Timeline for Why should we believe in the axiom of regularity?
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
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| Jun 25, 2018 at 2:26 | comment | added | Timothy | I think ZF without regularity is a subtheory of ZF. There's an injection from the theorems of ZF to the theorems of ZF without regularity. The formal system that only lets you prove those theorems, although it is equivalent to ZF is actually a subtheory of ZF without regularity, not a supertheory of it like ZF is. I guess some mathematicians agree that the real meaning of the theorems of ZF talk only about well-founded sets whereas others don't because the axiom of regularity and inexistence of urelements aren't disprovable either. | |
| May 13, 2018 at 8:47 | history | edited | Peter LeFanu Lumsdaine | CC BY-SA 4.0 |
Added further discussion of the “harmlessness” issue
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| Jan 26, 2017 at 19:59 | comment | added | Timothy | Does the axiom of regularity just insist on the inexistence of elements other than those that can be defined in a very constructive way like from taking the power set of a set, from which it can be proven without the axiom of choice that all cardinal numbers are smaller than some cardinal number with an aleph number? | |
| Jan 26, 2017 at 19:53 | comment | added | Timothy | What if there are strictly more sets and urelements than well founded objects, then we have no way of redefining the hierarchy to contain all sets and urelements. I know there is actually no such thing as a proper class since it cannot be an element. There is only such a thing as a statement that mentions a proper class. | |
| Sep 30, 2015 at 14:00 | vote | accept | Wojowu | ||
| Sep 29, 2015 at 19:38 | comment | added | Todd Trimble | Much the same explanation is given in Kunen's Set Theory: that all of mathematics takes place anyway within the universe of well-founded sets. It also leads naturally to the intuitive picture of sets called the cumulative hierarchy which, if I understand the message from this MO post and the thread underneath -- mathoverflow.net/a/208729/2926 -- imparts to ZF much of its "ontological" force. | |
| Sep 29, 2015 at 19:37 | comment | added | Wojowu | Your second paragraph gives an interesting perspective. I have never thought of it this way. | |
| Sep 29, 2015 at 19:24 | history | answered | Peter LeFanu Lumsdaine | CC BY-SA 3.0 |