20:03 | comment | added | user76284 | @ZuhairAl-Johar Do you have a link to that FOM thread? | |
| Nov 8, 2020 at 19:57 | comment | added | Zuhair Al-Johar | @user76284 yep! | |
| Nov 8, 2020 at 19:34 | comment | added | user76284 | Also, does dropping extensionality, foundation, and/or choice retain the consistency strength (as they do for ZFC)? | |
| Nov 8, 2020 at 19:28 | comment | added | user76284 | Do you recall which schema or FOM post you’re referring to in your last comment? | |
| S Nov 8, 2020 at 13:04 | history | suggested | Master | CC BY-SA 4.0 |
Clarified the problem
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| Nov 7, 2020 at 23:05 | review | Suggested edits | |||
| S Nov 8, 2020 at 13:04 | |||||
| Aug 2, 2019 at 22:34 | answer | added | Master | timeline score: 2 | |
| Jun 21, 2019 at 21:01 | history | edited | Zuhair Al-Johar | CC BY-SA 4.0 |
added 8 characters in body
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| Jun 21, 2019 at 8:29 | history | edited | Zuhair Al-Johar | CC BY-SA 4.0 |
edited title
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| Jun 18, 2019 at 14:53 | history | edited | user44143 | CC BY-SA 4.0 |
removed "exact", since an answer was accepted providing only a lower bound; removed "very"; made minor clarifications and correations to main text
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| Jun 18, 2019 at 14:24 | vote | accept | Zuhair Al-Johar | ||
| Jun 21, 2019 at 8:29 | |||||
| Jun 18, 2019 at 14:15 | comment | added | Zuhair Al-Johar | @NajibIdrissi, to my trivial knowledge I've first presented here as a question. I don't know if it was on table before this attempt. I've presented before a similar axiomatization to FOM but with implication instead of the biconditional in set comprehension, and it was only the two schemas. | |
| Jun 18, 2019 at 14:13 | comment | added | Najib Idrissi | Who is this axiom system due to? | |
| Jun 18, 2019 at 14:12 | comment | added | Noah Schweber | @ZuhairAl-Johar Ah, d'oy. | |
| Jun 18, 2019 at 14:12 | answer | added | Will Sawin | timeline score: 5 | |
| Jun 18, 2019 at 14:03 | comment | added | Zuhair Al-Johar | @NoahSchweber, take $\varphi$ to be $x=x_1$, and substitute it in set comprehension. | |
| Jun 18, 2019 at 13:46 | comment | added | Zuhair Al-Johar | @NoahSchweber, the "bi-conditional" in set comprehension grant transitivity of $W$, but even if we weaken it to "implication" still we can get to interpret $ZFC$ over the class of all Hereditarily $\in W$ sets. | |
| Jun 18, 2019 at 13:36 | comment | added | Zuhair Al-Johar | @NoahSchweber, yes this follows from transitivity of $W$, and Extensionality of course. Actually I think that even if I remove Extensionality, still it would be interpretable in this system. But this is another story. | |
| Jun 18, 2019 at 13:33 | comment | added | Noah Schweber | @ZuhairAl-Johar I'm not sure this theory is appropriately complete (I mean with respect to the intuition): can you show that extensionality holds in $W$? (That is, that if $x,y$ are distinct elements of $W$, then there is some $z\in W$ with $z\in x\Delta y$.) This seems like something you'd want ... | |
| Jun 18, 2019 at 8:20 | history | edited | Zuhair Al-Johar | CC BY-SA 4.0 |
edited body
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| Jun 18, 2019 at 5:29 | comment | added | Zuhair Al-Johar | @EricWofsey, this is an aspect that it shares with Ackermann's. Both theories have many classes that they don't speak about a lot. But this theory speaks of those sets more than do Ackermann's do, since class comprehension here is more expressive. | |
| Jun 18, 2019 at 5:24 | comment | added | Zuhair Al-Johar | @WillSawin I believe it can prove that. For instance this theory interprets Ackermann's, so it interpret ZFC. However Ackermann's set theory doesn't prove its universe of sets being a model of ZFC, so that interpretation alone is not enough for proving $W \models ZFC$. Yet this theory is stronger than Ackermann's, so I tend to think it can prove $W \models ZFC$, but I'm not sure. | |
| Jun 18, 2019 at 2:54 | comment | added | Eric Wofsey | These axioms are weird in that they imply the existence of many elements that are not sets (since $W$ needs to not be definable with parameters from $W$), but say very little about such elements. For instance, if you have any model and a class $C$ in the model which is not an element, it seems you can get a new model by just making $C$ an element: i.e., for every class $D$ that you had, add a new class $D\cup\{C\}$. (Although I suppose if $C$ is not definable with parameters from $W$, this might somehow mess up set comprehension.) | |
| Jun 17, 2019 at 22:24 | comment | added | Eric Wofsey | If I'm not mistaken, you can construct a model of your theory from ZFC with an inaccessible. If $\kappa$ is inaccessible, you let the universe be $V_{\kappa+1}$ and let $W$ be $V_\alpha$ for $\alpha<\kappa$ such that any ordinal less than $\kappa$ definable in $V_{\kappa+1}$ with parameters from $V_\alpha$ is less than $\alpha$. (You can prove such an $\alpha$ exists using inaccessibility of $\kappa$.) | |
| Jun 17, 2019 at 20:53 | comment | added | Will Sawin | Have you proved that $W$ is a model of ZFC in your theory? Or do you believe you can prove this? | |
| Jun 17, 2019 at 20:19 | comment | added | Noah Schweber | (Turning my misinterpretation into a comment): Note that if $W$ were a relation symbol, this theory would be quite weak - in particular, consistent relative to PA. Let $M$ be a nonstandard model of PA, $\alpha$ a nonstandard $Ackermann(M)$-ordinal, consider the the expansion of $(V_{\alpha+1})^{Ackermann(M)}$ by interpreting $W$ as the well-founded part of $Ackermann(M)$, and note that set comprehension follows from overspill. Having $W$ an actual object changes things: e.g. $W$ satisfies Infinity since it's the smallest adjunction-closed set. | |
| Jun 17, 2019 at 20:14 | history | edited | Zuhair Al-Johar | CC BY-SA 4.0 |
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| Jun 17, 2019 at 20:07 | comment | added | Zuhair Al-Johar | @AsafKaragila, I personally think this theory is STRONGER than ZFC\NBG\MK. it is much stronger than what it looks. | |
| Jun 17, 2019 at 19:57 | comment | added | Zuhair Al-Johar | @AsafKaragila, the idea is to interpret $ZFC$ over $W$ and not over $V$ the class of all elements (which Noah calls as $E$). | |
| Jun 17, 2019 at 19:13 | comment | added | Noah Schweber | @AsafKaragila I think powerset falls out of set and class comprehension. Roughly: set comprehension gives that any subclass of a set is a set; class comprehension gives that the class of all subsets of a set exists; and set comprehension says that that class is a set. | |
| Jun 17, 2019 at 18:37 | comment | added | Asaf Karagila♦ | Ideal? That seems to be like Pocket set theory more than a ZF/NBG/KM variant... Where's the power set? | |
| Jun 17, 2019 at 18:15 | history | edited | Zuhair Al-Johar | CC BY-SA 4.0 |
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| Jun 17, 2019 at 18:09 | history | asked | Zuhair Al-Johar | CC BY-SA 4.0 |