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Joel David Hamkins

mathematics and philosophy of the infinite

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The Book of Infinity

Recent Comments

Joel David Hamkins on The global choice principle in Gödel-Bernays set theory
Nick DiBella on The global choice principle in Gödel-Bernays set theory
Joel David Hamkins on The Math Tea argument: must there be numbers we can neither describe nor define? Barcelona March 2023
Peter Gerdes on The Math Tea argument: must there be numbers we can neither describe nor define? Barcelona March 2023
Peter Gerdes on The Math Tea argument: must there be numbers we can neither describe nor define? Barcelona March 2023

JDH on Twitter

Mathoverflow activity

Comment by Joel David Hamkins on Are all constructible from below sets parameter free definable?
So much for intended models, I say. People talk about intended models only when they haven't managed to characterise the models by the theory. It is an admission of failure...
Comment by Joel David Hamkins on does this relation associated with a poset have a name?
@მამუკაჯიბლაძე You are right. I retract my comment (and have deleted).
Comment by Joel David Hamkins on Complexity of |a| < |b| for ordinal notations?
Why do you say "fully $\Pi^1_1$"? The natural upper bound seems to be $\Sigma^1_1$, since the order coded by $a$ will be shorter than that of $b$ if and only if there exists an order-embeding of the $a$ order with a bounded suborder of the $b$ order.
Comment by Joel David Hamkins on Are all constructible from below sets parameter free definable?
That's right, it proves the inference rule you mention, but it doesn't prove that every object is definable. Those are not the same, for the reasons I explain in the first half of my answer.
Comment by Joel David Hamkins on Are all constructible from below sets parameter free definable?
Your theory includes V=L, which implies that there is a definable well-ordering, and so yes, the theory proves the rule, because for every statement, if it has a counterexample, it has a definable counterexample.
Comment by Joel David Hamkins on Higher-order oracle computation of reals and axiom of constructibility
The question of which reals have which order depends a lot on how the ordinals are coded by natural numbers. You haven't specified the hierarchy sufficiently to determine a precise hierarchy. For example, any given real can be made to have order $\omega+1$ or whatever, if you use a coding of this ordinal that enables […]
Answer by Joel David Hamkins for Are all constructible from below sets parameter free definable?
There are two issues with your question. First, your statement "in other words" is not correct, since there are theories whose models have the property that whenever a statement holds of every parameter-free definable element, then it holds of every element, but not all models are pointwise definable. Theorem. In any model of ZFC+V=HOD, if […]
Comment by Joel David Hamkins on Extending a first-order deductive system with satisfaction relation
Yes, that's right. The first approach is what is adopted in effect in any standard logic text. For an account of definitional expansion, see en.wikipedia.org/wiki/Extension_by_definitions and the references there.

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