Timeline for Why is there a need for ordinal analysis?
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24 events
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| Dec 23, 2015 at 17:21 | comment | added | Thomas Benjamin | @NoahSchweber: But is is relevant to the question, "How do you know the natural numbers satisfy the Peano Axioms?". We have a 'semblance' of a model for the Peano Axioms--if Nelson was successful, then contentual number theory (the standard model) would exhibit the same inconsistency, right? But the inconsistency proof for $PA$ failed, and $PA$ is safe (for now...). | |
| Dec 22, 2015 at 18:03 | comment | added | Noah Schweber | @ThomasBenjamin Yes, this was found very quickly (for those interested, see the comment thread at golem.ph.utexas.edu/category/2011/09/…) but I don't see how that's relevant. I don't think the error being discovered changed Nelson's beliefs about PA significantly (although obviously I don't know this for certain - more accurately, I know of no evidence that it did). | |
| Dec 22, 2015 at 17:39 | comment | added | Thomas Benjamin | @NoahSchweber: Are you aware that Terry Tao found a flaw in Nelson's 'proof' of the inconsistency of $PA$? | |
| Dec 9, 2015 at 6:49 | comment | added | Thomas Benjamin | Note: You might find Jeremy Avigad's paper "Ordinal Analysis Without Proofs" of particular interest because, as he states in his abstract, "An approach to ordinal analysis is presented which is finitary, but highlights the semantic content of the theories under consideration, rather than the syntactic structure of their proofs...." | |
| Dec 9, 2015 at 6:33 | comment | added | Thomas Benjamin | (cont.) $PRA$). Do you? I ask because there are those (Richard Zach being one of them, particularly in his paper "The Practice of Finitism: Epsilon Calculus and and Consistency Proofs in Hilbert's Program (arXiv: math0102819v1 [mathLO]), in particular section 3.2 titled "The Consistency Proof For Primitive Recursive Arithmetic.") who do not. If I were to give you an answer along these lines, would you find such an answer particularly helpful? | |
| Dec 9, 2015 at 6:11 | comment | added | Thomas Benjamin | @user3730940: Regarding your question "Why is Gentzen's proof necessary?", is there any deeper reason for your question other than '$PA$ is a first-order theory. By Goedel's completeness theorem, every first-order theory $T$ is consistent iff it has a model. $PA$ has a model' should suffice? The deeper reason I am concerned with is the fact that many (Andreas Blass being one of them) hold to the view that $\epsilon_0$-induction, being an example of 'transfinite induction', is ostensibly 'non-finitary' (similarly for the use of $\omega^{\omega}$-induction to prove the consistency of | |
| Dec 4, 2015 at 18:37 | comment | added | Asaf Karagila♦ | math.stackexchange.com/questions/1557664/… | |
| Dec 4, 2015 at 16:25 | answer | added | Henry Towsner | timeline score: 13 | |
| Dec 3, 2015 at 21:36 | comment | added | Timothy Chow | Related: mathoverflow.net/questions/66121/is-pa-consistent-do-we-know-it | |
| Dec 3, 2015 at 20:32 | comment | added | Noah Schweber | @ThomasBenjamin Also Edward Nelson, if I understand his views correctly. | |
| Dec 3, 2015 at 16:06 | comment | added | Steven Landsburg | @ThomasBenjamin: I believe Voevodsky is an example of someone who both has doubts and is much smarter than me. (Though, per Qiaochu's comment, it would be clearer to say that his doubts concern the consistency of Peano arithmetic and hence the existence of the natural numbers). | |
| Dec 3, 2015 at 13:16 | vote | accept | user3730940 | ||
| Dec 3, 2015 at 11:20 | answer | added | Andreas Blass | timeline score: 26 | |
| Dec 3, 2015 at 10:06 | comment | added | Thomas Benjamin | @user3730940: Also, is the "model existence theorem" the following form of Goedel's Completeness Theorem: For any first-order theory $T$, $T$ is consistent iff $T$ has a model? I think your question, "Why is Gentzen's proof necessary?", is a good research-level question from a mathematical- philosophy point of view because the answer would help clarify the conceptual presuppositions surrounding the proof. | |
| Dec 3, 2015 at 9:52 | comment | added | Thomas Benjamin | @StevenLandsburg: "...but some people have doubts..." Who, exactly? | |
| Dec 3, 2015 at 8:01 | comment | added | Mauro ALLEGRANZA | It may help if you locate Gentzen's proof in the context of "foundational debate" : see Hilbert's program and The consistency of arithmetic and analysis. If you are interested to the question : "how may I prove that there exists a "structure" satisfying Peano's axioms", then you cannot invoke the model existence theorem, because it presuppose the consistency of the theory, and this is "hard" to prove (if we do not assume the existence of the sought structure). | |
| Dec 3, 2015 at 6:38 | review | Close votes | |||
| Dec 4, 2015 at 2:29 | |||||
| Dec 3, 2015 at 6:31 | comment | added | Qiaochu Yuan | I think it's reasonable to take, say, "the unique model of second-order Peano arithmetic" as a definition of the natural numbers; hence they satisfy the first-order Peano arithmetic axioms by definition. The problem is then: how do you know the natural numbers exist? | |
| Dec 3, 2015 at 5:54 | comment | added | Steven Landsburg | How do you know the natural numbers satisfy the Peano axioms? (Personally I have no doubt about this, but some people do have doubts, and some of those are much smarter than I am.) | |
| S Dec 3, 2015 at 4:35 | history | suggested | user82740 |
I added some related tags.
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| Dec 3, 2015 at 4:34 | review | Suggested edits | |||
| S Dec 3, 2015 at 4:35 | |||||
| Dec 3, 2015 at 4:08 | history | edited | user3730940 | CC BY-SA 3.0 |
added 44 characters in body
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| Dec 3, 2015 at 3:56 | review | First posts | |||
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| Dec 3, 2015 at 3:53 | history | asked | user3730940 | CC BY-SA 3.0 |