Timeline for Is there a compendium of the consistency strength between the most important formal theories?
Current License: CC BY-SA 3.0
17 events
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| Sep 6, 2023 at 7:10 | comment | added | Jonathan Julian | @plm Thanks, I was an undergrad when I posted the question. The last part remains true, if you have a link/reference to these classifications, it would be good to have them in one place for the rest of the internet to find it. | |
| Sep 1, 2023 at 18:05 | comment | added | plm | This is a nice question and i'm disappointed that we only see the well known set theory results. A classification-by-consistency of subsystems of second-order arithmetic is also easy to find. But then focus should be on drawing less well-exposed types of theories: how do algebraic theories fare compared to arithmetic -Hilbert's tenth pb, representability, is relevant ? Theories of orderings ? Of special types of graphs ? Other combinatorial structures. Wrt extensions, interpretations, consistency, or representation. FO theories are type-0 languages. We'd like to see invariants, a big picture. | |
| Mar 12, 2021 at 15:33 | answer | added | Vincent R.B. Blazy | timeline score: 3 | |
| Feb 10, 2015 at 7:05 | vote | accept | Jonathan Julian | ||
| Feb 10, 2015 at 7:05 | history | edited | Jonathan Julian | CC BY-SA 3.0 |
added a final note
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| Jan 24, 2015 at 5:01 | history | edited | Mohammad Golshani |
edited tags
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| Jan 24, 2015 at 4:55 | answer | added | Mohammad Golshani | timeline score: 4 | |
| Jan 22, 2015 at 11:51 | comment | added | Emil Jeřábek | @Joel: Sure. But on the level of generality of the question, a formal system may be anything from large cardinal hypotheses down to propositional logic, so I wanted to know what is the OP's perspective. | |
| Jan 22, 2015 at 2:00 | comment | added | Andrés E. Caicedo | For subsystems of second order arithmetic, there is the Reverse Mathematics Zoo. rmzoo.math.uconn.edu | |
| Jan 22, 2015 at 1:45 | comment | added | Joel David Hamkins | @EmilJeřábek I suppose the phrase, "strong theories like ZFC" is a matter of perspective! In the large cardinal hierarchy, where of course consistency strength issues loom large, one views ZFC as a weak theory. :-) | |
| Jan 22, 2015 at 1:43 | answer | added | Joel David Hamkins | timeline score: 19 | |
| Jan 21, 2015 at 21:41 | comment | added | Jonathan Julian | Ok, I could take two points of view here to answer my question and both would be fine for me: The first one is to take as a metatheory the weakest system you want, maybe PRA. The second one is just to take a relativistic point of view and ask for a compendium of the form: In metatheory T, the consistency of X implies the consistency of Y. | |
| Jan 21, 2015 at 21:32 | comment | added | Emil Jeřábek | Once again: in normal mathematics, the axiomatic second order arithmetic is provably consistent, and of course Robinson arithmetic is, hence it doesn’t make much sense to consider relative consistency of these theories, unless you severely restrict the metatheory. So, what is your metatheory? | |
| Jan 21, 2015 at 21:25 | comment | added | Jonathan Julian | Well, I know there are some random systems out there that very few people study. ZFC and its extensions are welcomed but not just them. For example NBG and MK Set Theories can be included and as examples of somewhat weak theories that should be included might be Second Order Arithmetic and Robinson Arithmetic. | |
| Jan 21, 2015 at 21:17 | comment | added | Emil Jeřábek | The vast majority of systems encountered in the wild can be outright proved consistent, which makes the question moot. Does this mean you are only interested in strong theories like ZFC and its extensions, or do you have in mind provability of the implication in some weak metatheory? | |
| Jan 21, 2015 at 18:33 | review | First posts | |||
| Jan 21, 2015 at 18:39 | |||||
| Jan 21, 2015 at 18:28 | history | asked | Jonathan Julian | CC BY-SA 3.0 |