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Joel David Hamkins

mathematics and philosophy of the infinite

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Recent Comments

Yu LI on Did Turing prove the undecidability of the halting problem?
GJ on All triangles are isosceles
Joel David Hamkins on The set-theoretical multiverse: a natural context for set theory, Japan 2009
Hugo Valadão on The set-theoretical multiverse: a natural context for set theory, Japan 2009
Alan Turing submits "On Computable Numbers" for publication - Design History Today on Alan Turing, On computable numbers

JDH on Twitter

Mathoverflow activity

Comment by Joel David Hamkins on Why is it so difficult to define constructive cardinality?
The title is ungrammatical, so I am unsure what you are asking. Also, the text of the question itself is not grammatical. Perhaps fixing these issues might help your question to have a better reception?
Comment by Joel David Hamkins on Can ZFC minus Powersets be captured in this theory?
Could you explain the formal meaning of $z=Y$ in this logic? Usually in multi-sorted logic, individuals in different sorts are never identical. Do you mean just that they have the same elements, that, is taking the cross-sort = as a defined notion? If so, then extensionality for classes would not seem to imply extensionality for […]
Comment by Joel David Hamkins on For a given theory $T$, do all theorems provable in $T$ have "short" proofs using stronger theories?
In this case, I think you should clarify that you are not talking about all theories, but rather theories capable of expressing some some arithmetic, in order to know that those consistency assertions are expressible. But also, I would point out that if you intend $\alpha$ to be transfinite, then the lose uniqueness for $T_\alpha$, […]
Comment by Joel David Hamkins on For a given theory $T$, do all theorems provable in $T$ have "short" proofs using stronger theories?
But keep in mind that we could add $P\wedge Q$, for any suitable $Q$, and so you won't escape the objection so easily.
Comment by Joel David Hamkins on For a given theory $T$, do all theorems provable in $T$ have "short" proofs using stronger theories?
Of course, if we should add any given theorem as an axiom, then it would have a very short one-line proof in this new theory, and it would do so furthermore without changing the consequences of the theory. Could you explain your question in light of that observation?
Answer by Joel David Hamkins for Computation Via Infinite Sums
Let me point out that one can improve your argument about the primitive recursive functions to press it into the feasible realm—we can take $S$ as the collection of polynomial-time functions. Simply run your argument for the primitive recursive functions, except take $p_i(n)$ to run the computations for only $|n|$ steps instead of $n$ steps, […]
Comment by Joel David Hamkins on Measuring subsets of a cardinal definable in infinitary logic
Well, if you've killed the Mahloness of $\kappa$, that shows how much it has come down. But there are softer ways to do it (and there is the whole killing-them-softly program, by which one seeks to destroy a large cardinal property, but as little as possible). The general picture I have is that weak compactness […]
Answer by Joel David Hamkins for Measuring subsets of a cardinal definable in infinitary logic
The answer is negative, because we can destroy the weak compactness of $\kappa$ by forcing that adds no small sets, and so the old measures will still work. That is, assume $\kappa$ is weakly compact in $V$, and let $\mathbb{P}$ be the forcing to shoot a club through the non-inaccessible cardinals beow $\kappa$. If $C$ […]

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