Compactness of Loeb Spaces

In this paper we show that the compactness of a Loeb space depends on its cardinality, the nonstandard universe it belongs to and the underlying model of set theory we live in. In section 1 we prove that Loeb spaces are compact under various assumptions, and in section 2 we prove that Loeb spaces are not compact under various other assumptions. The results in section 1 and section 2 give a quite complete answer to a question of D. Ross

A multiverse perspective on the axiom of constructiblity

I shall argue that the commonly held V not equal L via maximize position, which rejects the axiom of constructibility V = L on the basis that it is restrictive, implicitly takes a stand in the pluralist debate in the philosophy of set theory by presuming an absolute background concept of ordinal. The argument appears to lose its force, in contrast, on an upwardly extensible concept of set, in light of the various facts showing that models of set theory generally have extensions to models of V = L inside larger set-theoretic universes.Comment: 21 pages. This article expands on an argument that I made during my talk at the Asian Initiative for Infinity: Workshop on Infinity and Truth, held July 25--29, 2011 at the Institute for Mathematical Sciences, National University of Singapore. Commentary concerning this paper can be made at http://jdh.hamkins.org/multiverse-perspective-on-constructibilit

Automorphism Groups of Countable Arithmetically Saturated Models of Peano Arithmetic

If M,N are countable, arithmetically saturated models of Peano Arithmetic and Aut(M) is isomorphic to Aut(N), then the Turing-jumps of Th(M) and Th(N) are recursively equivalent.Comment: This version is a complete revision of the previous version. The main result of this version greatly improves the main result of the earlier versio

Solutions to the "General Grand Unification Problem," and the Questions "How Did Our Universe Come Into Being?" and "Of What is Empty Space Composed?"

Using mathematical techniques to model one of the most simplistic of human linguistic processes, it is rationally predicted that within the nonstandard physical world (NSP-world) there exists a force-like (logical) operator *S and an entity w' such that *S{w'} sequentially generates each of the Natural systems that comprise a Universe. This model shows specifically that within the NSP-world the behavior of each Natural world Natural system is related logically. Further, the model predicts the rational existence of a single type of entity within the NSP-world's substratum that can be used to construct, by means of an exceptionally simple process, all of the fundamental Natural world particles used within particle physics. In section 11.2, it is shown how (Natural law) allowable perturbations in Natural system behavior are also included within this mathematical model. These results solve the pre-geometry problem of Wheeler. In general, the model predicts that when the behavior of these Universe creating processes is viewed globally, it can be described as apparently mirroring the behavior of an infinitely powerful computer or mind.Comment: Plain Tex, 74 pages. For this final version, a discussion of the continuous verses the discrete development is given. Numerously many additional and recently published references are given and their application is noted within the appropriate section. arXiv admin note: substantial text overlap with arXiv:math/990308

Nonstandard model categories and homotopy theory

In order to apply nonstandard methods to questions of algebraic geometry we continue our investigation from "Enlargements of categories" (Theory Appl. Categ. 14 (2005), No. 16, 357--398) and show how important homotopical constructions behave under enlargements.Comment: 15 page

Tanaka's Theorem Revisited

Tanaka (1997) proved a powerful generalization of Friedman's self-embedding theorem that states that given a countable nonstandard model $(\mathcal{M},\mathcal{A})$ of the subsystem $\mathrm{WKL}_{0}$ of second order arithmetic, and any element $m$ of $\mathcal{M}$, there is a self-embedding $j$ of $(\mathcal{M},\mathcal{A})$ onto a proper initial segment of itself such that $j$ fixes every predecessor of $m$. Here we extend Tanaka's work by establishing the following results for a countable nonstandard model $(\mathcal{M},\mathcal{A})$ of $\mathrm{WKL}_{0}$ and a proper cut $\mathrm{I}$ of $\mathcal{M}$: Theorem A. The following conditions are equivalent: (a) $\mathrm{I}$ is closed under exponentiation. (b) There is a self-embedding $j$ of $(\mathcal{M},\mathcal{A})$ onto a proper initial segment of itself such that $I$ is the longest initial segment of fixed points of $j$. Theorem B. The following conditions are equivalent: (a) $\mathrm{I}$ is a strong cut of $\mathcal{M}$ and $\mathrm{I}\prec _{\Sigma _{1}}\mathcal{M}.$ (b) There is a self-embedding $j$ of $(\mathcal{M},\mathcal{A})$ onto a proper initial segment of itself such that $\mathrm{I}$ is the set of all fixed points of $j$.Comment: 15 page

What Is Boolean Valued Analysis?

This is a brief overview of the basic techniques of Boolean valued analysis.Comment: 25 pages with a few improvement

Hahn Field Representation of A. Robinson's Asymptotic Numbers

Let $^*\mathbb{R}$ be a nonstandard extension of $\mathbb{R}$ and $\rho$ be a positive infinitesimal in $^*\mathbb{R}$. We show how to create a variety of isomorphisms between A. Robinson's field of asymptotic numbers $^\rho\mathbb{R}$ and the Hahn field $\hat{^\rho\mathbb{R}}(t^\mathbb{R})$, where $\hat{^\rho\mathbb{R}}$ is the residue class field of $^\rho\mathbb{R}$. Then, assuming that $^*\mathbb{R}$ is fully saturated we show that $\hat{^\rho\mathbb{R}}$ is isomorphic to $^*\mathbb{R}$ and so $^\rho\mathbb{R}$ contains a copy of $^*\mathbb{R}$. As a consequence (that is important for applications in non-linear theory of generalized functions) we show that every two fields of asymptotic numbers corresponding to different scales are isomorphic.Comment: 18 page

Categorical large cardinals and the tension between categoricity and set-theoretic reflection

Inspired by Zermelo's quasi-categoricity result characterizing the models of second-order Zermelo-Fraenkel set theory $\text{ZFC}_2$, we investigate when those models are fully categorical, characterized by the addition to $\text{ZFC}_2$ either of a first-order sentence, a first-order theory, a second-order sentence or a second-order theory. The heights of these models, we define, are the categorical large cardinals. We subsequently consider various philosophical aspects of categoricity for structuralism and realism, including the tension between categoricity and set-theoretic reflection, and we present (and criticize) a categorical characterization of the set-theoretic universe $\langle V,\in\rangle$ in second-order logic.Comment: 26 pages. Commentary about this article can be made on the first author's web page at http://jdh.hamkins.org/categorical-large-cardinal

Fixed Points of Self-embeddings of Models of Arithmetic

We investigate the structure of fixed point sets of self-embeddings of models of arithmetic. In particular, given a countable nonstandard model M of a modest fragment of Peano arithimetic, we provide complete characterizations of (a) the initial segments of M that can be realized as the longest initial segment of fixed points of a nontrivial self-embedding of M onto a proper initial segment of M; and (b) the initial segments of M that can be realized as the fixed point set of some nontrivial self-embedding of M onto a proper initial segement of M. Moreover, we demonstrate the the standard cut is strong in M iff there is a self-embedding of M onto a proper initial segment of itself that moves every element that is not definable in M by an existential formula.Comment: 36 page; this is a revised draft in which misprints of the previous draft are correcte