Timeline for In the von Neumann–Bernays–Gödel axiomatic system, the Axiom of Transposition can be simplified
Current License: CC BY-SA 4.0
29 events
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| Apr 12, 2020 at 18:10 | history | edited | Taras Banakh | CC BY-SA 4.0 |
added 7 characters in body
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| Apr 12, 2020 at 8:13 | history | edited | Taras Banakh | CC BY-SA 4.0 |
Added some history
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| Apr 11, 2020 at 2:16 | vote | accept | Taras Banakh | ||
| Apr 9, 2020 at 15:49 | history | edited | Taras Banakh | CC BY-SA 4.0 |
Corrected the axioms of transposition
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| Apr 9, 2020 at 15:26 | history | edited | Taras Banakh | CC BY-SA 4.0 |
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| Apr 9, 2020 at 13:35 | history | edited | Taras Banakh | CC BY-SA 4.0 |
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| Apr 9, 2020 at 13:35 | comment | added | Taras Banakh | @YCor Truly speaking I was very excited with the answer, so changed the question correspondingly. Now I removed the exclamation mark (as you suggested), but did not return the question mark since the answer is known already. | |
| Apr 9, 2020 at 11:52 | comment | added | YCor | This is a Q&A site, so I think it would be better to leave the title in a form of a question, or at least not in the form of an answer title (also better avoid exclamation marks in title) | |
| Apr 9, 2020 at 9:47 | comment | added | Taras Banakh | @AsafKaragila I will just cite the enernal phrase of Gellileo Galilei: "E pur si muove!" Optimization is helpful, when you present the axioms to students which just study the subject and will choose other fields of math. In this case the axioms should be short and natural (in order to be attractive and to be memorized). In this respect compacr the Axioms of Replacement in ZFC and NBG. Two very big differences (as say in Odesa). | |
| Apr 9, 2020 at 9:38 | history | edited | Taras Banakh | CC BY-SA 4.0 |
edited title
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| Apr 9, 2020 at 9:30 | history | edited | Taras Banakh | CC BY-SA 4.0 |
Changed the title to a more optimistic!
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| Apr 9, 2020 at 9:29 | answer | added | Taras Banakh | timeline score: 3 | |
| Apr 9, 2020 at 8:35 | comment | added | Asaf Karagila♦ | Taras, over-optimisation can be a bad thing. Redundancies exist to make things easier. We include Replacement and Separation as axioms in ZFC because it's easier, and we don't have to go through the proof that Replacement implies Separation. Likewise for Pairing, or Empty Set. | |
| Apr 8, 2020 at 20:54 | history | edited | Taras Banakh | CC BY-SA 4.0 |
extended the question
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| Apr 8, 2020 at 20:13 | answer | added | Pace Nielsen | timeline score: 4 | |
| Apr 8, 2020 at 19:39 | history | edited | Taras Banakh | CC BY-SA 4.0 |
edited title
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| Apr 8, 2020 at 19:28 | history | edited | Taras Banakh | CC BY-SA 4.0 |
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| Apr 8, 2020 at 18:41 | history | edited | Taras Banakh | CC BY-SA 4.0 |
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| Apr 8, 2020 at 18:39 | comment | added | Taras Banakh | In fact, I am writing an introductory text for students in order to define clearly all set-theoretic notions they use outside of Set Theory. In this respect NBG looks better than ZFC because NBG is finitely axiomatizable and allows to speak about classes which appear very often in Mathematics (for example in Category Theory). And the Axiom of Replacement is much more clear in NBG than in ZFC. But anyway the number of NBG axioms is 15 so I wanted to make some optimization, if possible. That is the reason for my question. | |
| Apr 8, 2020 at 18:34 | comment | added | Emil Jeřábek | You need infinitely many transpositions to generate the group of finitary permutations of $\omega$, hence it is a minor miracle that you can make do with only finitely many axioms at all; it certainly does not give a reason to think that you can reduce the two premutation axioms to one. | |
| Apr 8, 2020 at 18:33 | comment | added | Taras Banakh | Of course, if you have just one axiom of transposition, then then it will be difficult to generate other transpositions. But we have other axioms (product by V, domain). In the worst case one can replace the circular axiom by another transposition axiom (for example first and second). So there will be two tranposition axioms of the same type, which is just a bit simpler than have two different axioms (one for odd and the other for even permutations). | |
| Apr 8, 2020 at 18:32 | comment | added | Asaf Karagila♦ | Taras, any finite number of axioms can be replaced by one. Just take the conjunction... | |
| Apr 8, 2020 at 18:29 | comment | added | Noah Schweber | @TarasBanakh I think Asaf's point is that the transposition axiom only includes one transposition (swapping second and third coordinates), and more generally a "one permutation per axiom" set-up will require two axioms. (And if we drop that setup, then we can just take the conjunction of the axioms.) | |
| Apr 8, 2020 at 18:28 | comment | added | Gerhard Paseman | Linguistically you can combine them: forall A exists B exists C ( conjunct describing both types ). Gerhard "Reshape The Rules To Fit" Paseman, 2020.04.08. | |
| Apr 8, 2020 at 18:27 | comment | added | Taras Banakh | It may happen that the axiom of transposition plus the axiom of domain or replacement will give other versions of the permutation axioms? | |
| Apr 8, 2020 at 18:24 | comment | added | Taras Banakh | The question is not about the number of transpositions but about the number of axioms. Can two axioms be replaced by a single axiom? If yes, how this single axiom can look like? I hope that I am not the first person that asks this question. So I just would like to know what experts think on this question. | |
| Apr 8, 2020 at 18:17 | comment | added | Asaf Karagila♦ | How many generators do you need for $S_3$? | |
| Apr 8, 2020 at 18:13 | history | edited | YCor | CC BY-SA 4.0 |
unabbreviated, added tag
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| Apr 8, 2020 at 18:10 | history | asked | Taras Banakh | CC BY-SA 4.0 |