Timeline for In the von Neumann–Bernays–Gödel axiomatic system, the Axiom of Transposition can be simplified
Current License: CC BY-SA 4.0
24 events
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| Apr 9, 2020 at 20:41 | history | edited | Taras Banakh | CC BY-SA 4.0 |
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| Apr 9, 2020 at 20:40 | comment | added | Taras Banakh | @PaceNielsen Very good! Thank you. Following your suggestions I made the simplifications of the proofs. Now all of them are almost trivial. | |
| Apr 9, 2020 at 20:05 | comment | added | Pace Nielsen | Lemma 5 can similarly be simplified, as $\pi_3[pr_1]=\pi_2[p_1]\times V$. I'm sure Lemmas 6 and 7 have similar simplifications. | |
| Apr 9, 2020 at 19:40 | history | edited | Taras Banakh | CC BY-SA 4.0 |
Replaced proofs of Lemmas 3,4 by shorter proofs.
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| Apr 9, 2020 at 19:34 | history | edited | Taras Banakh | CC BY-SA 4.0 |
Replaced proofs of Lemmas 3,4 by shorter proofs.
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| Apr 9, 2020 at 19:25 | comment | added | Taras Banakh | @PaceNielsen Ups! Yes, you are right. Thank you for your comment and explanations. I will make the necessary changes to the proofs of those two lemmas. | |
| Apr 9, 2020 at 19:05 | comment | added | Pace Nielsen | Nice solution. The proofs of lemmas 3 and 4 can be simplified, since $p_1=\pi_3[I\times V]$ and $p_2=\pi_3^2[I\times V]$ (or vice versa, depending on which cycle $\pi$ represents). | |
| Apr 9, 2020 at 15:34 | comment | added | Taras Banakh | @EmilJeřábek Yes, it was too optimistic to try to remove unions and power-sets. Concerning "functions" $x\mapsto\bigcup x$ and $x\mapsto\mathcal P(x)$, I can only prove (it is not difficult) that the relations $\{(x,y):\exists y \;(y\in z\in x)\}$ and $\{(x,y):y\subseteq x\}$ exist. But those relations are not functions, so the Axiom of Replacement cannot be applied. Thank you for explanations. | |
| Apr 9, 2020 at 15:12 | comment | added | Emil Jeřábek | Functions only map sets to sets, hence if $x\mapsto\bigcup x$ is a function, that already implies that $\bigcup x$ is a set without using replacement. You didn’t justify your claim these functions are actually functions (i.e., total). | |
| Apr 9, 2020 at 15:07 | comment | added | Emil Jeřábek | It’s not clear to me how you intend to apply the axiom of replacement here, but note that the set $F[x]$ constructed by replacement has always cardinality at most that of another already known set (namely, $x$), whereas $\bigcup x$ or $\mathcal P(x)$ may be much larger than $x$. | |
| Apr 9, 2020 at 15:05 | comment | added | Emil Jeřábek | This cannot be right, as it would also make these axioms redundant in ZFC by the usual expand-a-model-with-definable-classes argument. Obviously, the operations $\bigcup X$ and $\mathcal P(X)$ are definable using only set quantifiers, hence the existence of $\bigcup X$ and $\mathcal P(X)$ as classes follows from the class existence axioms. But the point of the axioms of union and powerset is that if $x$ is a set, then the (already existing classes) $\bigcup x$ and $\mathcal P(x)$ are also sets. | |
| Apr 9, 2020 at 14:54 | comment | added | Taras Banakh | @EmilJeřábek By the same reason, the Axiom of Power-Set follows from the Axiom of Replacemnt and the fact that the correspondence $X\mapsto \mathcal P(X)$ is a function (which exists as a class). | |
| Apr 9, 2020 at 14:49 | comment | added | Taras Banakh | @EmilJeřábek By the way the existence of the function $X\mapsto \bigcup X$ follows from the Axioms of Membership, Inversion, Permutation, Complement and Domain. Consequently, the axiom of union in NGB follows from the Axiom of Replacement, and hence can be removed. I am not sure that it can be removed from the list of Godel operations though. | |
| Apr 9, 2020 at 14:38 | history | edited | Taras Banakh | CC BY-SA 4.0 |
Added the fact that the Axiom of Intersection follows from the axiom of complement.
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| Apr 9, 2020 at 14:34 | comment | added | Emil Jeřábek | Yes, but it means you can drop one of your axioms of NBG. | |
| Apr 9, 2020 at 14:24 | history | edited | Taras Banakh | CC BY-SA 4.0 |
Corrected Transposition' to Inversion
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| Apr 9, 2020 at 14:23 | comment | added | Taras Banakh | Good point! Thank you. But anyway the smallest number of (elementary) Godel's operations at the moment remains 8. | |
| Apr 9, 2020 at 14:11 | comment | added | Emil Jeřábek | You don’t need intersection if you have complement: $X\cap Y=X\setminus(X\setminus Y)$. | |
| Apr 9, 2020 at 11:29 | history | edited | Taras Banakh | CC BY-SA 4.0 |
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| Apr 9, 2020 at 11:20 | history | edited | Taras Banakh | CC BY-SA 4.0 |
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| Apr 9, 2020 at 11:14 | history | edited | Taras Banakh | CC BY-SA 4.0 |
Added info on Godel's operations.
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| Apr 9, 2020 at 10:19 | history | edited | Taras Banakh | CC BY-SA 4.0 |
Added Axiom of Extensionality
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| Apr 9, 2020 at 9:41 | history | edited | Taras Banakh | CC BY-SA 4.0 |
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| Apr 9, 2020 at 9:29 | history | answered | Taras Banakh | CC BY-SA 4.0 |