Joel David Hamkins

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Tag Archives: NWO

Modal logics in set theory, NWO grants, 2006 – 2008

Posted on April 30, 2006 by Joel David Hamkins

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Modal logics in set theory, (with Benedikt Löwe), Nederlandse Organisatie voor Wetenschappelijk (B 62-619), 2006-2008.

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Posted in Grants and Awards | Tagged Amsterdam, NWO | Leave a reply

Recent Comments

The surprising strength of reflection in second-order set theory with abundant urelements, CUNY Set Theory seminar, April 2022 | Joel David Hamkins on Reflection in second-order set theory with abundant urelements bi-interprets a supercompact cardinal
Reflection in second-order set theory with abundant urelements bi-interprets a supercompact cardinal | Joel David Hamkins on The surprising strength of reflection in second-order set theory with abundant urelements, CUNY Set Theory seminar, April 2022
Bokai Yao on The surprising strength of reflection in second-order set theory with abundant urelements, CUNY Set Theory seminar, April 2022
Jason Chen on The surprising strength of reflection in second-order set theory with abundant urelements, CUNY Set Theory seminar, April 2022
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Comment by Joel David Hamkins on Are there interesting examples of theorems proved using ‘height’ extensions?
In our paper, we prove essentially that all of the most common urelement theories are bi-interpretable with pure set theories. For example, ZFC with ZFCU + Ord many urelements or ℝ many, KM with KMU+omega many atoms, etc. A many for any class A of pure sets. We take this to explain on structuralist grounds […]
Answer by Joel David Hamkins for Are there interesting examples of theorems proved using ‘height’ extensions?
Here is another example. The maximality principle in forcing is the scheme asserting of every statement $\varphi$ in the language of set theory that if there is forcing extension $V[G]$ of the set-theoretic universe $V$ for which all further forcing extensions $V[G][H]$ satisfy $\varphi$, then $\varphi$ was already true in the original universe $V$. The […]
Answer by Joel David Hamkins for Are there interesting examples of theorems proved using ‘height’ extensions?
Here is another instance, which appears in my recent paper with Bokai Yao on second-order reflection in the context of KMU with abundant urelements. Joel David Hamkins and Bokai Yao, Reflection in second-order set theory with abundant urelemets bi-interprets a supercompact cardinal, 2022, arXiv:2204.09766. The following theorem is an immediate consequence of the main theorem. […]
Comment by Joel David Hamkins on Ordering of large cardinals by cardinality
To my knowledge all those instances are still open.
Comment by Joel David Hamkins on Church-Turing tests and quasi-computational models
My view, in keeping with my general attitude about the relation between mathematics and philosophy, is that one should undertake the philosophical task (which is how I had understood your question) in light of the related mathematical analysis. To analyze philosophically whether a given notion should count as computational, one should first know how it […]
Comment by Joel David Hamkins on Church-Turing tests and quasi-computational models
Indeed, these hierarchies are intensely studied, and one gains much more refined information from them than just a yes/no answer as to whether the concept should count as computational.
Comment by Joel David Hamkins on Church-Turing tests and quasi-computational models
@SamHopkins Yes, my answer is that one gains such insight by fitting it into the hierarchies I mention. This is how these hierarchies are often used.
Answer by Joel David Hamkins for Church-Turing tests and quasi-computational models
Computability theory overflows with hierarchies of computability notions, which allow us precisely to compare the computational strength of diverse notions. We have the hierarchy of Turing degrees, which measures the strength of relative computability by oracles. We have the hierarchy of complexity theory and the complexity zoo, which measures the strength of diverse resource-limited computation. […]

New York Logic

Title TBA
Resource Sharing Linear Logic, I
Sub-Kripkean epistemic models I
Generic logical semantics of justifications III
Generic logical semantics of justifications II
Generic logical semantics of justifications.
Cut Elimination
Functors and infinitary interpretations of structures
Recursive Reducts of PA III
Kripke completeness of strictly positive modal logics and related problems

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