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
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jul 20, 2019 at 14:59 | comment | added | Thomas Benjamin | [Note that 'in' in the statement "if one assumes that this $\omega$- rule in a primitive recursive $\omega$-rule" should be "is"--sorry for the typo.] | |
| Jul 20, 2019 at 14:46 | comment | added | Thomas Benjamin | As regards your example (since the second example seems to be a variation of the first): consider the example of an arithmetic statement $\forall$ $x$$\varphi$($x$) which is not provable in $PA$, but for each individual natural number $n$ the instance $\varphi$($n$) is provable in $PA$; since this seems an example of an $\omega$-rule, if one assumes assumes that this $\omega$-rule in a primitive recursive $\omega$-rule, though the primitive recursive $\omega$-rule cannot be proven in $PA$, is there some reason a primitive recursive $\omega$-rule cannot be deemed 'finitary'? | |
| Jul 20, 2019 at 14:28 | comment | added | Thomas Benjamin | (cont.) ...Not very infinitary at all.". Given that in an earlier paragraph he states that the list-notation for $\omega$ is "((( )))", why can there not be a finite list notation for $\epsilon_0$ (and can it be proven that there is no finite list notation for $\epsilon_0$)? | |
| Jul 20, 2019 at 14:18 | comment | added | Thomas Benjamin | (cont.) statement of Theorem 1); second, though you are entirely correct that in his answer to IamMeeoh's (matteo's) question, Prof. Chow does state that the term "ordinal" means "ordinal below $\epsilon_0$", in a later paragraph he states, "My personal opinion is that the usual ways of defining $\epsilon_0$ make it seem mind-bogglingly infinitary, but the above definition [the list notation for ordinals below $\epsilon_0$] makes it clear that any particular ordinal [below $\epsilon_0$, presumably] is just a special type of finite list of finite lists of finite lists of ...of finite lists... | |
| Jul 20, 2019 at 13:55 | comment | added | Thomas Benjamin | Thank you for your answer; I will take it under advisement (in general it is helpful but I have some questions about your examples). A couple of quibbles though: you attribute the view that the proof of the consistency of $PA$ is "infinitary" because "its proof must somehow assume 'assume the existence of infinite sets' " to me when in fact it is correctly attributable to Prof. Chow ("The alert reader will notice that the statement of Theorem 1 presupposes the concept of an arbitrary infinite sequence and hence it is not finitary"--this from the paragraph directly below his | |
| Jul 16, 2019 at 19:58 | history | edited | Noah Schweber | CC BY-SA 4.0 |
deleted 2 characters in body
|
| Jul 16, 2019 at 19:43 | history | answered | Noah Schweber | CC BY-SA 4.0 |