Physical Sciences and Mathematics Commons^™

Countable Short Recursively Saturated Models Of Arithmetic, Erez Shochat

Dissertations, Theses, and Capstone Projects

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Short recursively saturated models of arithmetic are exactly the elementary initial segments of recursively saturated models of arithmetic. Since any countable recursively saturated model of arithmetic has continuum many elementary initial segments which are already recursively saturated, we turn our attention to the (countably many) initial segments which are not recursively saturated. We first look at properties of countable short recursively saturated models of arithmetic and show that although these models cannot be cofinally resplendent (an expandability property slightly weaker than resplendency), these models have non-definable expansions which are still short recursively saturated.

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The Ground Axiom, Jonas Reitz

Dissertations, Theses, and Capstone Projects

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A new axiom is proposed, the Ground Axiom, asserting that the universe is not a nontrivial set-forcing extension of any inner model. The Ground Axiom is first-order expressible, and any model of ZFC has a class-forcing extension which satisfies it. The Ground Axiom is independent of many well-known set-theoretic assertions including the Generalized Continuum Hypothesis, the assertion V=HOD that every set is ordinal definable, and the existence of measurable and supercompact cardinals. The related Bedrock Axiom, asserting that the universe is a set-forcing extension of a model satisfying the Ground Axiom, is also first-order expressible, and its negation is consistent. …

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Infinitely Often Dense Bases And Geometric Structure Of Sumsets, Jaewoo Lee

Dissertations, Theses, and Capstone Projects

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We'll discuss two problems related to sumsets.

Nathanson constructed bases of integers with prescribed representation functions, then asked how dense bases for integers can be in such cases. Let A(-x, x) be the number of elements of A whose absolute value is less than or equal to x, then it's easy to see that A(-x, x) << x1/2 if its representation function is bounded, giving us a general upper bound. Chen constructed unique representation bases for integers with A(-x, x) ≥ x1/2-epsilon infinitely often. In the first chapter, we'll construct bases for integers with a prescribed representation function with A(-x, x) > x1/2/&phis;(x) infinitely often where &phis;(x) is any nonnegative real-valued function which tends to infinity.

In the second chapter, we'll see how sumsets appear geometrically. Assume A is a finite set of lattice points and h*D=h˙x:x∈conv A is a full dimensional polytope. Then we'll see …

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