Timeline for In the von Neumann–Bernays–Gödel axiomatic system, the Axiom of Transposition can be simplified
Current License: CC BY-SA 4.0
13 events
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| Apr 12, 2020 at 6:35 | comment | added | Taras Banakh | I have just discovered that the Axioms of Inversion and Associativity were included to the original axiom system of Benays, see his axioms (c2),(c3) on page 7 of this paper of Kanamori about Barnays: file:///D:/taras/coauthors/banakh/mybooks/Topology1/Bernays_and_Set_Theory.pdf | |
| Apr 11, 2020 at 2:27 | comment | added | Taras Banakh | Concerning the concrete form of the Axiom of Associativity in NBG I would suggest to postulate the existence of the class $\{(((x,y),z),(x,(y,z))):x,y,z\in V\}$, which a function transforming the triple $((x,y),z)$ into the triple $(x,(y,z))$. Maybe the same should be done with the inversion: just to postulate the existence of the function $\{((x,y),(y,x)):x,y\in V\}$ that transforms a pair $(x,y)$ into the pair $(y,x)$. And rename the corresponding axiom to Axiom of Symmetry. Then then two Axioms (of Symmetry and Associativity) are indeed axioms for handling tuples, as written in Wikipedia. | |
| Apr 11, 2020 at 2:16 | vote | accept | Taras Banakh | ||
| Apr 9, 2020 at 21:20 | history | edited | Pace Nielsen | CC BY-SA 4.0 |
added 2172 characters in body
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| Apr 9, 2020 at 19:21 | comment | added | Taras Banakh | By the minimality I mean the minimality in a very human sense: just imagine that you should teach student elements of Set Theory and you have a restricted amount of time (two or three lectures). And you do not have time to prove everything for the scratch. Just to mention and explain the most important things. And to not lie, but not say all the truth (because of the lack of time). Which list of axioms will you choose? This is my understanding of minimality. | |
| Apr 9, 2020 at 19:13 | comment | added | Pace Nielsen | @TarasBanakh I think you are misunderstanding my point about minimality and Godel operations. As an analogy, there is a one axiom Hilbert deduction system for propositional logic. (In fact, there are many of them.) That one axiom is not merely the conjunction of the usual axioms. It isn't too long, so it meets your requirement of using fewer symbols. But that axiom does too much work to be "nice" for us humans to use. | |
| Apr 9, 2020 at 19:10 | comment | added | Taras Banakh | I have in mind the minimality of elementary operations (those that cannot be decomposed into simpler pieces). If you want, the minimality of the total number of symbols necessary for writing these axioms, so each artificial conjunction increases the number of such symbols. But let us leave this minimality discussion as fruitless. | |
| Apr 9, 2020 at 18:42 | comment | added | Pace Nielsen | Eight is not minimal, since one can always arrange for operations to be extremely complicated and do more than one job at once. | |
| Apr 9, 2020 at 13:53 | history | edited | LSpice | CC BY-SA 4.0 |
Authors of paper; link to paper and book
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| Apr 9, 2020 at 4:25 | comment | added | Taras Banakh | Thank you for this link. I mentions the book of Jech who defined 10 Godel's operations but in Exercise 13.4 Jech remarks that those 10 can be reduced to 8. The number of NBG axioms allowing to construct classes is also 8. Maybe 8 is this optimal number? What I will try to realize now is if the 9-th Godel's operation of Jech (en.wikipedia.org/wiki/G%C3%B6del_operation) is expressible via 8-th and other operations. | |
| Apr 8, 2020 at 21:27 | comment | added | Pace Nielsen | @TarasBanakh You might be interested to know that Godel's original set of operations has three different permutation rules. en.wikipedia.org/wiki/G%C3%B6del_operation | |
| Apr 8, 2020 at 20:38 | comment | added | Taras Banakh | Thank you for your answer. Ok, let me ask a concrete question: can the axiom of transposition be simplied to the form: $\forall A\exists B\;\forall x\forall y \;((x,y)\in A\Leftrightarrow (y,x)\in B)$? As I understood the axiom of cyclic permutation composed with the axiom of domain is necessary for the proof of the existence of projections on each coordinate. And the axiom of transposition makes the main job generating all the permutations. Right? | |
| Apr 8, 2020 at 20:13 | history | answered | Pace Nielsen | CC BY-SA 4.0 |