We develop a unified framework for dealing with constructibility and absoluteness in set theory, decidability of relations in effective structures (like the natural numbers), and domain independence of queries in database theory. Our framework and results suggest that domain-independence and absoluteness might be the key notions in a general theory of constructibility, predicativity, and computability. 1

Are minds subject to laws of physics? Are the laws of physics computable? Are conscious thought processes computable? Currently there is little agreement as to what are the right answers to these questions. Penrose ([41], p. 644) goes one step further and asserts that: a radical new theory is indeed needed, and I am suggesting, moreover, that this theory, when it is found, will be of an essentially non-computational character. The aim of this paper is three fold: 1) to examine the incompatibility between the hypothesis of strong determinism and computability, 2) to give new examples of uncomputable physical laws, and 3) to discuss the relevance of Gödel’s Incompleteness Theorem in refuting the claim that an algorithmic theory—like strong AI—can provide an adequate theory of mind. Finally, we question the adequacy of the theory of computation to discuss physical laws and thought processes. 1

In this thesis we introduce a general notion of a D-ring generalizing that of a differential or difference ring. In Chapter 3, this notion is specialized to consider valued fields D-fields: valued fields K having an operator D : K ! K and a fixed element e 2 K satisfying D(x + y) = Dx + Dy, D(1) = 0, D(xy) = xDy + yDx + eDxDy, v(e) 0, and v(Dx) v(x). Upon a further specialization, namely that the residue field has characteristic zero and v(e) > 0, we present axioms for the model completion and prove a version of the Ax-Kochen-Ershov principle. Using the general model theory of valued D-fields and results of Hrushovski on groups definable in separably closed fields we prove a characteristic p analog of Buium's abc theorem for semi-abelian varieties. Using the same general results on estimates in valued D-fields together with results of Chatzidakis and Hrushovski on groups definable in transformally closed fields we prove a version of a conjecture of Tate and Voloch on the p-adic distance from t...

After defining Axiom of Monotonicity, it is used along with Zermelo-Fraenkel set theory to derive Cantor's Continuum Hypothesis. Several related theorems are also proved.

Distributing the elements of # 1 within a unit interval, intuitive arguments are given to justify the Continuum Hypothesis, suggesting that it should be accepted.

Abstract. We consider the problem of transition from a weakly separated family of probability measures to a strictly separated family.

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Mathematics is in a dramatic and massive process of changing, mainly due to the advent of computers and computer science. Our aim is to present a pocket image of this phenomenon; a "case study" will give us the opportunity to describe some of these new ideas, problems, and techniques. Particularly, we will be concerned with foreseeable mutations in the interaction between deductive and experimental trends.

The concept of a set is familiar to most mathematicians and we refer the reader to [2] for details of the various axiomatizations of set theory. Fuzzy sets have been studied by Zadeh and others (see [4] and [9], for instance) and they attempt to model the notion of classes which do not have precisely defined criteria of membership. Multisets

We suggest a new basic framework for the Weyl-Feferman predicativist program by constructing a formal predicative set theory PZF which resembles ZF. The basic idea is that the predicatively acceptable instances of the comprehension schema are those which determine the collections they define in an absolute way, independent of the extension of the “surrounding universe”. This idea is implemented using syntactic safety relations between formulas and sets of variables. These safety relations generalize both the notion of domain-independence from database theory, and Godel notion of absoluteness from set theory. The language of PZF is type-free, and it reflects real mathematical practice in making an extensive use of statically defined abstract set terms. Another important feature of PZF is that its underlying logic is ancestral logic (i.e. the extension of FOL with a transitive closure operation). 1