102,197 reputation
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bio website jdh.hamkins.org
location New York City
age
visits member for 4 years, 11 months
seen 3 hours ago

I am a professor at the City University of New York, at the College of Staten Island and the CUNY Graduate Center, living in midtown Manhattan. My main research interest lies in mathematical logic, particularly set theory, focusing on the mathematics and philosophy of the infinite. A principal concern has been the interaction of forcing and large cardinals, two central concepts in set theory. I have worked in group theory and its interaction with set theory in the automorphism tower problem, and in computability theory, particularly the infinitary theory of infinite time Turing machines. Recently, I am preoccupied with the set-theoretic multiverse, engaging with the emerging field known as the philosophy of set theory.


1d
comment Concept of synchronizability
You might be interested in the Henkin quantifier (see en.wikipedia.org/wiki/Branching_quantifier), which is an instance of simultaneity in first-order logic: for every $a$ there is $b$, and simultaneously, for every $c$ there is $d$ (that is, depending only on $c$), such that $\phi(a,b,c,d)$.
2d
revised Decidability of decidability
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2d
comment Decidability of decidability
@sure, there is no such statement; see my update. I see now that Emil made the same observation.
2d
revised Decidability of decidability
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2d
comment Decidability of decidability
Thanks, fixed! Do you agree with that point? I find it confusing to argue in those iterated consistency theories, without knowing that they are true in the standard model.
2d
revised Decidability of decidability
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2d
revised Decidability of decidability
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2d
revised Decidability of decidability
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2d
answered Decidability of decidability
2d
comment Decidability of decidability
I don't follow your remark, because Dec (A) asserts exactly that A is provable or not A is provable. Perhaps your answer would be more clear with a greater distinction between your meta theoretic consistency assumptions and the theory to which Dec is referring.
2d
comment Decidability of decidability
Your statement "a sufficiently powerful theory has to have $\infty$-undecidable statements" is not correct, since if $F$ is sufficiently powerful and consistent, then $F+\neg\text{Con}(F)$ is even more powerful, but has no $\infty$-undecidable statements, since this theory thinks that everything is provable in it, and so it has $\text{Dec}(A)$ for every $A$, even though it is actually consistent by the incompleteness theorem.
2d
comment Decidability of decidability
Regarding your final remark, you haven't said it correctly. It isn't that "a proof of $\neg\text{Dec}(A)$" implies $\text{Con}(F)$, but rather just $\neg\text{Dec} (A)$ itself implies $\text{Con}(F)$. This is obvious, since if a statement is independent, then not everything is provable, and so the theory must be consistent. And regarding your first statement, yes, I would think you care most about the background in which those iterated consistency assumptions hold. Without that, your hierarchy will trivialize at some $n$.
Oct
28
comment Decidability of decidability
But meanwhile, with $F+\neg\text{Con}(F)$ in the background, then all $\text{Dec}(A)$ come out true, and so also $\text{Dec}(\text{Dec}(A))$ and so on, trivializing the hierarchy.
Oct
28
comment Decidability of decidability
For example, if $F$ is the formal system for your proofs, and we have F+Con(F) in our background, then the second case of Dec(Dec(A)) cannot occur, since we can have no proof in F that $\neg\text{Dec}(A)$, as this provably implies Con(F), which would violate the second incompleteness theorem under our assumption of Con(F). So in this case Dec(Dec(A)) amounts to "there is a proof of Dec(A)".
Oct
28
comment Decidability of decidability
It seems to me that your question and the discussion would benefit from a greater level of precision in provability logic (e.g. see csc.villanova.edu/~japaridz/Text/prov.pdf), since the assertion Dec(A) is $\square A\vee\square\neg A$, and Dec(Dec(A)) is $\square(\square A\vee\square\neg A)\vee\square\neg(\square A\vee\square\neg A)$, and so on. Thus, your question is really about modal reasoning in the provability logic of your formal system. Which validities come out true will depend on iterated consistency assumptions in the background theory.
Oct
25
awarded  Good Answer
Oct
24
comment Reinhardt cardinals and iterability
Andres, why doesn't the usual argument show that all iterates are well-founded? Let $\xi$ be least such that $j_{0,\lambda}(\xi)$ is in the ill-founded part of some $M_\lambda$, with $\lambda$ a limit. But in some $M_\alpha$ with $0<\alpha<\lambda$, there will be born a smaller $\xi'<\xi$ with $j_{\alpha,\lambda}(\xi')$ also in the ill-founded part. This argument seems to use only ZF in the language with j, in order that we may refer in any model to the corresponding iterates of $j$.
Oct
24
comment Reinhardt cardinals and iterability
Yair, we may apply $j$ to any $j\upharpoonright V_\alpha$, and take the union to form $j_{1,2}=j(j)=\bigcup_\alpha j(j\upharpoonright V_\alpha)$. This is presumably what Mohammad means by iterating.
Oct
23
comment Are there examples of families of objects which are canonically isomorphic, but where diagrams of canonical isomorphisms don't commute?
@Dirk Yes, I meant to rotate the square about its center.
Oct
23
answered Are sums of 0-1 Pareto efficient vectors Pareto efficient?