Mathematical Logic Seminar Abstracts Winter 2008

This paper investigates ambiguities between modals and definite descriptions, and in particular, the treatment of these ambiguities in dynamic semantics. Such ambiguities are a mainstay of the Russellian analysis of definite descriptions, which treats definite descriptions as quantifiers entering widely into scope ambiguities with modal operators. In contrast, classical dynamic theories of definites (such as Kamp's discourse representation theory and Heim's file change semantics) treat definites as fundamentally like free variables, and hence as non-scope-taking. These theories are thus challenged by the appearance of ambiguities with modal operators. This paper shows that in fact, the range of such ambiguities is far more limited than is commonly noted (building on work of Rothschild). It then explores how the ambiguities actually to be found can be accounted for in a dynamic framework, without making definite descriptions themselves scope-taking operators. It shows how the extant ambiguities can be explained in terms of how the presuppositions of definites map to the restrictors of modals, rather than in terms of operator scope.

In the logic seminar meetings for January 22 and January 29 we will be working through the paper, "Finite sets and Gödel's incompleteness theorems," by S. Swierczkowski, in which a fully detailed proof is given of both incompleteness theorems for a theory HF of finite sets equivalent in strength to Peano Arithmetic, PA. The detailed proof of the second incompleteness theorem, which is in principle mechanizable, may be the first of its kind. In order to cover all the material in this paper, we need to extend the time period of the seminar beyond its usual length to 6:05. Copies of the paper will be made available to all participants. The presentation of the material will be divided up, according to the following schedule:

In the logic seminar meetings for January 22 and January 29 we will be working through the paper, "Finite sets and Gödel's incompleteness theorems," by S. Swierczkowski, in which a fully detailed proof is given of both incompleteness theorems for a theory HF of finite sets equivalent in strength to Peano Arithmetic, PA. The detailed proof of the second incompleteness theorem, which is in principle mechanizable, may be the first of its kind. In order to cover all the material in this paper, we need to extend the time period of the seminar beyond its usual length to 6:05. Copies of the paper will be made available to all participants. The presentation of the material will be divided up, according to the following schedule:

There is a certain standard myth about Frege's interest in identity statements, centering on the notion that his interest was motivated by a desire to justify a particular philosophical theory of meaning. On this accounting, answering why "Hesperus is Phosphorus" and "Hesperus is Hesperus" don’t mean the same thing led Frege to posit a substantive account of the meaning of expressions - the doctrine of sense and reference - and it is this doctrine that constitutes Frege's enduring contribution. There is of course more than a grain of truth to this, but at heart it is misleading as to Frege's real interest in identity statements; namely the essential role they play in logicism, especially in establishing the fundamental tenet that numbers are "self-subsistent" logical objects. For Frege, the puzzle of identity statements called for solution just in order to maintain this result. In this paper, I will explore how this plays out in the development of Frege's views on identity statements, proposing that the evolution from a view which countenances both metalinguistic identity and mathematical equality, to one in which there is a single notion of objectual identity, tracks a change in Frege's conception of the relation of language and content, from language structuring content to language representing content. In the course of the presentation, I will explore the answers to the following questions, among others: Why did Frege initially adopt a metalinguistic view? What caused him to change his view to one in which identity statements express objectual identity? What role do identity statements play in the logicist program? What is the significance of the puzzle, and what is its origin? And how does "On Sense and Reference" fit into Frege's oeuvre?

ZFVLIF stands for

ZF + V=L + ∀α∃λ>α("λ is inaccessible") + "every symmetric partial order has a symmetric generic filter"

Starting with a model of this theory, I build a model of NF. A crucial ingredient in this construction is a significant improvement of the interpretation of NF given in [Boffa88]. Although the theory ZFVLIF itself might be inconsistent (as well as some other theories in the respected literature), I think the interpretation of NF I present has more independent value. It remains a question whether symmetric generic filters do exist, or whether I really need them.

References

• [Boffa88] Boffa, M. ZFJ and the consistency problem for NF, Jarhbuch der Kurt-Gödel-Gesellschaft, vol. 1988, pp. 102–106.

Two of the main sources of ineffectivity of proofs in ordinary mathematics are the uses of (1) Heine-Borel compactness and (2) sequential compactness. From reverse mathematics, (1) is known to correspond to the binary (`weak') Koenig's lemma WKL while (2) corresponds to arithmetical comprehension CA_ar. Whereas techniques such as monotone functional interpretation show that WKL does not contribute to the growth of extractable bounds from proofs, CA_ar in general strongly does contribute. However, experience with concrete proofs based on sequential compactness shows that usually this contribution is very limited. We will present some recent results which explain this to a large extent and also discuss some applications to proofs in nonlinear analysis.

A new consistency proof for first-order arithmetic PA is presented. The proof uses the schema of bar recursion. The goal of further work is an extension to analysis (second-order arithmetic).

The Dodd-Jensen core model is of the form K=L^E where E is a sequence of measures. We structure the model L^E by a continuous fine hierarchy (F^E_α)_α∈Ord. Each F^E_α is a structure of the form F^E_α = (F^E_α,∈E,S^E,…), which contains a Skolem function S^E and other basic constructible operations. The next level F^E_α+1 is the collection of all subsets of F^E_α which are definable over the structure F^E_α by quantifier-free formulas. The hierarchy satisfies condensation theorems and other finestructural laws.

The sequence E consists of measures E_α which are represented as elementary maps extenders E_α: F^E_δ→F^E_α. Core model theory can be developed with the fine hierarchy. One can canonically define finestructural extensions (ultrapowers) of levels F^E_γ by extenders in E. If all proper initial segments of F^E_γ are finestructurally sound then this is inherited by finestructural extensions. Iterated finestructural extensions can be used to compare structures F^E_γ and F^E′_γ^′. The unique predicate E defining K consists of measures for which the formation of finestructural extensions can be iterated arbitrarily (iterability).

The use of the fine hierarchy instead of standard fine structure theory circumvents the complications of iterated projecta and reducts and simplifies the construction of finestructural extensions.