Timeline for Does every set admit a ring structure or a field structure?
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jul 13, 2023 at 22:44 | comment | added | Asaf Karagila♦ | Since you're listing algebraic structures, left-cancellative magma is enough for the proof to get choice. | |
| Jul 13, 2023 at 22:17 | history | edited | Joel David Hamkins | CC BY-SA 4.0 |
updated with abelian group structure
|
| Jul 12, 2023 at 19:22 | comment | added | Peter LeFanu Lumsdaine | @KevinCasto: I think both versions are enlightening, in slightly different ways. As Joel says, Löwenheim–Skolem fits it most clearly into a big general picture. On the other hand your example (and similar ones for other theories) give concrete structures, algebraically natural and tractable, with the invocation of choice confined to the construction of a bijection with the original set. | |
| Jul 12, 2023 at 14:02 | comment | added | Lee Mosher | I would also quibble with the statement that Lowenheim-Skolem is "sophisticated". It's a basic result which, I suspect, is presented in any book/course on model theory (I learned it as an undergraduate in a course using Monk's book). | |
| Jul 11, 2023 at 18:19 | comment | added | Joel David Hamkins | Kevin, thanks for the more specific construction. I agree. However, my view also is that the Löwenheim-Skolem theorem is valuable because it avoids the need for ad hoc constructions realizing different kinds of (first-order) mathematical structure in different cardinalities. It is an elementary result in model theory, but sweeping, in that it answers all such cardinality questions at once, for any desired kind of mathematical structure. It is part of the framework of my understanding of how mathematics works. | |
| Jul 11, 2023 at 18:10 | history | edited | Joel David Hamkins | CC BY-SA 4.0 |
added 96 characters in body
|
| Jul 11, 2023 at 17:57 | comment | added | Kevin Casto | Fwiw you don't need anything as sophisticated as Lowenheim-Skolem, you can just take say $\mathbb F_2[S]$, where you mean the free associative $\mathbb F_2$-algebra generated by elements of $S$. This is evidently a countable union of finite powers of $S$, so under choice has the same cardinality as $S$. | |
| Jul 11, 2023 at 17:56 | history | edited | Joel David Hamkins | CC BY-SA 4.0 |
added 11 characters in body
|
| Jul 11, 2023 at 17:50 | history | edited | Joel David Hamkins | CC BY-SA 4.0 |
added 64 characters in body
|
| Jul 11, 2023 at 17:45 | history | answered | Joel David Hamkins | CC BY-SA 4.0 |