[bibtex key=Hamkins2005:TheGroundAxiom]
This is an extended abstract for a talk I gave at the 2005 Workshop in Set Theory at the Mathematisches Forschungsinstitut Oberwolfach.
Oberwolfach Research Report 55/2005 | Ground Axiom on Wikipedia
[bibtex key=Hamkins2005:TheGroundAxiom]
This is an extended abstract for a talk I gave at the 2005 Workshop in Set Theory at the Mathematisches Forschungsinstitut Oberwolfach.
Oberwolfach Research Report 55/2005 | Ground Axiom on Wikipedia
[bibtex key=Hamkins2005:InfinitaryComputabilityWithITTM]
Infinite time Turing machines extend the operation of ordinary Turing machines into transfinite ordinal time. By doing so, they provide a theoretical model of infinitary computability, while remaining close in spirit to many of the methods and concepts of classical computability. The model gives rise to a robust theory of infinitary computability on the reals, such as notions of computability for functions
[bibtex key=Hamkins2005:TourlakisBookReview]
Review of George Tourlakis, Lectures in Logic and Set Theory, volumes 1 and 2, Cambridge studies in advanced mathematics, vol. 83. Cambridge University Press, Cambridge, UK, 2003. This is a detailed two-volume development of mathematical logic and set theory, written from a formalist point of view, aimed at a spectrum of students from the third-year undergraduate to junior graduate level. Volume 1 presents the heart of mathematical logic, including the Completeness and Incompleteness theorems along with a bit of computability theory and accompanying ideas. Tourlakis aspires to include “the absolutely essential topics in proof, model and recursion theory” (vol. 1, p. ix). In addition, for the final third of the volume, Tourlakis provides a proof of the Second Incompleteness Theorem “right from Peano’s axioms,…gory details and all,” which he conjectures “is the only complete proof in print [from just Peano arithmetic] other than the one that was given in Hilbert and Bernays (1968)” (vol. 1, p. x). In the opening
page of Chapter II, Tourlakis provides a lucid explanation of the proof in plain language, before diving into the details and emerging a hundred pages later with the provability predicate, the derivability conditions and a complete proof. Tempering his formalist tendencies, Tourlakis speaks “the formal language with a heavy `accent’ and using many `idioms’ borrowed from `real’ (meta)mathematics and English,” in a mathematical argot (vol. 1, p. 39). In his theorems and proofs, therefore, he stays close to the formal language without remaining inside it.
But let me focus on volume 2, a stand-alone development of axiomatic set theory, containing within it a condensed version of volume 1. The book emphasizes the formal
foundations of set theory and, like the first volume, gives considerable attention to the details of the elementary theory. Tourlakis is admirably meticulous in maintaining
the theory/metatheory distinction, with a careful explanation of the role of inductive arguments and constructions in the metatheory (vol. 2, p. 20) and a correspondingly precise treatment of axioms, theorems and their respective schemes throughout. What is more, he sprinkles the text with philosophical explanations of the theory/metatheory interaction, giving a clear account, for example, of how it is that we may use apparently set theoretic arguments in the metatheory without circularity (vol. 1, p. 10-12). After developing the logical background, he paints the motivating picture of the cumulative hierarchy, the process by which we imagine sets to be built, with Russell’s paradox as a cautionary tale. In Chapter III, the axioms of set theory march forward in succession. He presents them gradually, motivating them from the cumulative hierarchy and deriving consequences as they appear. This treatment includes the Axiom of Choice, which he motivates, impressively, by developing Goedel’s constructible universe
Interestingly, Tourlakis’ version of ZFC set theory, like Zermelo’s, allows for (without insisting on) the existence of urelements, atomic objects that are not sets, but which
can be elements of sets. His reason for this is philosophical and pedagogical: he finds “it extremely counterintuitive, especially when addressing undergraduate audiences, to tell them that all their familiar mathematical objects — the `stuff of mathematics’ in
Barwise’s words — are just perverse `box-in-a-box-in-a-box\dots’ formulations built from an infinite supply of empty boxes” (vol. 2, p. xiii). The enrichment of the theory to allow
urelements requires only minor modifications of the usual ZFC axioms, such as the restriction of Extensionality to the sets and not the urelements. The application of the
definition
Comprehension Axiom from Tourlakis’ version of the Power Set Axiom, which asserts that for every object
At times, the author employs what some might take as an exaggerated formal style. For example, after introducing the Pairing Axiom, stating that for any
is
steps, with duly noted uses of the Leibniz axiom and modus ponens. To be sure, he later adopts what he calls a “relaxed” proof style, but even so, in the “Informal”
Example III.9.2, he fills a page with tight reasoning and explicit appeals to the deduction theorem, the principle of auxiliary constants and more, to show merely that if
a set and
The preface of volume 2 explains that the book weaves a middle path between those set theory books that merely build set-theoretic tools for use elsewhere and those that
aim at research in set theory. But I question this assessment. Many of the topics constituting what I take to be the beginnings of the subject appear only very late in
the book. For example, the von Neumann ordinals appear first on page 331; Cantor’s theorem on the uncountability of
diamond principle and even ultrafilters. Large cardinals are not mentioned beyond the inaccessible cardinals. The omission of ultrafilters is particularly puzzling, given
the author’s claim to have included “all the fundamental tools of set theory as needed elsewhere in the mathematical sciences” (vol. 2, p.~xii). Certainly ultrapowers are one
of the most powerful and successful such tools, whose fundamental properties remain deeply connected with logic.
In the final chapter, the author provides a formal account of the foundations of forcing, with useful explanations again of the important theory/metatheory interaction
arising in connection with it. Because his account of forcing is based on countable transitive models, some set theorists may find it old-fashioned. This way of forcing
tends to push much of the technique into the metatheory, which Tourlakis adopts explicitly (vol. 2, p. 519), and can sometimes limit forcing to its role in independence
results. A more contemporary view of forcing makes sense within ZFC of forcing over
sensibly to discuss the possibilities achievable by forcing over any given model of set theory.
Despite my reservations, I welcome Tourlakis’ addition to the body of logic texts. Readers with a formalist bent especially will gain from it.
[bibtex key=Hamkins2004:SupertaskComputation]
Infinite time Turing machines extend the classical Turing machine concept to transfinite ordinal time, thereby providing a natural model of infinitary computability that sheds light on the power and limitations of supertask algorithms.
[bibtex key=Hamkins2003:ExtensionsWithApproximationAndCoverProperties]
If an extension
[bibtex key=HamkinsWelch2003:PfneqNPf]
Abstract. We discuss the question of Ralf-Dieter Schindler whether for infinite time Turing machines
[bibtex key=ApterHamkins2003:ExactlyControlling]
We summarize the known methods of producing a non-supercompact strongly compact cardinal and describe some new variants. Our Main Theorem shows how to apply these methods to many cardinals simultaneously and exactly control which cardinals are supercompact and which are only strongly compact in a forcing extension. Depending upon the method, the surviving non-supercompact strongly compact cardinals can be strong cardinals, have trivial Mitchell rank or even contain a club disjoint from the set of measurable cardinals. These results improve and unify previous results of the first author.
[bibtex key=Hamkins2003:MaximalityPrinciple]
In this paper, following an idea of Christophe Chalons, I propose a new kind of forcing axiom, the Maximality Principle, which asserts that any sentence
[bibtex key=Hamkins2001:HowTall?]
The automorphism tower of a group is obtained by computing its automorphism group, the automorphism group of that group, and so on, iterating transfinitely by taking the natural direct limit at limit stages. The question, known as the automorphism tower problem, is whether the tower ever terminates, whether there is eventually a fixed point, a group that is isomorphic to its automorphism group by the natural map. Wielandt (1939) proved the classical result that the automorphism tower of any finite centerless group terminates in finitely many steps. This was generalized to successively larger collections of groups until Thomas (1985) proved that every centerless group has a terminating automorphism tower. Here, it is proved that every group has a terminating automorphism tower. After this, an overview is given of the author’s (1997) result with Thomas revealing the set-theoretic essence of the automorphism tower of a group: the very same group can have wildly different towers in different models of set theory.
[bibtex key=ApterHamkins2002:LevelByLevel]
Can a supercompact cardinal
[bibtex key=HamkinsLewis2002:PostProblem]
Recently we have introduced a new model of infinite computation by extending the operation of ordinary Turing machines into transfinite ordinal time. In this paper we will show that the infinite time Turing machine analogue of Post’s problem, the question whether there are supertask degrees between
[bibtex key=Hamkins2002:Turing]
This is a survey of the theory of infinite time Turing machines.
Infinite time Turing machines extend the operation of ordinary Turing machines into transfinite ordinal time. By doing so, they provide a natural model of infinitary computability, a theoretical setting for the analysis of the power and limitations of supertask algorithms.
[bibtex key=FishkindHamkinsMontero2002:NewInconsistencies]
In the context of worlds with infinitely many bearers of utility, we argue that several collections of natural Utilitarian principles–principles which are certainly true in the classical finite Utilitarian context and which any Utilitarian would find appealing–are inconsistent.
[bibtex key=ApterHamkins2001:IndestructibleWC]
We show that if the weak compactness of a cardinal is made indestructible by means of any preparatory forcing of a certain general type, including any forcing naively resembling the Laver preparation, then the cardinal was originally supercompact. We then apply this theorem to show that the hypothesis of supercompactness is necessary for certain proof schemata.
[bibtex key=Hamkins2001:UnfoldableCardinals]
Introducing unfoldable cardinals last year, Andres Villaveces ingeniously extended the notion of weak compactness to a larger context, thereby producing a large cardinal notion, unfoldability, with some of the feel and flavor of weak compactness but with a greater consistency strength. Specifically,