Timeline for Why should we believe in the axiom of regularity?
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| Jun 25 at 18:35 | history | edited | Martin Sleziak | CC BY-SA 4.0 |
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| Sep 30, 2015 at 22:12 | comment | added | Thomas Benjamin | @Wojowu: Please take a look at Zermelo's 'proof', If you can find a copy of van Heijenoort's book. | |
| Sep 30, 2015 at 22:07 | comment | added | Thomas Benjamin | (cont.) hindsight. Certainly they were vaguely aware unrestricted Comprehension was the cause ot the paradoxes, but were at odds of how to restrict the principle adequately. To quote Zermelo from the paper I mentioned in my answer (pg. 203) in the paragraph following of his 'proof' of Regularity from the Axiom of Separation: "It follows from the theorem that not all objects $x$ of the domain $\mathfrak B$ can be elements of one and the same set ; that is, the domain $\mathfrak B$ is itself not a set, and this disposes of the Russell antinomy so far as we are concerned." | |
| Sep 30, 2015 at 21:57 | comment | added | Thomas Benjamin | @Wojowu: You are partially correct in that, I think. To quote Russell (from his letter to Frege: "Let $w$ be the predicate: 'to be a predicate that cannot be predicated of itself. Can $w$ be predicated of itself? From each answer its opposite follows. Therefore we must conclude that $w$ is not a predicate. Likewise there is no class (as a totality) of those classes which, each taken as a totality, do not belong to themselves. From this I conclude that under certain circumstances a definable collection [[Menge]] does not form a totality." One should beware of engaging in 20/20 | |
| Sep 30, 2015 at 13:57 | comment | added | Wojowu | "One can certainly understand the early set theorists' concern over the existence of a set y such that y∈y." I slightly disagree to that. As far as I can see, and what mathematicians back in the day probably saw as well, is that what causes the paradox is not a (supposed) existence of self-containing sets, but clearly the unrestricted comprehension - after all, Russell's paradox isn't avoided if we assume regularity in any form. | |
| Sep 30, 2015 at 7:21 | history | edited | David Roberts♦ | CC BY-SA 3.0 |
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| Sep 30, 2015 at 6:35 | history | answered | Thomas Benjamin | CC BY-SA 3.0 |