Timeline for Predicative definition and existence of ordinal numbers
Current License: CC BY-SA 3.0
11 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Apr 13, 2017 at 12:19 | history | edited | CommunityBot |
replaced http://math.stackexchange.com/ with https://math.stackexchange.com/
|
|
| Nov 16, 2016 at 22:27 | vote | accept | Damian | ||
| Nov 1, 2016 at 16:24 | comment | added | Nik Weaver | $\Sigma_1$ constructions generally are not seen as predicative, for obvious reasons. $\Delta_0$ replacement would certainly be accepted. But if you adopt intuitionistic semantics, for any $\phi$, having a proof that $(\forall x \in A)(\exists!y)\phi(x,y)$ would entail knowing how to construct such a $y$ for every $x \in A$, and this would allow the predicative construction of the set of all $y$'s, assuming $A$ was predicatively constructible. | |
| Nov 1, 2016 at 16:02 | comment | added | Asaf Karagila♦ | I would imagine that perhaps some limited form of Replacement is considered predicative. Maybe something like $\Sigma_1$? | |
| Nov 1, 2016 at 14:54 | comment | added | Nik Weaver | Powerset is certainly not considered predicatively acceptable, but I don't recall ever seeing any doubts about replacement expressed in the predicativist literature. | |
| Oct 31, 2016 at 19:57 | history | edited | Noah Schweber | CC BY-SA 3.0 |
added 863 characters in body
|
| Oct 31, 2016 at 19:50 | history | edited | Noah Schweber | CC BY-SA 3.0 |
added 863 characters in body
|
| Oct 31, 2016 at 19:44 | history | edited | Noah Schweber | CC BY-SA 3.0 |
added 863 characters in body
|
| Oct 31, 2016 at 0:40 | comment | added | Noah Schweber | On the philosophical side of things, I believe that neither Replacement nor Powerset are usually viewed as predicatively acceptable. (I could be wrong.) But to address the question "Is there a predicative proof that $\omega_1$ exists?", I would need a precise definition of predicativity. (In particular, I don't think getting rid of Replacement entirely is a good idea - without Replacement, we can't even prove that the ordinal $\omega+\omega$ exists! So it's really a sensitive question, axiomatically speaking.) | |
| Oct 31, 2016 at 0:35 | history | edited | Noah Schweber | CC BY-SA 3.0 |
added 905 characters in body
|
| Oct 31, 2016 at 0:26 | history | answered | Noah Schweber | CC BY-SA 3.0 |