Joel David Hamkins

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Tag Archives: Science News

Quoted in Science News

Posted on August 31, 2003 by Joel David Hamkins

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I was quoted briefly in Infinite Wisdom: A new approach to one of mathematics’ most notorious problems, Science News, by Erica Klarrreich, August 30, 2003, in an article about Woodin’s attempted solution of the continuum hypothesis.

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Posted in Asides | Tagged CH, in the news, Science News, W. Hugh Woodin | Leave a reply

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Comment by Joel David Hamkins on Are the vertical sections of the Ackermann function primitive recursive?
Thank you very much for this soft answer, which is how I shall prefer to think about it. What I find curious about the situation is that this fact shows the typical emphasis on the diagonal is somewhat misplaced. One proves the horizontal sections are cofinal in PR by induction on the PR definitions, and […]
Comment by Joel David Hamkins on Are the vertical sections of the Ackermann function primitive recursive?
And do you mean $A^3$ for my preferred version, or for the Ackermann–Péter function?
Comment by Joel David Hamkins on Are the vertical sections of the Ackermann function primitive recursive?
Great! But can you post a proof? But also, in that case, why do we talk about the diagonal, rather than just $A^3$?
Are the vertical sections of the Ackermann function primitive recursive?
The Ackermann function $A(m,n)$ is a binary function on the natural numbers defined by a certain double recursion, famous for exhibiting extremely fast-growing behavior. One finds various slightly different formulations of the Ackermann function, with slightly different initial conditions. I prefer the following version: \begin{eqnarray*} A(0,n) &=& n+1 \\ A(m,0) &=& 1, \quad\text{ except for […]
Comment by Joel David Hamkins on Is this notion of finiteness closed under unions?
I think one might hope to prove that the finite union of amorphous sets is psuedo-finite, using the same model theoretic arguments that Noah mentioned.
Comment by Joel David Hamkins on Every mathematician has only a few tricks
Unfortunately, it seems not to be a good question, because it seems we won't learn much from the answers.
Comment by Joel David Hamkins on If we have a class like $L$ but allowing a set number of unbounded quantifiers, is it strict superset of $L$?
If X is transitive, the the OP's Def(X) is transitive, for the same reason as in L. Every element of X gets added, and also some subsets.
Answer by Joel David Hamkins for If we have a class like $L$ but allowing a set number of unbounded quantifiers, is it strict superset of $L$?
Every $L^{\Sigma_n}$, for $n\geq 2$ will be the same as HOD, the class of hereditarily ordinal-definable sets. This is a consequence of the Myhill-Scott theorem, which asserts that if you form the constructible universe using second-order logic (which means you allow quantifiers over subsets of $X$ only in your set-up), then you get exactly the […]

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Generic logical semantics of justifications.
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Recursive Reducts of PA III
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