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 3.0
30 events
| when toggle format | what | by | license | comment | |
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| Dec 28, 2021 at 19:50 | vote | accept | Zuhair Al-Johar | ||
| May 2, 2018 at 21:48 | comment | added | Zuhair Al-Johar | The answer did show something, that foundation is dismissed by this method! I still think it can be salvaged by upgrading the principle to include transfer from the pure countable world as well. I'll make a rewrite at a separate post to exposit that in a better manner | |
| May 2, 2018 at 21:40 | history | bounty ended | Zuhair Al-Johar | ||
| May 2, 2018 at 17:10 | comment | added | Jim Conant | @JoelDavidHamkins: I admire your patience! | |
| May 1, 2018 at 17:57 | comment | added | Zuhair Al-Johar | by the way I think your answer doesn't touch the ternary transfer principle. I've edited the post to explicitly mention this principle. Actually I think the ternary transfer principle is consistent relative to ZFC | |
| May 1, 2018 at 10:19 | comment | added | Joel David Hamkins | OK, I have edited to mention foundation. | |
| May 1, 2018 at 10:18 | history | edited | Joel David Hamkins | CC BY-SA 3.0 |
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| May 1, 2018 at 6:59 | comment | added | Zuhair Al-Johar | your answer as it stands is wrong, so it needs to be corrected! by adding that foundation is assumed, or otherwise fix it by some other approach | |
| May 1, 2018 at 6:45 | comment | added | Zuhair Al-Johar | I mean you can define a pure set theory (I mean a theory without classes, like ZF, NF, etc..) and you can add to it a modification of the transfer principle above, and also you can prove from it separation, replacement, union and power, however the approach would be more complex. | |
| Apr 30, 2018 at 22:12 | comment | added | Joel David Hamkins | But without classes, then every $y$ is in $V$, so I don't really follow your remarks about not needing classes. | |
| Apr 30, 2018 at 20:47 | comment | added | Zuhair Al-Johar | Try to see it the other way. IF there no inconsistency involved with it [I myself greatly doubt that], then it would provide a motivation for ZF axioms that doesn't rely on foundation, actually it relies on ANTI-FOUNDATION. | |
| Apr 30, 2018 at 20:44 | comment | added | Zuhair Al-Johar | foundation is not a big problem, it can be interpreted anyway from those axioms. | |
| Apr 30, 2018 at 20:42 | comment | added | Zuhair Al-Johar | I think I can re-write this principle in the first-order language of set theory [in the usual sense]. The whole point is the TRANSFER principle and not to have classes as objects, this later piece is a just a tool to exposit that principle | |
| Apr 30, 2018 at 20:39 | comment | added | Joel David Hamkins | Well, I've already shown that they contradict foundation, which is something that many people would want. And I would think you would mention GB or KM rather than ZF, since the whole point of your project is to have classes as objects. | |
| Apr 30, 2018 at 20:35 | comment | added | Zuhair Al-Johar | They might be few, but they are substantial, actually they are the formulas that can interpret the constructive axioms of the standard set theory ZF [union, power, separation, replacement], what else one would want really [perhaps infinity, but this is built-in it is what the transfer is all about, so its axiomatization is part of the background idea, so its justification "i.e. of infinity" is already "built-in" this line of thought]. However, I'm myself still suspicious of my own transfer principles, the chances are high that an inconsistency is there, but I don't know how, thus the question. | |
| Apr 30, 2018 at 20:29 | comment | added | Joel David Hamkins | Yes, it seems unfortunately that your transfer principle applies to very few formulas of interest. | |
| Apr 30, 2018 at 20:21 | comment | added | Zuhair Al-Johar | "easily" expressed? perhaps, I'm not sure, but I thought it means that every non empty subset of the transitive closure of x must have an element that is disjoint of it. This takes more than 8 atomic formulas, and you already have five in the above formula, a total of more than 13, so it is about the length of the shortest definition of finiteness. | |
| Apr 30, 2018 at 20:14 | comment | added | Joel David Hamkins | Ah, you don't have foundation. In that case, you can modify the formula simply to say that $x$ is nonempty and well-founded, rather than just nonempty (and this is easily expressed). This in effect moves to the well-founded part of your set-theoretic universe. | |
| Apr 30, 2018 at 20:07 | comment | added | Zuhair Al-Johar | what about a Quine atom? | |
| Apr 30, 2018 at 19:56 | comment | added | Joel David Hamkins | Every nonempty HF set has a $\in$-maximal element, since otherwise, you could keep going up in $\in$ inside the set, and get infinitely many elements that way. | |
| Apr 30, 2018 at 19:54 | comment | added | Zuhair Al-Johar | Also wrong, but I generally think it can be modified to reach into your goals. The error is that not every non empty set in HF has a maximal element in the sense you've defined, but I think this can be salvaged really, and I think a modification of your argument would work. I'll look into it. | |
| Apr 30, 2018 at 18:42 | comment | added | Joel David Hamkins | This answer works by using not a definition of finiteness, but rather a necessary property of all finite sets that is not necessary for all sets. And the point is that there is a such a property (having an $\in$-maximal element) that is expressible by a short formula, so that the overall formula is still short. | |
| Apr 30, 2018 at 18:22 | comment | added | Joel David Hamkins | I posted a modified example, which I think is what you want. | |
| Apr 30, 2018 at 18:22 | history | undeleted | Joel David Hamkins | ||
| Apr 30, 2018 at 18:22 | history | edited | Joel David Hamkins | CC BY-SA 3.0 |
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| Apr 30, 2018 at 18:10 | history | deleted | Joel David Hamkins | via Vote | |
| Apr 30, 2018 at 15:43 | comment | added | Joel David Hamkins | Ah, I understand a bit better, I think. Let me modify the example. | |
| Apr 30, 2018 at 15:40 | comment | added | Zuhair Al-Johar | No this won't affect this method since not every x that fulfills $\phi(x)$ is finite for example take the set $\omega$ union the triple singleton of $\empty$ | |
| Apr 30, 2018 at 15:38 | history | edited | Joel David Hamkins | CC BY-SA 3.0 |
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| Apr 30, 2018 at 15:22 | history | answered | Joel David Hamkins | CC BY-SA 3.0 |