In this presidential address, I use the metaphor of “reclaiming my time” as a framework that highlights the ways black women are playing an essential role transforming workplaces, media, and politics in the current moment. I consider how black feminist thought provides a useful starting point for assessing these efforts, and I examine how black women’s leadership offers a blueprint for how other groups also can restructure social institutions in an era of increasing polarization and inequality.

Many men who work in women's professions experience a glass escalator effect that facilitates their advancement and upward mobility within these fields. Research finds that subtle aspects of the interactions, norms, and expectations in women's professions push men upward and outward into the higher-status, higher-paying, more “masculine” positions within these fields. Although most research includes minority men, little has explicitly considered how racial dynamics color these men's encounters with the mechanisms of the glass escalator. In this article, the author examines (...) how intersections of race and gender combine to shape experiences for minority men in the culturally feminized field of nursing and finds that the upward mobility implied by the glass escalator is not uniformly available to all men who do “women's work.” The author concludes that the glass escalator is a racialized concept and a gendered one and considers the implications of this for future studies of men in feminized occupations. (shrink)

This study applies the concept of intersectionality to Black women's entrepreneurial activity. Specifically, the author addresses the ways in which race, gender, and class intersect to inform working-class Black women's decisions and experiences as hair salon owners. By placing Black women at the center of analysis, the author explores business ownership from the perspective of a group that has frequently been overlooked in sociology of entrepreneurship research. The findings indicate that race, gender, and class inequalities shape working-class Black women's entrepreneurship (...) in two areas: the process of becoming entrepreneurs and relationships with stylists. (shrink)

The naive theory of properties states that for every condition there is a property instantiated by exactly the things which satisfy that condition. The naive theory of properties is inconsistent in classical logic, but there are many ways to obtain consistent naive theories of properties in nonclassical logics. The naive theory of classes adds to the naive theory of properties an extensionality rule or axiom, which states roughly that if two classes have exactly the same members, they are identical. In (...) this paper we examine the prospects for obtaining a satisfactory naive theory of classes. We start from a result by Ross Brady, which demonstrates the consistency of something resembling a naive theory of classes. We generalize Brady’s result somewhat and extend it to a recent system developed by Andrew Bacon. All of the theories we prove consistent contain an extensionality rule or axiom. But we argue that given the background logics, the relevant extensionality principles are too weak. For example, in some of these theories, there are universal classes which are not declared coextensive. We elucidate some very modest demands on extensionality, designed to rule out this kind of pathology. But we close by proving that even these modest demands cannot be jointly satisfied. In light of this new impossibility result, the prospects for a naive theory of classes are bleak. (shrink)

Two countable well orderings are weakly comparable if there is an order preserving injection of one into the other. We say the well orderings are strongly comparable if the injection is an isomorphism between one ordering and an initial segment of the other. In [5], Friedman announced that the statement “any two countable well orderings are strongly comparable” is equivalent to ATR 0 . Simpson provides a detailed proof of this result in Chapter 5 of [13]. More recently, Friedman has (...) proved that the statement “any two countable well orderings are weakly comparable” is equivalent to ATR 0 . The main goal of this paper is to give a detailed exposition of this result. (shrink)

Based on the relevant logic R, the system R# was proposed as a relevant Peano arithmetic. R# has many nice properties: the most conspicuous theorems of classical Peano arithmetic PA are readily provable therein; it is readily and effectively shown to be nontrivial; it incorporates both intuitionist and classical proof methods. But it is shown here that R# is properly weaker than PA, in the sense that there is a strictly positive theorem QRF of PA which is unprovable in R#. (...) The reason is interesting: if PA is slightly weakened to a subtheory P+, it admits the complex ring C as a model; thus QRF is chosen to be a theorem of PA but false in C. Inasmuch as all strictly positive theorems of R# are already theorems of P+, this nonconservativity result shows that QRF is also a nontheorem of R#. As a consequence, Ackermann's rule γ is inadmissible in R#. Accordingly, an extension of R# which retains its good features is desired. The system R##, got by adding an omega-rule, is such an extension. Central question: is there an effectively axiomatizable system intermediate between R# and R##, which does formalize arithmetic on relevant principles, but also admits γ in a natural way? (shrink)

Reflection, in the sense of [Fr03a] and [Fr03b], is based on the idea that a category of classes has a subclass that is “similar” to the category. Here we present axiomatizations based on the idea that a category of classes that does not form a class has extensionally different subclasses that are “similar”. We present two such similarity principles, which are shown to interpret and be interpretable in certain set theories with large cardinal axioms.

We study the formalization within sybsystems of second-order arithmetic of theorems concerning periodic points in dynamical systems on the real line. We show that Sharkovsky's theorem is provable in WKL0. We show that, with an additional assumption, Sharkovsky's theorem is provable in RCA0. We show that the existence for all n of n-fold iterates of continuous mappings of the closed unit interval into itself is equivalent to the disjunction of Σ02 induction and weak König's lemma.

We now fix A ⊆ Q. We study a fundamental class of digraphs associated with A, which we call the A-digraphs. An A,kdigraph is a digraph (Ak,E), where E is an order invariant subset of A2k in the following sense. For all x,y ∈ A2k, if x,y have the same order type then x ∈ E ↔ y ∈ E.

The subtle, almost ineffable, and ineffable cardinals were introduced in an unpublished 1971 manuscript of R. Jensen and K. Kunen. The concepts were extended to that of k-subtle, k-almost ineffable, and k-ineffable cardinals in 1975 by J. Baumgartner. In this paper we give a self contained treatment of the basic facts about this level of the large cardinal hierarchy, which were established by J. Baumgartner. In particular, we give a proof that the k-subtle, k-almost ineffable, and k-ineffable cardinals define three (...) properly intertwined hierarchies with the same limit, lying strictly above “total indescribability” and strictly below “arrowing ω”. The innovation here is presented in Section 2, where we take a distinctly minimalist approach. Here the subtle cardinal hierarchy is characterized by very elementary properties that do not mention closed unbounded or stationary sets. This development culminates in a characterization of the hierarchy by means of a striking universal second-order property of linear orderings. (shrink)

We consider a number of statements involving restrictions and extensions of algebras, and derive connections with large cardinal axioms.

The kind of unknowability I will discuss concerns the count of certain natural finite sets of objects. Even the situation with regard to our present strong formal systems is rather unclear. One can just profitably focus on that, putting aside issues of general unknowability.

Most of my intellectual efforts have focused around a single general question in the foundations of mathematics (f.o.m.). I became keenly aware of this question as a student at MIT around 40 years ago, and readily adopted it as the principal driving force behind my research.

We present some examples of mathematically natural nonrecursive sets of integers and relations on integers by combining results from Part 1, recursion theory, and from the negative solution to Hilbert’s 10th Problem ([3], [1], and [2]).

An unprovable theorem is a mathematical result that can-not be proved using the com-monly accepted axioms for mathematics (Zermelo-Frankel plus the axiom of choice), but can be proved by using the higher infinities known as large cardinals. Large car-dinal axioms have been the main proposal for new axioms originating with Gödel.

An extreme kind of logic skeptic claims that "the present formal systems used for the foundations of mathematics are artificially strong, thereby causing unnecessary headaches such as the Gödel incompleteness phenomena". The skeptic continues by claiming that "logician's systems always contain overly general assertions, and/or assertions about overly general notions, that are not used in any significant way in normal mathematics. For example, induction for all statements, or even all statements of certain restricted forms, is far too general - mathematicians (...) only use induction for natural statements that actually arise. If logicians would tailor their formal systems to conform to the naturalness of normal mathematics, then various logical difficulties would disappear, and the story of the foundations of mathematics would look radically different than it does today. In particular, it should be possible to give a convincing model of actual mathematical practice that can be proved to be free of contradiction using methods that lie within what Hilbert had in mind in connection with his program”. Here we present some specific results in the direction of refuting this point of view, and introduce the Strict Reverse Mathematics (SRM) program. (shrink)

It has been accepted since the early part of the Century that there is no problem formalizing mathematics in standard formal systems of axiomatic set theory. Most people feel that they know as much as they ever want to know about how one can reduce natural numbers, integers, rationals, reals, and complex numbers to sets, and prove all of their basic properties. Furthermore, that this can continue through more and more complicated material, and that there is never a real problem.

In a modal system of arithmetic, a theory S has the modal disjunction property if whenever $S \vdash \square\varphi \vee \square\psi$ , either $S \vdash \square\varphi$ or $S \vdash \square\psi. S$ has the modal numerical existence property if whenever $S \vdash \exists x\square\varphi(x)$ , there is some natural number n such that $S \vdash \square\varphi(\mathbf{n})$ . Under certain broadly applicable assumptions, these two properties are equivalent.

We axiomatize EFA in strictly mathematical terms, involving only the ring operations, without extending the language by either exponentiation, finite sets of integers, or polynomials.

We can equivalently present this by the recursion equations f1(n) = 2n, fk+1(1) = fk(1), fk+1(n+1) = fk(fk+1(n)), where k,n ≥ 1. We define A(k,n) = fk(n).

In mathematics, we back up our discoveries with rigorous deductive proofs. Mathematicians develop a keen instinctive sense of what makes a proof rigorous. In logic, we strive for a *theory* of rigorous proofs.

Most of the research in foundations of mathematics that I do in some way or another involves the use of nonstandard models. I will give a few examples, and indicate what is involved.

Each of these theorems and concepts arose from very specific considerations of great general interest in the foundations of mathematics (f.o.m.). They each serve well defined purposes in f.o.m. Naturally, the preferred way to formulate them for mathe-matical logicians is in terms that are close to their roots in f.o.m.

We present several selection theorems for Borel relations, involving only Borel sets and functions, all of which can be obtained as consequences of closely related theorems proved in [DSR 96,99,01,01X] involving coanalytic sets. The relevant proofs given there use substantial set theoretic methods, which were also shown to be necessary. We show that none of our Borel consequences can be proved without substantial set theoretic methods. The results are established for Baire space. We give equivalents of some of the main (...) results for the reals. (shrink)

We give a complete proof that all 3 quantifier sentences in the primitive notation of set theory (Œ,=), are decided in ZFC, and in fact in a weak fragment of ZF without the power set axiom. We obtain information concerning witnesses of 2 quantifier formulas with one free variable. There is a 5 quantifier sentence that is not decided in ZFC (see [Fr02]).

We present two forms of “sentential reflection”, which are shown to be mutually interpretable with Z2 and ZFC, respectively.

A lot of the well known impact of the Gödel phenomena is in the form of painful messages telling us that certain major mathematical programs cannot be completed as intended. This aspect of Gödel – the delivery of bad news –is not welcomed, and defensive measures are now in place.

1. Transfer principles from N to On. A. Mahlo cardinals. B. Weakly compact cardinals. C. Ineffable cardinals. D. Ramsey cardinals. E. Ineffably Ramsey cardinals. F. Subtle cardinals. G. From N to (...) 4. Decidability of statements on N. 5. Decidability of statements on shrink)

This is widely accepted, inside and outside philosophy, but one can spend an entire career clarifying, justifying, and amplifying on this statement. Certainly a graduate student career.

Provide a formal system that is a conservative extension of PA for Π02 sentences, and even a conservative extension of HA, that supports the worry free smooth development of constructive analysis in the style of Errett Bishop.

NOTE: This is an expanded version of my lecture at the special session on reverse mathematics, delivered at the Special Session on Reverse Mathematics held at the Atlanta AMS meeting, on January 6, 2005.

This distinction between logic and mathematics is subject to various criticisms and can be given various defenses. Nevertheless, the division seems natural enough and is commonly adopted in presentations of the standard foundations for mathematics.

Mathematical Logic had a glorious period in the 1930s, which was briefly rekindled in the 1960s. Any Shock Value, such as it is, has surrounded unprovability from ZFC.

An extreme kind of logic skeptic claims that "the present formal systems used for the foundations of mathematics are artificially strong, thereby causing unnecessary headaches such as the Gödel incompleteness phenomena". The skeptic continues by claiming that "logician's systems always contain overly general assertions, and/or assertions about overly general notions, that are not used in any significant way in normal mathematics. For example, induction for all statements, or even all statements of certain restricted forms, is far too general - mathematicians (...) only use induction for natural statements that actually arise. If logicians would tailor their formal systems to conform to the naturalness of normal mathematics, then various logical difficulties would disappear, and the story of the foundations of mathematics would look radically different than it does today. In particular, it should be possible to give a convincing model of actual mathematical practice that can be proved to be free of contradiction using methods that lie within what Hilbert had in mind in connection with his program”. Here we present some specific results in the direction of refuting this point of view, and introduce the Strict Reverse Mathematics (SRM) program. (shrink)

I don’t remember if I got as high as 2-390, but I distinctly remember taking my first logic course - as a Freshman - with Hartley Rogers, in Fall 1964 - here in 2-190. Or was it in 2-290?

We say that R is strictly dominating if and only if for all x,yŒ[1,n], if R(x,y) then max(x) 3k ¥ [1,n], there exists A Õ [1,n] such that R = A. Furthermore, A Õ [1,n] is unique.

We present some new decision and comparison problems of unusually high computational complexity. Most of the problems are strictly combinatorial in nature; others involve basic logical notions. Their complexities range from iterated exponential time completeness to (0 time completeness to ((((,0) time completeness to ((((,,0) time completeness. These three ordinals are well known ordinals from proof theory, and their associated com- plexity classes represent new levels of computational complexity for natural decision problems. Proofs will appear in an extended version of (...) this manuscript to be published elsewhere. (shrink)

We have been particularly interested in the demonstrable unremovability of machinery, which is a theme that can be pursued systematically starting at the most elementary level - the use of binary notation to represent integers; the use of rational numbers to solve linear equations; the use of real and complex numbers to solve polynomial equations; and the use of transcendental functions to solve differential equations.

In fact, Godel gave an important model of pure predication, where he showed that restricted comprehension without parameters is valid, but where restricted comprehension with parameters is not (although this invalidity was not established until Cohen). This is the model based on ordinal definability in set theory.