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See also

The full text of each of my articles listed here is available in pdf and other formats—just follow the links provided to the math arxiv for preprints or to the journal itself for the published version, if this is available.

- Large cardinals need not be large in HOD
- Strongly uplifting cardinals and the boldface resurrection axioms
- Satisfaction is not absolute
- The foundation axiom and elementary self-embeddings of the universe
- Resurrection axioms and uplifting cardinals
- Superstrong and other large cardinals are never Laver indestructible
- The least weakly compact cardinal can be unfoldable, weakly measurable and nearly $\theta$-supercompact
- Algebraicity and implicit definability in set theory
- Transfinite game values in infinite chess
- A multiverse perspective on the axiom of constructiblity
- A question for the mathematics oracle
At the Workshop on Infinity and Truth in Singapore last year, we had a special session in which the speakers were asked to imagine that they had been granted an audience with an all-knowing mathematical oracle, given the opportunity to ask …

- Moving up and down in the generic multiverse
- Structural connections between a forcing class and its modal logic
- Every countable model of set theory embeds into its own constructible universe
- Well-founded Boolean ultrapowers as large cardinal embeddings
- Singular cardinals and strong extenders
- Is the dream solution of the continuum hypothesis attainable?
- The mate-in-n problem of infinite chess is decidable
- Inner models with large cardinal features usually obtained by forcing
- What is the theory ZFC without power set?
- The hierarchy of equivalence relations on the natural numbers under computable reducibility
- Set-theoretic geology
- The rigid relation principle, a new weak choice principle
- Generalizations of the Kunen inconsistency
- Pointwise definable models of set theory
- Effective Mathemematics of the Uncountable
- Infinite time Turing machines and an application to the hierarchy of equivalence relations on the reals
- The set-theoretical multiverse
- Infinite time decidable equivalence relation theory
- The set-theoretical multiverse: a natural context for set theory, Japan 2009
- A natural model of the multiverse axioms
- Indestructible strong unfoldability
- Some second order set theory
- Post’s problem for ordinal register machines: an explicit approach
- Degrees of rigidity for Souslin trees
- Tall cardinals
- The proper and semi-proper forcing axioms for forcing notions that preserve $\aleph_2$ or $\aleph_3$
- Infinite time computable model theory
- Changing the heights of automorphism towers by forcing with Souslin trees over $L$
- The ground axiom is consistent with $V\ne{\rm HOD}$
- The modal logic of forcing
- Large cardinals with few measures
- A survey of infinite time Turing machines
- The complexity of quickly decidable ORM-decidable sets
- Post’s Problem for Ordinal Register Machines
- The halting problem is decidable on a set of asymptotic probability one
- Diamond (on the regulars) can fail at any strongly unfoldable cardinal
- ${\rm P}\neq{\rm NP}\cap\textrm{co-}{\rm NP}$ for infinite time Turing machines
- The necessary maximality principle for c.c.c. forcing is equiconsistent with a weakly compact cardinal
- The Ground Axiom
- Infinitary computability with infinite time Turing machines
- Book review of G. Tourlakis, Lectures in Logic and Set Theory I & II
- Supertask computation
- Extensions with the approximation and cover properties have no new large cardinals
- ${\rm P}^f\neq {\rm NP}^f$ for almost all $f$
- Exactly controlling the non-supercompact strongly compact cardinals
- A simple maximality principle
- How tall is the automorphism tower of a group?
- Indestructibility and the level-by-level agreement between strong compactness and supercompactness
- Post’s problem for supertasks has both positive and negative solutions
- Infinite time Turing machines
- New inconsistencies in infinite utilitarianism
- Indestructible weakly compact cardinals and the necessity of supercompactness for certain proof schemata
- Unfoldable cardinals and the GCH
- Gap forcing
- Infinite time Turing machines with only one tape
- The wholeness axioms and $V=\rm HOD$
- Infinite time Turing machines
- The lottery preparation
- Changing the heights of automorphism towers
- Small forcing creates neither strong nor Woodin cardinals
- With infinite utility, more needn’t be better
- Utilitarianism in infinite worlds
- Book review of The Higher Infinite, Akihiro Kanamori
- Gap forcing: generalizing the Lévy-Solovay theorem
- Universal indestructibility
- Superdestructibility: a dual to Laver’s indestructibility
- Small forcing makes any cardinal superdestructible
- Destruction or preservation as you like it
- Every group has a terminating transfinite automorphism tower
- Book review of Notes on Set Theory, Moschovakis
- Canonical seeds and Prikry trees
- Fragile measurability
- Lifting and extending measures; fragile measurability
- A class of strong diamond principles
- Pointwise definable models of set theory, extended abstract, Oberwolfach 2011

It seems that infinite time Turing machines are fully deterministic except at the limit stages when cells on the tape are updated to the lim sup of a countable infinity of values that occurred in the cells at non-limit stages. What kind of hardware would we have to imagine added to an ordinary Turing machine to complete the analysis that determines this lim sup for each cell? In other words, how could a machine be built to discover whether the series of cell contents converges to 1 or oscillates continually between 0 and 1?

The limit cell values are deterministic in the sense that the value at the limit is logically determined by earlier values, and it couldn’t be a different value (contrast with non-deterministic computation in finite time, where there is a sense in which the next step of computation is not logically determined by the previous states, since there are multiple paths of computation that all accord with the computational rules). As for the physical implementation, this is an issue for the physicists. I imagine some kind of biased magnetic cell memory, which will show a positive value at a limit if it was unboundedly often positive going in to the limit. Ultimately, of course, what I am interested in is the purely mathematical theory of the resulting class of functions, and so the issue of physical implementation doesn’t actually matter.