Results 1  10
of
10
Gödel's program for new axioms: Why, where, how and what?
 IN GODEL '96
, 1996
"... From 1931 until late in his life (at least 1970) Gödel called for the pursuit of new axioms for mathematics to settle both undecided numbertheoretical propositions (of the form obtained in his incompleteness results) and undecided settheoretical propositions (in particular CH). As to the nature of ..."
Abstract

Cited by 16 (6 self)
 Add to MetaCart
From 1931 until late in his life (at least 1970) Gödel called for the pursuit of new axioms for mathematics to settle both undecided numbertheoretical propositions (of the form obtained in his incompleteness results) and undecided settheoretical propositions (in particular CH). As to the nature of these, Gödel made a variety of suggestions, but most frequently he emphasized the route of introducing ever higher axioms of in nity. In particular, he speculated (in his 1946 Princeton remarks) that there might be a uniform (though nondecidable) rationale for the choice of the latter. Despite the intense exploration of the "higher infinite" in the last 30odd years, no single rationale of that character has emerged. Moreover, CH still remains undecided by such axioms, though they have been demonstrated to have many other interesting settheoretical consequences. In this paper, I present a new very general notion of the "unfolding" closure of schematically axiomatized formal systems S which provides a uniform systematic means of expanding in an essential way both the language and axioms (and hence theorems) of such systems S. Reporting joint work with T. Strahm, a characterization is given in more familiar terms in the case that S is a basic
Does Mathematics Need New Axioms?
 American Mathematical Monthly
, 1999
"... this article I will be looking at the leading question from the point of view of the logician, and for a substantial part of that, from the perspective of one supremely important logician: Kurt Godel. From the time of his stunning incompleteness results in 1931 to the end of his life, Godel called f ..."
Abstract

Cited by 11 (2 self)
 Add to MetaCart
this article I will be looking at the leading question from the point of view of the logician, and for a substantial part of that, from the perspective of one supremely important logician: Kurt Godel. From the time of his stunning incompleteness results in 1931 to the end of his life, Godel called for the pursuit of new axioms to settle undecided arithmetical problems. And from 1947 on, with the publication of his unusual article, "What is Cantor's continuum problem?" [11], he called in addition for the pursuit of new axioms to settle Cantor's famous conjecture about the cardinal number of the continuum. In both cases, he pointed primarily to schemes of higher infinity in set theory as the direction in which to seek these new principles. Logicians have learned a great deal in recent years that is relevant to Godel's program, but there is considerable disagreement about what conclusions to draw from their results. I'm far from unbiased in this respect, and you'll see how I come out on these matters by the end of this essay, but I will try to give you a fair presentation of other positions along the way so you can decide for yourself which you favor.
A First Glance at NonRestrictiveness
, 1999
"... Maddy's notion of restrictiveness has many problematic aspects, one of them being that it is almost impossible to show that a theory is not restrictive. In this note the author addresses a crucial question of Martin Goldstern and shows some positive aspects of Maddy's notion. ..."
Abstract
 Add to MetaCart
Maddy's notion of restrictiveness has many problematic aspects, one of them being that it is almost impossible to show that a theory is not restrictive. In this note the author addresses a crucial question of Martin Goldstern and shows some positive aspects of Maddy's notion.
VISUALIZATION OF ORDINALS ∗
"... We describe the pictorial representations of infinite ordinals used in teaching set theory, and discuss a possible use in naturalistic foundations of mathematics. 1 ..."
Abstract
 Add to MetaCart
We describe the pictorial representations of infinite ordinals used in teaching set theory, and discuss a possible use in naturalistic foundations of mathematics. 1
Large Cardinals and Determinacy
, 2011
"... The developments of set theory in 1960’s led to an era of independence in which many of the central questions were shown to be unresolvable on the basis of the standard system of mathematics, ZFC. This is true of statements from areas as diverse as analysis (“Are all projective sets Lebesgue measura ..."
Abstract
 Add to MetaCart
The developments of set theory in 1960’s led to an era of independence in which many of the central questions were shown to be unresolvable on the basis of the standard system of mathematics, ZFC. This is true of statements from areas as diverse as analysis (“Are all projective sets Lebesgue measurable?”), cardinal arithmetic (“Does Cantor’s Continuum Hypothesis hold?”), combinatorics(“DoesSuslin’sHypotheseshold?”), andgrouptheory (“Is there a Whitehead group?”). These developments gave rise to two conflicting positions. The first position—which we shall call pluralism—maintains that the independence results largely undermine the enterprise of set theory as an objective enterprise. On this view, although there are practical reasons that one might give in favour of one set of axioms over another—say, that it is more useful for a given task—, there are no theoretical reasons that can be given; and, moreover, this either implies or is a consequence of the fact—depending on the variant of the view, in particular, whether it places realism before reason,
SET THEORY FOR CATEGORY THEORY
, 810
"... Abstract. Questions of settheoretic size play an essential role in category theory, especially the distinction between sets and proper classes (or small sets and large sets). There are many different ways to formalize this, and which choice is made can have noticeable effects on what categorical co ..."
Abstract
 Add to MetaCart
Abstract. Questions of settheoretic size play an essential role in category theory, especially the distinction between sets and proper classes (or small sets and large sets). There are many different ways to formalize this, and which choice is made can have noticeable effects on what categorical constructions are permissible. In this expository paper we summarize and compare a number
Members of the Thesis Committee:
, 2012
"... The topic of this thesis is the relationship between formal and informal proofs. Chapter One opens the discussion by examining what a proof is, when two proofs are identical, what the purpose of proving is and how to distinguish the two categories of proof. Chapters Two and Three focus in on informa ..."
Abstract
 Add to MetaCart
The topic of this thesis is the relationship between formal and informal proofs. Chapter One opens the discussion by examining what a proof is, when two proofs are identical, what the purpose of proving is and how to distinguish the two categories of proof. Chapters Two and Three focus in on informal and formal proof respectively, with the latter also including descriptions of various computational proof checkers and the Formalist family of positions in the Philosophy of Mathematics. In Chapter Four I look at the Formalisability Thesis, that every informal proof corresponds to a formal proof, and argue that this breaks apart into a weak and a strong reading. In Chapter Five, I outline a simple mathematical problem and attempt the practical process of formalisation, following which I consider the decisions that were involved in doing so. Finally, in Chapter Six, I use what was learned from the practicalities of formalisation to argue in favour of the weak reading of the Formalisability Thesis, which I take to be closely related to Carnapian explication, but against the strong reading, which corresponds to the Formalists ’ approach to formalisation. Thanks
Justin ClarkeDoane Monash University [Note: This is the penultimate draft of a paper that is forthcoming in Noûs.] Moral Epistemology: The Mathematics Analogy *
"... There is a long tradition of comparing moral knowledge to mathematical knowledge. Plato compared mathematical knowledge to knowledge of the Good. 1 In recent years, metaethicists have found the comparison to be illuminating. 2 Sometimes the comparison is supposed to show that moral realism is peculi ..."
Abstract
 Add to MetaCart
There is a long tradition of comparing moral knowledge to mathematical knowledge. Plato compared mathematical knowledge to knowledge of the Good. 1 In recent years, metaethicists have found the comparison to be illuminating. 2 Sometimes the comparison is supposed to show that moral realism is peculiarly problematic. For example, James Rachels writes, “[H]ow do we know moral facts?....In mathematics there are proofs….But moral facts are not accessible by…these familiar methods [1998, p. 3].” But other times the comparison is supposed to show that moral realism is no more problematic than mathematical realism. For example, Hilary Putnam writes, “[A]rguments for “antirealism ” in ethics are virtually identical with arguments for antirealism in the philosophy of mathematics; yet philosophers who resist those arguments in the latter case often capitulate in the former [2004, p. 1].”