Akihiro Kanamori | Boston University (original) (raw)

Papers by Akihiro Kanamori

Cambridge University Press eBooks, Oct 8, 2011

The work of Stanley Tennenbaum in set theory was centered on the investigation of Suslin's Hypoth... more The work of Stanley Tennenbaum in set theory was centered on the investigation of Suslin's Hypothesis (SH), to which he made crucial contributions. In 1963 Tennenbaum established the relative consistency of ¬SH, and in 1965, together with Robert Solovay, the relative consistency of SH. In the formative period after Cohen's 1963 founding of forcing when set theory was transmuting into a modern, sophisticated field of mathematics, this work on SH exhibited the power of forcing for elucidating a classical problem of mathematics and stimulated the development of new methods and areas of investigation. §1 discusses the historical underpinnings of SH. §2 then describes Tennenbaum's consistency result for ¬SH and related subsequent work. §3 then turns to Tennenbaum's work with Solovay on SH and the succeeding work on iterated forcing and Martin's Axiom. To cast an amusing sidelight on the life and the times, I relate the following reminiscence of Gerald Sacks from this period, no doubt apocryphal: Tennenbaum let it be known that he had come into a great deal of money, 30,000,000itwassaid,andstartedtoborrowmoneyagainstit.GeraldconvincedhimselfthatTennenbaumseriouslybelievedthis,butnonethelessaskedSimonKochenaboutit.Kochenreplied,"Well,withStanhemightbeonepercentright.Butthen,that′sstill30,000,000 it was said, and started to borrow money against it. Gerald convinced himself that Tennenbaum seriously believed this, but nonetheless asked Simon Kochen about it. Kochen replied, "Well, with Stan he might be one percent right. But then, that's still 30,000,000itwassaid,andstartedtoborrowmoneyagainstit.GeraldconvincedhimselfthatTennenbaumseriouslybelievedthis,butnonethelessaskedSimonKochenaboutit.Kochenreplied,"Well,withStanhemightbeonepercentright.Butthen,that′sstill300,000." §1. Suslin's Problem In 1920, at the end of the first volume of Fundamenta Mathematicae there appeared a list of problems with one attributed to Mikhail Suslin [1920], a problem that would come to be known as Suslin's Problem. After the reunification of Poland in 1918, there was a deliberate decision by its aspiring mathematicians to focus on set theory and related areas and to bring out a new journal to promote international research in this area. 2 This was the origin of Fundamenta Mathematicae, which became the main conduit of scholarship in 'fundamental mathematics' during the 1920s and 1930s. That first list of problems had to do with possible consequences of the Continuum Hypothesis (CH) or issues in the emerging descriptive set theory. These problems would be solved, but by contrast Suslin's Problem would grow in significance through its irresolution. Georg Cantor, the founder of set theory, had famously characterized the ordertypes of the rationals and reals in his Beiträge [1895], his mature presentation of his theory of the transfinite. The ordering of the reals is that unique dense linear ordering with no endpoints which is order-complete (i.e. every bounded set has a least upper bound 3) and separable (i.e. has a countable dense subset). Suslin's Problem asks whether this last condition can be weakened to the countable chain condition (c.c.c.): every disjoint family of open intervals is countable. Although Suslin himself did not hypothesize it, the affirmative answer has come to be known as Suslin's Hypothesis (SH). For a dense linear order, deleting endpoints and

Modern mathematics is, to my mind, a complex edifice based on conceptual constructions. The subje... more Modern mathematics is, to my mind, a complex edifice based on conceptual constructions. The subject has undergone something like a biological evolution, an opportunistic one, to the point that the current subject matter, methods, and proce-dures would be patently unrecognizable a century, certainly two centuries, ago. What has been called “classical mathematics ” has indeed seen its day. With its richness, variety, and complexity any discussion of the nature of modern mathematics cannot but accede to the primacy of its history and practice. As I see it, the applicability of mathematics may be a driving motivation, but in the end mathematics is au-tonomous. Mathematics is in a broad sense self-generating and self-authenticating, and alone competent to address issues of its correctness and authority. What brings us mathematical knowledge? The carriers of mathematical knowl-edge are proofs, more generally arguments and constructions, as embedded in larger contexts.1 Mathematicians and ...

We revisit Putnam’s constructivization argument from his Models and Reality, part of his model-th... more We revisit Putnam’s constructivization argument from his Models and Reality, part of his model-theoretic argument against metaphysical realism. We set out how it was initially put, the commentary and criticisms, and how it can be specifically seen and cast, respecting its underlying logic and in light of Putnam’s contributions to mathematical

Most of the chapters of this volume present the historical development of aspects of modern, ZFC ... more Most of the chapters of this volume present the historical development of aspects of modern, ZFC set theory as a field of mathematics, and a couple of chapters at the end, categorical logic as a foundation of mathematics. Be that as it may, as part of the Handbook of the History of Logic the volume can be construed as the one focusing on extensions vis-` a-vis the intension vs. extension distinction. That distinction, traditionally attributed to Antoine Arnauld in the 1662 Logique de Port Royal, is exemplified by " featherless biped " , a conceptualization, and the corresponding extension, the collection of all homo sapiens sapiens. However this distinction was historically developed and worked in logic, it was in the mathematical development of set theory that extensions became explicit through the mathematization of infinitary concepts and the further development of mathematics , itself increasingly based on informal concepts of set. How many points are there on the line...

Kurt Gödel Philosopher-Scientist, 2016

Gödel's work from the beginning to his first substantive explorations in philosophy would to a si... more Gödel's work from the beginning to his first substantive explorations in philosophy would to a significant extent be contextualized by, reactive to, and reflective of, Russell's. Russell was the towering figure who set the stage for analytic philosophy in the first two decades of the 20 th Century; Gödel insisted that his mathematical work was substantively motivated by his own philosophical outlook; and so it becomes especially pertinent to draw out and highlight the interconnections between the two. What follows is a narrative that focuses on the interplay of several arching motifs: Russell's theory of types and Axiom of Reducibility; definability as analysis; Gödel's incompleteness theorems and constructible sets; and Russell's and Gödel's respective construals of the nature of truth as a, if not the, philosophical problem. More specifically, Gödel's reflections on Russell's so-called "multiple relation theory of judgment" (MRTJ), the theory of truth at work in the Principia Mathematica and later writings of Russell, especially his William James Lectures (1940), will be set forth and framed. This will allow us to draw out, in a provisional way, some of the philosophical significance of Gödel's recently transcribed Max Phil notebooks. Russell's MRTJ was an idiosyncratic version of the correspondence theory of truth, framed in terms of complex (non-dual) relations between judgers, the objects of their judgments, and their environment. Our view is that it was this specific theory of truth that preoccupied Gödel by 1929, and again in 1942-1943, when he returned to reading Russell in earnest. We offer a narrative of influence, reaction, and confluence, of suggested thematic causalities discernible in Gödel's intellectual development. With the conviction that there is still much to be said about Gödel vis-à-vis Russell, we forge a particular path through now well-known aspects of Gödel's work and thought, putting new weight on some lesser-known aspects. In particular, we give a nuanced reading of Gödel's mathematical platonism as emergent according to mathematical advances and, by his lights, increasingly indispensable. We also insist, especially given the evidence of the Max Phil notebooks, on the increasingly serious importance to Gödel (as also evidently to Russell throughout) of philosophical questions of the most classical and fundamental kind concerning the relation of logic to grammar and to mathematics, of truth to perception, and these questions' relevance to foundational questions about the very possibility of conceptual analysis. Others have weighed in here, in an increasingly informative and fine-grained way; our aim is primarily to broach the bearing of the

Annals of the Japan Association for Philosophy of Science, 2013

thanks to all those whe made helpful suggestions, especlally Juliet Floyd, i In a perceptive pape... more thanks to all those whe made helpful suggestions, especlally Juliet Floyd, i In a perceptive paper advocating similar themes Rav [15] used the term "bearer" instead of `Ccarrier", The former is more passive, as in :`bearer of good tidings", 2 Here, "statement" is being used to suggest mere prose expression, in preference to the weightier `Cproposition", which often translates Frege's `CSatz" and was also used by Russell. the JapanAssociation forPhUosophy of Science 24 Akihiro KANAMORI Vbl.

Zermelo in his remarkable 1930a offered his final axiomatization of set theory as well as a strik... more Zermelo in his remarkable 1930a offered his final axiomatization of set theory as well as a striking, synthetic view of a procession of natural models that would have a modern resonance. Appearing only six articles after Skolem 1930 in Fundamenta mathematicae, Zermelo’s article seemed strategically placed as a response, an aspect that we will discuss below, but its dramatically new picture of set theory reflects gained experience and suggests the germination of ideas over a prolonged period. The subtitle, “New investigations in the foundations of set theory”, evidently recalls his axiomatization article 1908b, differing only in the “New” from the title of that article. The new article is a tour de force which sets out principles that would be adopted in the further development of set theory and draws attention to the cumulative hierarchy picture, dialectically enriched by initial segments serving as natural models. In Section 1, Zermelo formulates his axiom system, the “constitutive...

Annals of the Japan Association for Philosophy of Science, 2012

I address t･he historical emergence of the mathematical infinite, and how we are to take the infi... more I address t･he historical emergence of the mathematical infinite, and how we are to take the infinite in and out ot' rriat･hematics,

Archive for Mathematical Logic, 2016

Annals of Mathematical Logic, 1980

Annals of Mathematical Logic, 1978

Resumen. El infinito como método en la teoría de conjuntos y la matemática Este artículo da cuent... more Resumen. El infinito como método en la teoría de conjuntos y la matemática Este artículo da cuenta de la aparición histórica de lo infinito en la teoría de conjuntos, y de cómo lo tratamos dentro y fuera de las matemáticas. La primera sección analiza el surgimiento de lo infinito como una cuestión de método en la teoría de conjuntos. La segunda sección analiza el infinito dentro y fuera de las matemáticas, y cómo deben adoptarse.

I address the historical emergence of the mathematical infinite, and how we are to take the infin... more I address the historical emergence of the mathematical infinite, and how we are to take the infinite in and out of mathematics. The thesis is that the mathematical infinite in mathematics is a matter of method.

Springer Monographs in Mathematics, 2003

The use of general descriptive names, registered names, trademarks, etc. in this publication does... more The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use.

The Bulletin of Symbolic Logic, 2014

Paul Erdős (26 March 1913—20 September 1996) was a mathematician par excellence whose results and... more Paul Erdős (26 March 1913—20 September 1996) was a mathematician par excellence whose results and initiatives have had a large impact and made a strong imprint on the doing of and thinking about mathematics. A mathematician of alacrity, detail, and collaboration, Erdős in his six decades of work moved and thought quickly, entertained increasingly many parameters, and wrote over 1500 articles, the majority with others. His modus operandi was to drive mathematics through cycles of problem, proof, and conjecture, ceaselessly progressing and ever reaching, and his modus vivendi was to be itinerant in the world, stimulating and interacting about mathematics at every port and capital.

Philosophy of Mathematics, 2009

Set theory is an autonomous and sophisticated field of mathematics, enormously successful not onl... more Set theory is an autonomous and sophisticated field of mathematics, enormously successful not only at its continuing development of its historical heritage but also at analyzing mathematical propositions and gauging their consistency strength. But set theory is also distinguished by having begun intertwined with pronounced metaphysical attitudes, and these have even been regarded as crucial by some of its great developers. This has encouraged the exaggeration of crises in foundations and of metaphysical doctrines in general. However, set theory has proceeded in the opposite direction, from a web of intensions to a theory of extension par excellence, and like other fields of mathematics its vitality and progress have depended on a steadily growing core of mathematical proofs and methods, problems and results. There is also the stronger contention that from the beginning set theory actually developed through a progression of mathematical moves, whatever and sometimes in spite of what has been claimed on its behalf. What follows is an account of the development of set theory from its beginnings through the creation of forcing based on these contentions, with an avowedly Whiggish emphasis on the heritage that has been retained and developed by the current theory. The whole transfinite landscape can be viewed as having been articulated by Cantor in significant part to solve the Continuum Problem. Zermelo's axioms can be construed as clarifying the set existence commitments of a single proof, of his Well-Ordering Theorem. Set theory is a particular case of a field of mathematics in which seminal proofs and pivotal problems actually shaped the basic concepts and forged axiomatizations, these transmuting the very notion of set. There were two main junctures, the first being when Zermelo through his axiomatization shifted the notion of set from Cantor's range of inherently structured sets to sets solely structured by membership and governed and generated by axioms. The second juncture was when the Replacement and Foundation Axioms were adjoined and a first-order setting was established; thus transfinite recursion was incorporated and results about all sets could established through these means, including results about definability and inner models. With the emergence of the cumulative hierarchy picture, set theory can be regarded as becoming a theory of well-foundedness, later to expand to a study of consistency strength. Throughout, the subject has not only been sustained by the axiomatic tradition through Gödel and Cohen but also fueled by Cantor's two legacies, the extension of number into the transfinite as transmuted into the theory of large cardinals and the investigation of definable sets of reals as transmuted into descriptive set theory. All this can be regarded as having a historical and mathematical logic internal to set theory, one that is often misrepresented at critical junctures in textbooks (as will be pointed out). This view, from inside set theory and about itself, serves to shift the focus to

Springer Monographs in Mathematics

This chapter describes the first advances using Cohen's method of forcing that involved large car... more This chapter describes the first advances using Cohen's method of forcing that involved large cardinals and the first applications of large cardinals in descriptive set theory. Cohen's creation transformed set theory, and large cardinal hypotheses played an increasingly prominent role as a consequence. §10 discusses the development of forcing, reviews the basic theory, and then focuses on mild extensions and the Levy collapse. §11 is devoted to Solovay's inspiring result that if there is an inaccessible cardinal, then in an inner model of a forcing extension, every set of reals is Lebesgue measurable. §12 reviews the historical development of descriptive set theory and establish a working context, one in which the classical results are established in §13 through to a delimitation established by Gödel with L. This sets the stage for the further results about large cardinals and projective sets, a major direction of set-theoretic research from the mid-1960's onwards. §14 describes Solovay's germinal work on Σ 1 2 sets that grew out of his Lebesgue measurability result, and his results and conjectures on the definability of sharps. Then §15 describe how Martin used sharps to extend the methods of classical descriptive set theory to the analysis of Σ 1 3 sets.