[bibtex key=Hamkins2001:GapForcing]
Many of the most common reverse Easton iterations found in the large cardinal context, such as the Laver preparation, admit a gap at some small delta in the sense that they factor as
[bibtex key=Hamkins2001:GapForcing]
Many of the most common reverse Easton iterations found in the large cardinal context, such as the Laver preparation, admit a gap at some small delta in the sense that they factor as
[bibtex key=HamkinsSeabold2001:OneTape]
Infinite time Turing machines with only one tape are in many respects fully as powerful as their multi-tape cousins. In particular, the two models of machine give rise to the same class of decidable sets, the same degree structure and, at least for functions
[bibtex key=Hamkins2001:WholenessAxiom]
The Wholeness Axioms, proposed by Paul Corazza, axiomatize the existence of an elementary embedding
[bibtex key=HamkinsLewis2000:InfiniteTimeTM]
This is the seminal paper introducing infinite time Turing machines.
We extend in a natural way the operation of Turing machines to infinite ordinal time, and investigate the resulting supertask theory of computability and decidability on the reals. The resulting computability theory leads to a notion of computation on the reals and concepts of decidability and semi-decidability for sets of reals as well as individual reals. Every
[bibtex key=Hamkins2000:LotteryPreparation]
The lottery preparation, a new general kind of Laver preparation, works uniformly with supercompact cardinals, strongly compact cardinals, strong cardinals, measurable cardinals, or what have you. And like the Laver preparation, the lottery preparation makes these cardinals indestructible by various kinds of further forcing. A supercompact cardinal
[bibtex key=HamkinsThomas2000:ChangingHeights]
If
[bibtex key=HamkinsWoodin2000:SmallForcing]
After small forcing, almost every strongness embedding is the lift of a strongness embedding in the ground model. Consequently, small forcing creates neither strong nor Woodin cardinals.
[bibtex key=HamkinsMontero2000:MoreBetter]
[bibtex key=HamkinsMontero2000:InfiniteWorlds]
Recently in the philosophical literature there has been some effort made to understand the proper application of the theory of utilitarianism to worlds in which there are infinitely many bearers of utility. Here, we point out that one of the best, most inclusive principles proposed to date contradicts fundamental utilitarian ideas, such as the idea that adding more utility makes a better world.
[bibtex key=Hamkins2000:BookReviewKanamori]
Akihiro Kanamori. The Higher Infinite. Large cardinals, stealing upwards through the clouds of imagined limitation like the steel skyscrapers of a ever-growing set-theoretic skyline, reach towards the stratosphere of Cantor’s absolute. In this century we have axiomatized larger and larger notions of infinity, and as we live amongst these giants, the formerly tall now seem small. Weakly inaccessible cardinals, for example, first considered by Hausdorff as a natural transfinite limit of set-theoretic operations, now occupy a floor at the entryway to the large cardinal hierarchy. In time over the past century we had Mahlo cardinals, strongly inaccessible cardinals, measurable cardinals, indescribable cardinals, weakly-compact cardinals, strongly-compact cardinals, super-compact cardinals, huge cardinals, almost huge cardinals, superhuge cardinals, and so on. And while when it comes to naming these enormous magnitudes, words have perhaps failed us, the mathematics is perfectly precise and fascinating.
Professor Kanamori has written—beautifully so—the book we large cardinal set-theorists have been lacking, a book spanning the possibilities from inaccessible to superhuge cardinals and beyond, a book full of historical insight, clear writing, interesting theorems and elegant proofs. This book is destined to become, if it has not already become, the standard reference in its field.
Finding that “a genetic account through historical progression…provides the most coherent exposition of the mathematics and holds the key to any epistemological concerns,” (p. XI) Kanamori weaves a historical perspective into the mathematics, deepening our understanding and appreciation of it. He sprinkles the text with quotations of Gödel and others, giving their mathematical-philosophical views on the mathematical developments. The introduction stands alone as a non-technical essay introducing the entire subject. From there, Kanamori begins with the smaller large cardinals, inaccessible and Mahlo cardinals, and then moves in time up to the strongest hypotheses.
So let me begin to explain a little about large cardinals. A cardinal
The point is that such a transcendence over ZFC in consistency strength is exactly what we want and what we need. In the decades since the invention of Cohen’s forcing technique, set theorists have set marching an infinite parade of independence results; indeed, it often seems as though almost all the interesting set-theoretic questions are independent of our ZFC axioms. We all know now that the cardinality of the set
Let me give one example. Almost all mathematicians are familiar with Vitali’s construction of a non-Lebesgue measurable set of reals and furthermore believe that the construction makes an essential use of the Axiom of Choice AC. But what does this mean exactly? The impossibility of removing AC from the Vitali construction is equivalent to the consistency (without AC) that every set of reals is Lebesgue measurable. Now of course we need some choice principle to develop a satisfactory theory of Lebesgue measure at all, so let us keep in the base theory the principle of Dependent Choices DC, which allows us to make countably many choices in succession. Thus, we are led to consider the consistency of the theory
After the beginnings, Kanamori moves swiftly through a chapter on partition properties, weak compactness, indiscernibles and
Kanamori’s book has already been translated into Japanese by S. Fuchino, and judging by the graduate students I saw last year in Japan pouring over it, the translation seems destined to create a new generation of large cardinal set theorists in Japan.
I do have one reservation about Kanamori’s book, namely, that he didn’t include much material on the interaction between forcing and large cardinals. Admittedly, this being the focus of much of my own work, I harbor some bias in its favor, but the topics of forcing and large cardinals are two major set theoretic research areas, and the intersection is rich. It would have been relatively easy for Kanamori to include a presentation, for example, of the landmark Laver preparation, by which any supercompact cardinal
My overall evaluation is entirely positive, and I recommend the book in the strongest possible terms to anyone with an interest in large cardinals. I can hardly wait for the subsequent volume!
[84] Saharon Shelah, “Can you take Solovay’s inaccessible away?” IJM 48 (1984), 1-47.
[65] Robert M. Solovay, “The measure problem,” NAMS 12 (1965), 217.
[bibtex key=Hamkins99:GapForcingGen]
The landmark Levy-Solovay Theorem limits the kind of large cardinal embeddings that can exist in a small forcing extension. Here I announce a generalization of this theorem to a broad new class of forcing notions. One consequence is that many of the forcing iterations most commonly found in the large cardinal literature create no new weakly compact cardinals, measurable cardinals, strong cardinals, Woodin cardinals, strongly compact cardinals, supercompact cardinals, almost huge cardinals, huge cardinals, and so on.
[bibtex key=ApterHamkins99:UniversalIndestructibility]
From a suitable large cardinal hypothesis, we provide a model with a supercompact cardinal in which universal indestructibility holds: every supercompact and partially supercompact cardinal kappa is fully indestructible by kappa-directed closed forcing. Such a state of affairs is impossible with two supercompact cardinals or even with a cardinal which is supercompact beyond a measurable cardinal.
[bibtex key=HamkinsShelah98:Dual]
After small forcing, any
[bibtex key=Hamkins98:SmallForcing]
[bibtex key=Hamkins98:AsYouLikeIt]
The Gap Forcing Theorem, a key contribution of this paper, implies essentially that after any reverse Easton iteration of closed forcing, such as the Laver preparation, every supercompactness measure on a supercompact cardinal extends a measure from the ground model. Thus, such forcing can create no new supercompact cardinals, and, if the GCH holds, neither can it increase the degree of supercompactness of any cardinal; in particular, it can create no new measurable cardinals. In a crescendo of what I call exact preservation theorems, I use this new technology to perform a kind of partial Laver preparation, and thereby finely control the class of posets which preserve a supercompact cardinal. Eventually, I prove the ‘As You Like It’ Theorem, which asserts that the class of