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Jan
29
comment Taller models of ZFC
In your answer to Question 1, I think you meant to say only that $\alpha$ is the only possible height for a transitive model in $L(\beta)$, not that $L(\alpha)$ is the only possible model. Since $L(\beta)$ thinks $L(\alpha)$ is countable, it contains al sorts of forcing extensions of $L(\alpha)$.
Jan
14
comment Is there an uncountable Borel almost disjoint family?
Asaf's example is Noah's example when the language is a propositional language with countably many propositional atoms.
Dec
29
awarded  Enlightened
Dec
29
awarded  Good Answer
Dec
29
awarded  Nice Answer
Dec
7
comment “Let” versus “for all”
Checking the nearest dictionary, I find that (as I expected) one of the meanings of "then" is "in that case." Are you claiming that this meaning is not appropriate at the beginning of a sentence?
Dec
4
awarded  Nice Answer
Dec
4
comment Why is there a need for ordinal analysis?
I don't think Hilbert's school had what they'd consider a finitary proof of the consistency of PRA. By today's standards, "finitary" would mean "formalizable in PRA", and PRA can't prove its own consistency. As far as I know, Gentzen did not try to prove the consistency of PRA + induction for $\varepsilon_0$; rather he just used this theory to prove the consistency of PA.
Dec
4
awarded  lo.logic
Dec
3
awarded  Enlightened
Dec
3
awarded  Nice Answer
Dec
3
comment Why is there a need for ordinal analysis?
Yes, from the viewpoint of most mathematicians, first-order Peano arithmetic is obviously consistent because all its axioms are true under the standard interpretation (natural numbers with the usual addition and multiplication).
Dec
3
comment “Clubiness” of projective sets of ordinals
In view of my previous comment, it suffices to have enough determinacy to carry out, for $\Pi^1_n$ formulas, Solovay's proof that AD makes the club filter ultra. So you could check the complexity of the game that Solovay used, under the additional assumption that the target set is $\Pi^1_n$; presumably it will be be at worst a little higher than $\Pi^1_n$ in the projective hierarchy. So something in the neighborhood of $n$ Woodin cardinals would suffice. (I have no guess at the moment about whether significantly smaller cardinals will suffice; that's why this comment isn't an answer.)
Dec
3
comment “Clubiness” of projective sets of ordinals
Any $\Pi^1_n$ sentence with real parameters is absolute between $L(\mathbb R)$ and $V$ because the quantifiers in the sentence range only over reals, and $L(\mathbb R)$ contains all of the reals.
Dec
3
answered Why is there a need for ordinal analysis?
Dec
1
comment Some questions regarding an alteration of Grzegorczyk's theory of concatenation, $\operatorname{TC}$
Emil has answered Questions 1 and 2. As for Question 3, considering that the theory is true (in the intended interpretation), it remains consistent when any true sentences are added. That would apply, in particular, to any theory of "the primitive recursive functions (appropriately recast in the language of concatentation)". I'm assuming here that false axioms would not count as appropriate recasting.
Nov
30
comment Was “arithmetical translation” (coding in the Goedel sense) ever a part of Hilbert's Program?
For example, the usual definition of the Ackermann function can easily be reformulated as a higher-type primitive recursion. I'd expect the type-0-to-type-0 functions obtainable in Gödel's system to be the same as those that are provably recursive in PA.
Nov
30
comment Was “arithmetical translation” (coding in the Goedel sense) ever a part of Hilbert's Program?
The following is not guaranteed; I'm working from old memories. Gödel's Dialectica system has, at the lowest level, the natural numbers, not numerals, though I doubt it makes much difference. I don't recall there being any Gödel numbering in this paper, but if there is then I'd expect the Gödel numbers to be numbers, not another sort of entity. (That is, after all, one of the main points of Gödel numbering.) The primitive recursive functionals of finite type are that happen to map type 0 to type 0 not just the primitive recursive functions. (Continued in next comment)
Nov
27
comment Was “arithmetical translation” (coding in the Goedel sense) ever a part of Hilbert's Program?
I can't imagine a genuinely higher-type system that would be finitary in Hilbert's sense, but one could view Gödel's Dialectica paper as an attempt in that direction. The title of that paper describes the higher type system there as an extension of the finitary position. (Of course, one could argue that all sorts of non-finitary things are extensions of finitism, but I think Gödel's intention was to suggest that he has not deviated too much from finitism, while conceding that his system is not strictly finitary.)
Nov
26
comment Was “arithmetical translation” (coding in the Goedel sense) ever a part of Hilbert's Program?
I've heard that the "much less" part of my previous comment, i.e., the observation that Hilbert's finitary reasoning can be formalized in second-order (or even first-order) arithmetic, was due not to Gödel but to von Neumann. Apparently Gödel originally said that his result doesn't destroy Hilbert's program, but I don't know whether that was really his opinion or just a very junior Gödel being polite to superstar Hilbert.