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bio website jdh.hamkins.org
location New York City
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visits member for 4 years, 5 months
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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.


11h
comment Formal definition of function: equality
See related mathoverflow.net/a/30397/1946
21h
comment Generic Ultrapower as a Class
I agree with this, and in any case, having a predicate for the ground model, as Jech does, also works fine, and avoids the need for localization. Nevertheless, to my way of thinking, the real answer to the question, "How does $M[G]$ talk about $M$?" is the ground model definability theorem.
1d
comment Generic Ultrapower as a Class
@Ashutosh, I added some references. A complete proof is found in Laver's paper, and also in Reitz's work and elsewhere.
1d
revised Generic Ultrapower as a Class
added 395 characters in body
1d
answered Generic Ultrapower as a Class
1d
comment infinitary logic and partial fixed point logic
@NoahS, the advantage of your method, even though it uses infinitely many variables, is that it works in any language (assuming $=$ is available), whereas I needed something special about the language.
1d
comment infinitary logic and partial fixed point logic
If one objects that the language here is infinite, then we may use instead the language with one constant $0$ and a successor operation $S$, defining the class of finite models $M$ for which the least $n$ for which $S^n(0)=0$ is in $X$.
1d
comment infinitary logic and partial fixed point logic
We could also insist that the models have only two elements, and so what we really have is a partition of the natural numbers into at most two pieces, depending on how the constants are interpreted.
1d
answered infinitary logic and partial fixed point logic
2d
comment infinitary logic and partial fixed point logic
Are you interested only in finite models? And do you mean $L_{\infty,\omega}$ rather than $L_{\omega,\infty}$?
Apr
11
comment Is there an intuitionistic generalized boolean algebra (of Stone)?
The topless Boolean algebras (a Boolean algebra with top element removed) arise in the characterization of the modal logic of forcing for certain forcing classes, such as in arxiv.org/abs/1207.5841. (But that use is purely classical, rather than intuistionistic.)
Apr
11
comment Does a BCL algebra define a partial order?
You seem to have an error in (2); do you mean $y*x=0$?
Apr
10
comment Comparing numbers $a \uparrow^b c$ and $d \uparrow^e f$
See my related question on math.SE math.stackexchange.com/questions/72646/…, concerning in effect iterations of such expressions.
Apr
10
comment Forcing with Nontransitive Models
If $H_\theta\models\varphi(x,\tau_G)$, with $\tau\in M$ and $\tau_G\in H_\theta$, then there is an antichain of possible values of $\tau$, and for each possible $y\in H_\theta$ that it might be, we have an $x$ for which $H_\theta\models\varphi(x,y)$. Now, by mixing $\check x$ along the antichain, we find a name $\dot x$ such that $H_\theta\models\varphi(\dot x_G,\tau_G)$. By elementarity $M\prec H_\theta$, there is such a name $\dot x$ inside $M$. And so $M[G]$ has the witness $\dot x_G$, which is one of the $x$'s that we mixed.
Apr
10
revised Quantifier elimination vs decidability
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Apr
10
answered Quantifier elimination vs decidability
Apr
10
revised Forcing with Nontransitive Models
added 155 characters in body
Apr
10
revised Forcing with Nontransitive Models
added 155 characters in body
Apr
10
revised Forcing with Nontransitive Models
added 155 characters in body
Apr
10
comment Forcing with Nontransitive Models
Vika, I think your claim that "The argument that $M[G]\prec H_\theta[G]$ works only in the case that $G$ is both fully $H_\theta$-generic and also $M$-generic," is not actually correct, in light of the theorem in my answer. You don't actually need that $G$ is $M$-generic for this conclusion.