Timeline for Ontological status of some "sets" in ZFC
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jun 5, 2013 at 3:33 | comment | added | Noah Schweber | I don't particularly mind, but I'm curious: why the downvote? | |
| Jun 5, 2013 at 0:30 | comment | added | The User | Indeed, it does not matter, but I was a bit surprised about your words “no finite description” for a recursively enumerable set. | |
| Jun 5, 2013 at 0:00 | comment | added | Noah Schweber | Okay, then take the set of (codes for) Diophantine equations with infinitely many solutions, or the set of (codes for) recursive subtrees of $\omega^{<\omega}$ which are well-founded; or the real coding the true theory of arithmetic. It really doesn't matter. | |
| Jun 4, 2013 at 22:39 | comment | added | The User | Well, “description by an algorithm” is ambiguous. But I think we all know what we mean and agree that recursively enumerable sets are very “constructive”, but you might want to insist on recursive sets, which are even more “constructive”. | |
| Jun 4, 2013 at 22:22 | comment | added | François G. Dorais | User, I would have agreed with you if you hadn't added the (deliberately?) ambiguous phrase "algorithmic description" since Noah's point is that this is a set that has no algorithmic decision procedure. | |
| Jun 4, 2013 at 21:54 | comment | added | Noah Schweber | Also, that seems to be the meaning of "finite description" most consistent with the OP's question (which is, however, extremely vague). | |
| Jun 4, 2013 at 21:53 | comment | added | Noah Schweber | I meant an algorithm; I'm just saying $Z$ is not recursive. My point is just that we can find sets with "somewhat nice" descriptions which provably have no "nice" descriptions, for almost any meanings of "somewhat nice" and "nice." :P | |
| Jun 4, 2013 at 21:52 | history | edited | Noah Schweber | CC BY-SA 3.0 |
added 262 characters in body
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| Jun 4, 2013 at 21:50 | comment | added | The User | What do you mean by a “finite description”?$Z$ is recursively enumerable, it has a finite, algorithmic description, there are very few notions that are even more constructive. | |
| Jun 4, 2013 at 21:43 | history | answered | Noah Schweber | CC BY-SA 3.0 |