The Role of the Foundation Axiom in the Kunen Inconsistency

We consider the role of the foundation axiom and various anti-foundation axioms in connection with the nature and existence of elementary selfembeddings of the set-theoretic universe.

2011

We show that the theory ZFC-, consisting of the usual axioms of ZFC but with the power set axiom removed-specifically axiomatized by extensionality, foundation, pairing, union, infinity, separation, replacement and the assertion that every set can be well-ordered-is weaker than commonly supposed and is inadequate to establish several basic facts often desired in its context. For example, there are models of ZFC-in which ω 1 is singular, in which every set of reals is countable, yet ω 1 exists, in which there are sets of reals of every size ℵn, but none of size ℵω, and therefore, in which the collection axiom fails; there are models of ZFC-for which the Loś theorem fails, even when the ultrapower is well-founded and the measure exists inside the model; there are models of ZFC-for which the Gaifman theorem fails, in that there is an embedding j : M → N of ZFC-models that is Σ 1 -elementary and cofinal, but not elementary; there are elementary embeddings j : M → N of ZFC-models whose cofinal restriction j : M → j " M is not elementary. Moreover, the collection of formulas that are provably equivalent in ZFC-to a Σ 1 -formula or a Π 1 -formula is not closed under bounded quantification. Nevertheless, these deficits of ZFC-are completely repaired by strengthening it to the theory ZFC − , obtained by using collection rather than replacement in the axiomatization above. These results extend prior work of Zarach [Zar96].

2021

We investigate systems of transitive models of ZFC which are elementarily embeddable into each other and the influence of definability properties on such systems. One of Ken Kunen’s best-known and most striking results is that there is no elementary embedding of the universe of sets into itself other than the identity. Kunen’s result is best understood in a theory that includes proper classes as genuine objects, such as von Neumann-Gödel-Bernays set theory (NBG), which we take in this article as our background theory. The argument proceeds by assuming there is a nontrivial j : V → V and using Replacement and Comprehension for formulas involving the parameter j, arriving at a contradiction. It does not exclude the possibility of the existence of an inner model M of Zermelo-Fraenkel set theory (ZFC) and a nontrivial elementary j : M → M , where M is not the class of sets of an NBG-universe that includes the class j. Indeed, Kunen’s early work on 0 showed an equivalence between the non...

2006

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. As many of these results rely on forcing with proper classes, an appendix is provided giving an exposition of the underlying theory of proper class forcing.

The universalist position in set theory maintains that there is only a single, maximal universe of sets, and all sentences about these objects are ideally verifiable. At this point, a prescriptive universalist will be committed to the claim that some mathematicians are wrong; while mathematicians may think that they are reasoning truthfully, they in fact fail to secure mathematical objects. In this paper, we engage with non-prescriptive universalists, those who attempt to account for the real diversity of mathematical practice. We will be analyzing the reduction strategies offered by universalists. Recently, Enayat in [Ena16] proved that no two models of ZF are bi-interpretable, while Hamkins and Freire in [FH20] proved that no two well-founded models of ZF are mutually interpretable. These results, we argue, put limitations on universalists' ability to produce isomorphic copies of alternative models of set theory. In orther words, adherents of a different universes have sufficient grounds to reject the alleged copy offered by the universalist as a faithful copy. Finally, we argue that the reasons for including additional elements to the multiverse should be specific instead of resulting from a reproduction of a model in a known universe.

2006

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.

Journal of Philosophical Logic, 1993

This note argues against Barwise and Etchemendy's claim that their semantics for self-reference requires use of Aczel's anti-foundational set theory, AFA, and that any alternative “would involve us in complexities of considerable magnitude, ones irrelevant to the task at hand” (The Liar, p. 35). Switching from ZF to AFA neither adds nor precludes any isomorphism types of sets. So it makes no difference to ordinary mathematics. I argue against the author's claim that a certain kind of ‘naturalness’ nevertheless makes AFA preferable to ZF for their purposes. I cast their semantics in a natural, isomorphism invariant form with self-reference as a fixed point property for propositional operators. Independent of the particulars of any set theory, this form is somewhat simpler than theirs and easier to adapt to other theories of self-reference.

Philosophia Mathematica, 2018

In recent work of McCallum a new large-cardinal axiom was formulated of strength intermediate between a totally indescribable cardinal and an ω-Erd˝ os cardinal, positing the existence of an " extremely reflective cardinal " , and it was demonstrated that the property of being extremely reflective was in fact equivalent to the property of being remarkable, and an argument was made that this axiom should be seen as intrinsically justified. This built on related earlier work in which the notion of an α-reflective cardinal was formulated. Welch and Roberts have recently put forward a family of reflection principles, Welch's principle implying the existence of a proper class of Shelah cardinals and provably consistent relative to a superstrong cardinal, and Roberts' principle implying the existence of a proper class of 1-extendible cardinals and provably consistent relative to a 2-extendible cardinal. Roberts tentatively argued that his principle should be seen as intrinsically justified (at least on the assumption that a weaker form of reflection involving reflection of second-order formulas with a second-order parameter should be seen as intrinsically justified). This work overlapped with previous work of Victoria Marshall's on reflection principles. We analyze the relationship between reflection principles equivalent to those studied by McCallum and stronger but similar reflection principles which are natural extensions of those of Welch and Roberts. We also show how a natural strengthening of Roberts' reflection principle yields the existence of supercompact cardinals, and in the process solve a question which Marshall left open, of whether her theory B 0 (V 0) is strong enough to imply the existence of supercompact cardinals. We also manage to resolve negatively her question of whether her theory B 1 (V 0) implies the existence of a huge cardinal.

Aequationes mathematicae, 2020

It is known that, if we take a countable model of Zermelo–Fraenkel set theory ZFC and “undirect” the membership relation (that is, make a graph by joining x to y if either $$x\in y$$ x ∈ y or $$y\in x$$ y ∈ x ), we obtain the Erdős–Rényi random graph. The crucial axiom in the proof of this is the Axiom of Foundation; so it is natural to wonder what happens if we delete this axiom, or replace it by an alternative (such as Aczel’s Anti-Foundation Axiom). The resulting graph may fail to be simple; it may have loops (if $$x\in x$$ x ∈ x for some x) or multiple edges (if $$x\in y$$ x ∈ y and $$y\in x$$ y ∈ x for some distinct x, y). We show that, in ZFA, if we keep the loops and ignore the multiple edges, we obtain the “random loopy graph” (which is $$\aleph _0$$ ℵ 0 -categorical and homogeneous), but if we keep multiple edges, the resulting graph is not $$\aleph _0$$ ℵ 0 -categorical, but has infinitely many 1-types. Moreover, if we keep only loops and double edges and discard single ed...

2018

This thesis concerns embeddings and self-embeddings of foundational structures in both set theory and category theory. The first part of the work on models of set theory consists in establishing a refined version of Friedman's theorem on the existence of embeddings between countable non-standard models of a fragment of ZF, and an analogue of a theorem of Gaifman to the effect that certain countable models of set theory can be elementarily end-extended to a model with many automorphisms whose sets of fixed points equal the original model. The second part of the work on set theory consists in combining these two results into a technical machinery, yielding several results about non-standard models of set theory relating such notions as self-embeddings, their sets of fixed points, strong rank-cuts, and set theories of different strengths. The work in foundational category theory consists in the formulation of a novel algebraic set theory which is proved to be equiconsistent to New ...