Automorphisms of models of set theory and extensions of NFU

@article{McKenzie2015AutomorphismsOM,
  title={Automorphisms of models of set theory and extensions of NFU},
  author={Zachiri McKenzie},
  journal={Ann. Pure Appl. Log.},
  year={2015},
  volume={166},
  pages={601-638}
}
  • Zachiri McKenzie
  • Published 18 September 2013
  • Materials Science
  • Ann. Pure Appl. Log.

INITIAL SELF-EMBEDDINGS OF MODELS OF SET THEORY

Abstract By a classical theorem of Harvey Friedman (1973), every countable nonstandard model $\mathcal {M}$ of a sufficiently strong fragment of ZF has a proper rank-initial self-embedding j, i.e.,

Largest initial segments pointwise fixed by automorphisms of models of set theory

The main achievement is the calculation of the theory of the Theory of C of structures of the form C as precisely MOST-Δ0P-MOST, a recursively saturated model of MOST that is closed under exponentiation and contains $\omega ^\mathcal {M}$$ωM.

Self-similarity in the Foundations

This thesis concerns embeddings and self-embeddings of foundational structures in both set theory and category theory, and the formulation of a novel algebraic set theory which is proved to be equiconsistent to New Foundations, and which can be modulated to correspond to intuitionistic or classical NF.

On the strength of a weak variant of the axiom of counting

It is shown that Ronald Jensen's modification of Quine's `New Foundations' Set Theory ($\mathrm{NF}$) fortified with a type-level pairing function but without the Axiom of Choice proves the consistency of the Simple Theory of Types with Infinity.

On the relative strengths of fragments of collection

This paper studies the relative strength of set theories obtained by adding fragments of the set-theoretic collection scheme to $\mathbf{M}$ and proves the consistency of Zermelo Set Theory plus strong $\Pi_n$-collection.

ALGEBRAIC NEW FOUNDATIONS

A novel categorical set theory, MLCat, is proved to be equiconsistent to New Foundations (NF), and which can be modulated to correspond to intuitionistic or classical NF, with atoms.

Largest initial segments pointwise fixed by automorphisms of models of set theory

Given a model M\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek}

References

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Automorphisms , Mahlo Cardinals , and NFU

In this paper we establish an intimate relationship between Mahlo cardinals and automorphisms of models of set theory. The models of set theory involved here are not well-founded since a routine

On the Consistency of a Slight (?) Modification of Quine’s New Foundations

Quine’s system of set theory, New Foundations (NF), can be conveniently formalized as a first-order theory containing two predicates ≡ (identity) and e (set membership). One of the most attractive

Strong axioms of infinity in NFU

This paper discusses a sequence of extensions of NFU, Jensen's improvement of Quine's set theory “New Foundations” (NF) of [16], which was to restrict extensionality to sets, allowing many non-sets (urelements) with no elements.

Finite Sets in Quine's New Foundations

Some axiomatic systems of set theory related to the system NF (New Foundations) of Quine are considered, including the possible relations of cardinality between a finite set x and its subset class SC(x) = { y | y ∩ x} and also between X and its unit set class USC(x).

FROM BOUNDED ARITHMETIC TO SECOND ORDER ARITHMETIC VIA AUTOMORPHISMS

In this paper we examine the relationship between automorphisms of models of I∆0 (bounded arithmetic) and strong systems of arithmetic, such as PA, ACA0 (arithmetical comprehension schema with

The strength of Mac Lane set theory

Permutations and wellfoundedness: the true meaning of the bizarre arithmetic of Quine's NF

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  • Mathematics, Philosophy
    Journal of Symbolic Logic
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It is shown that, according to NF, many of the assertions of ordinal arithmetic involving the T-function turn out to be equivalent to the truth-in-certain-permutation-models of assertions which have perfectly sensible ZF-style meanings.

End-extensions preserving power set

Abstract We consider the quantifier hierarchy of Takahashi [1972] and show how it gives rise to reflection theorems for some large cardinals in ZF, a new natural subtheory of Zermelo's set theory, a

The consistency strength of NFUB

We show that the consistency strength of the system NFUB, a variant of Quine's "New Foundations" recently introduced by Randall Holmes, is precisely that of [ZFC - Power Set] + "There is a weakly