Timeline for Can the Burgess-Hazen analysis of Predicative Arithmetic be extended to Transfinite Types?
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| May 20, 2019 at 4:50 | comment | added | user21820 | In particular, there is no philosophical issue with induction, but rather related principles that depend on LEM for unbounded quantification, such as the well-ordering principle. @UlrikBuchholtz: I'm interested to hear your opinion on my view as well. | |
| May 20, 2019 at 4:48 | comment | added | user21820 | @KeshavSrinivasan: It seems to me that skepticism in induction based on the view that the naturals are not a complete totality is actually ill-founded. That skepticism instead implies that we should not simply accept LEM for unbounded quantification. There are then two possible underlying logics that we may switch to, namely intuitionistic logic or 3-valued logic. In both cases, we can still justify having the rule ( ( A ⊢ B ; B ⊢ A ) ⊢ A∨¬A ) for any Σ1-sentence A and Π1-sentence B, and importantly we can justify the induction rule ( Q(0) ; ( k∈N ⊢ ( Q(k) ⊢ Q(k+1) ) ) ⊢ ∀k∈N ( Q(k) ) ). | |
| S May 31, 2018 at 15:13 | history | suggested | Ingo Blechschmidt | CC BY-SA 4.0 |
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| May 31, 2018 at 14:06 | review | Suggested edits | |||
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| Dec 2, 2013 at 17:47 | comment | added | Keshav Srinivasan | If it's true, that would mean that that theory has ordinal $\omega^7$, and then in the same way we can get levels for all the ordinals less than $\omega^{11}$ and so on, basically all the ordinals less than $\omega^\omega$, the proof-theoretic ordinal of PRA. That would be remarkable. | |
| Dec 2, 2013 at 17:38 | comment | added | Keshav Srinivasan | I'm not familiar with the "polymorphic lambda calculus", so tell me, does the result in that paper imply that EFA with predicative second-order logic with levels up to $\omega^3$ proves that all functions in $\mathscr E_{7}$ are total? That would be a very useful result if true. | |
| Dec 2, 2013 at 17:26 | comment | added | Keshav Srinivasan | It's true that when we talk about transfinite ordinals, we're going beyond finitistic reasoning. But that's perfectly acceptable: we're trying characterize a given philosophy of finitism from our perspective as Platonists. We're not trying to develop a characterization that finitists themselves would accept. What Burgess and Hazen are doing is developing a second-order system that would be unacceptable to strict finitists, but the goal is for each of the first-order consequences of that system to be acceptable to strict finitists. It's similar to what Feferman did with $ATR_0$. | |
| Dec 2, 2013 at 17:17 | comment | added | Keshav Srinivasan | What do you mean by "basic finitism"? And yes, there are different kind's of finitism. Tait's notion of finitism amounts to PRA, Kreisel's amounts to PRA plus quantifier-free transfinite induction up to $\epsilon_0$. But the "strict finitism" of Nelson and Parsons is much more extreme than either of those. Yes, including even a single existential quantifier would be prohibited in strict finitism, because it assumes that there's an existing totality of natural numbers that we can quantify over, as opposed to a mere potential infinity. That's why there's a skepticism of induction. | |
| Dec 2, 2013 at 7:23 | history | edited | Ulrik Buchholtz | CC BY-SA 3.0 |
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| Dec 2, 2013 at 4:02 | comment | added | Ulrik Buchholtz | I'm also aware that there are various approaches to finitism; e.g., under Kreisel's analysis it comes out to be equivalent with PA! But the system FA of Feferman-Strahm is fairly conservative: the logic is restricted to positive existential quantification over N. Maybe you would prefer a quantifier free presentation. In any case, with your proposal you run into the well-known problem with analyses of (any kind of) finitism that you want to go beyond the finite levels (!). Using unfolding avoids that quandary. | |
| Dec 2, 2013 at 3:56 | comment | added | Ulrik Buchholtz | First, yes I'm aware that the Feferman-Schütte analysis of predicative concerns predicativity given the natural numbers. The unfolding of NFA is one way to approach that, and Feferman proposed that it should also be able to capture other notions of predicative closure, for instance of basic finitism (and in my dissertation, I study the unfolding of ID$_1$ which can model the predicative closure of one positive arithmetical inductive definition). | |
| Dec 1, 2013 at 23:16 | comment | added | Keshav Srinivasan | Also, you're talking about finitism, which is what people like Kreisel and Tait talk about. But Edward Nelson and Parsons are talking about a more extreme version which Nelson calls "strict finitism" and which critics call "ultrafinitism" (although the term ultrafinitism more properly refers to someone like Essenin-Volpin who believes that there are only finitely many natural numbers). Unlike finitists, strict finitists don't accept mathematical induction, because they view it as impredicative. Are the papers you're discussing about finitism or strict finitism? | |
| Dec 1, 2013 at 22:55 | comment | added | Keshav Srinivasan | Are you aware that Feferman, Schutte, and Weyl are concerned with a different notion of predicativity than the one that Nelson and Parsons are dealing with? Feferman et al. are talking about "predicative given the natural numbers", i.e. we treat the set of natural numbers as a completed totality, but then we proceed predicatively after that. Nelson and Parsons are treating the natural numbers as a potential infinity, so they're just concerned with "predicativity", not "predicativity given the natural numbers". | |
| Dec 1, 2013 at 19:42 | history | answered | Ulrik Buchholtz | CC BY-SA 3.0 |