Timeline for What does "can almost be proven in PA" mean regarding Theorem 2 of Timothy Chow's expository article, "The Consistency of Arithmetic"?
Current License: CC BY-SA 4.0
11 events
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| Jul 20, 2019 at 15:06 | comment | added | Thomas Benjamin | @Jan-ChristophSchlage-Puchta: Thanks. Very interesting. As regards "which theory of arithmetic is needed to go the other way round", is there in fact such a theory, or would that be considered an open question? | |
| Jul 19, 2019 at 16:10 | comment | added | Jan-Christoph Schlage-Puchta | @Noah Schweber: It seems that T1 implies T1' and T1' implies T2 over RCA, so the real question which theory of arithmetic is needed to go the other way round? Induction on large countable ordinals implies the existence of fast growing functions, so I would assume that every "reasonable" theory of arithmetic can only prove T1 up to a certain point. If you can prove this and find a good theory that proves T1', which should also be possible, as PA almost proves it, then you have a precise mathematical meaning of "strength". | |
| Jul 16, 2019 at 23:36 | comment | added | Noah Schweber | @ThomasBenjamin It's pretty obvious that over any reasonable base theory T1 implies T1' and T1' implies T2 - "strength" seems to be used here in the simplest possible way. I really think you're making things much more complicated than they actually are. | |
| Jul 16, 2019 at 21:45 | comment | added | Thomas Benjamin | @EmilJeřábek: As opposed to $ZFC$ + 'Axiom A' has greater consistency strength than $ZFC$ + 'Axiom B'? So let me ask you a question, Prof. Jerabek, what do you think Prof. Chow means by Theorem $1^{'}$ being "intermediate in strength between Theorem 1 and Theorem 2"? How should "strength" be correctly interpreted here? What is its precise mathematical definition? | |
| Jul 16, 2019 at 19:43 | answer | added | Noah Schweber | timeline score: 5 | |
| Jul 16, 2019 at 19:39 | comment | added | Emil Jeřábek | Is there an actual mathematical question somewhere? “Is this thing more infinitary than that thing” certainly isn’t one. | |
| Jul 16, 2019 at 19:20 | comment | added | Thomas Benjamin | Why the downvote? | |
| Jul 16, 2019 at 19:20 | review | Close votes | |||
| Jul 19, 2019 at 16:10 | |||||
| Jul 16, 2019 at 19:04 | history | edited | LSpice | CC BY-SA 4.0 |
Link to paper and questions; proofreading and TeX
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| Jul 16, 2019 at 18:59 | comment | added | kodlu | @TimothyChow is a regular contributor to MO. He might read your question, I suppose. | |
| Jul 16, 2019 at 18:50 | history | asked | Thomas Benjamin | CC BY-SA 4.0 |