Set theory and truth

This thesis covers a number of related topics in the foundations of set theory. Section 1 contains an analysis of the paradoxes which suggests a way of looking at the axioms of any formalised set theory as to a certain extent legislating how 'is a member of' and 'set' are to be used. In section 2, two other points of view are examined and rejected. There are doubtless other views which directly contradict the view put forward in section 1, but clearly one cannot examine all the different solutions of the paradoxes in any one thesis. The positions examined in section 2 seem to represent views that are most opposed to the general tendency of this thesis. Section 3 consists of a critical examination of Tarski's work on the semantic conception of truth for a formalised calculus of classes. This section has two aspects. The first is a defence of Tarski against some of his critics; the second an attack on certain of Tarski's conclusions. Section 4 begins where section 3 leaves off. it examines Tarski's conclusions about a definition of truth for set theory. It contains an analysis of Godel's results on the incompleteness of formal systems and is particularly concerned with the transference of certain inferences made from Godel's theorem for a formalised arithmetic to a formalised set theory. The conclusion of section 4 returns to the theme of section 1 and utilises certain metatheorems recently proved about formalised set theories. Throughout the thesis no one axiom system of set theory is presupposed though reference will be made to several. The discussion is of a general nature and can be taken to be about any axiom system of set theory rather than some particular one

Existence, knowledge & truth in mathematics

This thesis offers an overview of some current work in the philosophy of mathematics, in particular of work on the metaphysical, epistemological, and semantic problems associated with mathematics, and it also offers a theory about what type of entities numbers are. Starting with a brief look at the historical and philosophical background to the problems of knowledge of mathematical facts and entities, the thesis then tackles in depth, and ultimately rejects as flawed, the work in this area of Hartry Field, Penelope Maddy, Jonathan Lowe, John Bigelow, and also some aspects of the work of Philip Kitcher and David Armstrong. Rejecting both nominalism and physicalism, but accepting accounts from Bigelow and Armstrong that numbers can be construed as relations, the view taken in this work is that mathematical objects, numbers in particular, are universals, and as such are mind dependent entities. It is important to the arguments leading to this conception of mathematical objects, that there is a notion of aspectual seeing involved in mathematical conception. Another important feature incorporated is the notion, derived from Anscombe, of an intentional object. This study finishes by sketching what appears to be a fruitful line of enquiry with some significant advantages over the other accounts discussed. The line taken is that the natural numbers are mind dependent intentional relations holding between intentional individuals, and that other classes of number - the rationals, the reals, and so on - are mind dependent intentional relations holding between other intentional relations. The distinction in type between the natural numbers and the rest, is the intuitive one that is drawn naturally in language between the objects referred to by the so-called count nouns, and the objects referred to by the so-called mass nouns

Elementary recursive complexity results in real algebraic geometry (Women in Mathematics)

I shall discuss two important results in real algebraic geometry - quantifier elimination, proving that the projection of a semi-algebraic set is semi-algebraic - Hilbert 17th problem, proving that a non negative polynomial is always a sum of squares of rational functions from the point of view of effectivity and complexity. The two problems look at first sight totally un related at all but it turns out that modern computer algebra techniques play a key role in proving elementary recursive complexity results for both these problems

Invariance and Logicality in Perspective

Although the invariance criterion of logicality first emerged as a criterion of a purely mathematical interest, it has developed into a criterion of considerable linguistic and philosophical interest. In this paper I compare two different perspectives on this criterion. The first is the perspective of natural language. Here, the invariance criterion is measured by its success in capturing our linguistic intuitions about logicality and explaining our logical behavior in natural-linguistic settings. The second perspective is more theoretical. Here, the invariance criterion is used as a tool for developing a theoretical foundation of logic, focused on a critical examination, explanation, and justification of its veridicality and modal force

The continuum hypothesis : independence and truth-value

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Philosophy, 1974.MIT Humanities Library copy: issued in two vols.Leaf number 84 used twice. Also issued as a two-volume set.Includes bibliographical references (leaves 217-258).by Thomas S. Weston.Ph.D

Tarski and Lesniewski on Languages with Meaning versus Languages without Use: A 60th Birthday Provocation for Jan Wolenski

Constructivism, Judgement and Meanin