Peter Koellner

Large Cardinals from Determinacy

This chapter gives an account of Woodin’s general technique for deriving large cardinal strength from determinacy hypotheses. These results appear here for the first time and the treatment is self-contained.

On reflection principles

Abstract Godel initiated the program of finding and justifying axioms that effect a significant reduction in incompleteness and he drew a fundamental distinction between intrinsic and extrinsic justifications. Reflection principles are the most promising candidates for new axioms that are intrinsically justified. Taking as our starting point Tait’s work on general reflection principles, we prove a series of limitative results concerning this approach. These results collectively show that general reflection principles are either weak (in that they are consistent relative to the Erdos cardinal κ ( ω ) ) or inconsistent. The philosophical significance of these results is discussed.

Truth in Mathematics: The Question of Pluralism

The discovery of non-Euclidean geometries (in the nineteenth century) undermined the claim that Euclidean geometry is the one true geometry and instead led to a plurality of geometries no one of which could be said (without qualification) to be “truer” than the others. In a similar spirit many have claimed that the discovery of independence results for arithmetic and set theory (in the twentieth century) has undermined the claim that there is one true arithmetic or set theory and that instead we are left with a plurality of systems no one of which can be said to be “truer” than the others. In this chapter I will investigate such pluralist conceptions of arithmetic and set theory. I will begin with an examination of what is perhaps the most sophisticated and developed version of the pluralist view to date—namely, that of Carnap in The Logical Syntax of Language—and I will argue that this approach is problematic and that the pluralism involved is too radical. In the remainder of the chapter I will investigate the question of what it would take to establish a more reasonable pluralism. This will involve mapping out some mathematical scenarios (using recent results proved jointly with Hugh Woodin) in which the pluralist could arguably maintain that pluralism has been secured.

Strong logics of first and second order

In this paper we investigate strong logics of first and second order that have certain absoluteness properties. We begin with an investigation of first order logic and the strong logics co-logic and /?-logic, isolating two facets of absoluteness, namely, generic invariance and faithfulness. It turns out that absoluteness is relative in the sense that stronger background assumptions secure greater degrees of absoluteness. Our aim is to investigate the hierarchies of strong logics of first and second order that are generically invariant and faithful against the backdrop of the strongest large cardinal hypotheses. We show that there is a close correspondence between the two hierarchies and we characterize the strongest logic in each hierarchy. On the first-order side, this leads to a new presentation of Woodins Q-logic. On the second-order side, we compare the strongest logic with full second-order logic and argue that the comparison lends support to Quines claim that second-order logic is really set theory in sheeps clothing. This paper is concerned with strong logics of first and second order. At the most abstract level, a strong logic of first-order has the following general form: Let L be a first-order language and let (x) be a formula that defines a class of L-structures. Then, for a recursively enumerable set T of sentences of L, and for a sentence ip of L set

Feferman on Set Theory: Infinity up on Trial

In this paper I examine Feferman’s reasons for maintaining that while the statements of first-order number theory are “completely clear” and “completely definite,” many of the statements of analysis and set theory are “inherently vague” and “indefinite.” I critique his five main arguments and argue that in the end the entire case rests on the brute intuition that the concept of subsets of natural numbers—along with the richer concepts of set theory—is not “clear enough to secure definiteness.” My response to this final, remaining point will be that the concept of “being clear enough to secure definiteness” is about as clear a case of an inherently vague and indefinite concept as one might find, and as such it can bear little weight in making a case against the definiteness of analysis and set theory.