The Ground Axiom

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.

arXiv: Logic, 2017

In light of the celebrated theorem of Vop\v{e}nka (1972), proving in ZFC that every set is generic over HOD, it is natural to inquire whether the set-theoretic universe $V$ must be a class-forcing extension of HOD by some possibly proper-class forcing notion in HOD. We show, negatively, that if ZFC is consistent, then there is a model of ZFC that is not a class-forcing extension of its HOD for any class forcing notion definable in HOD and with definable forcing relations there (allowing parameters). Meanwhile, S. Friedman (2012) showed, positively, that if one augments HOD with a certain ZFC-amenable class $A$, definable in $V$, then the set-theoretic universe $V$ is a class-forcing extension of the expanded structure $\langle\text{HOD},\in,A\rangle$. Our result shows that this augmentation process can be necessary. The same example shows that $V$ is not necessarily a class-forcing extension of the mantle, and the method provides counterexamples to the intermediate model property, n...

Acta Mathematica, 2013

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.

Archive for Mathematical Logic, 2016

We introduce and study the first-order Generic Vopěnka's Principle, which states that for every definable proper class of structures C of the same type, there exist B = A in C such that B elementarily embeds into A in some set-forcing extension. We show that, for n ≥ 1, the Generic Vopěnka's Principle fragment for Πn-definable classes is equiconsistent with a proper class of n-remarkable cardinals. The n-remarkable cardinals hierarchy for n ∈ ω, which we introduce here, is a natural generic analogue for the C (n)-extendible cardinals that Bagaria used to calibrate the strength of the first-order Vopěnka's Principle in [1]. Expanding on the theme of studying set theoretic properties which assert the existence of elementary embeddings in some set-forcing extension, we introduce and study the weak Proper Forcing Axiom, wPFA, which states that for every transitive model M in the language of set theory with some ω 1-many additional relations, if it is forced by a proper forcing P that M satisfies some Σ 1-property, then V has a transitive modelM, which satisfies the same Σ 1property, and in some set-forcing extension there is an elementary embedding fromM into M. This is a weakening of a formulation of PFA due to Schindler and Claverie [2], which asserts that the embedding fromM to M exists in V. We show that wPFA is equiconsistent with a remarkable cardinal and that wPFA implies PFA ℵ 2 , the proper forcing axiom for antichains of size at most ω 2 , but it is consistent with κ for all κ ≥ ω 2 , and therefore does not imply PFA ℵ 3 .

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].

The Logica 2020 Yearbook, 2021

In this paper, I argue that one of the arguments usually put forward in defence of universism is in tension with current set theoretic practice. According to universism, there is only one set theoretic universe, V , and when applying the method of forcing we are not producing new universes, but only simulating them inside V. Since the usual interpretation of set generic forcing is used to produce a "simulation" of an extension of V from a countable set inside V itself, the above argument is credited to be a strong defence of universism. However, I claim, such an argument does not take into account current mathematical practice. Indeed, it is possible to find theorems that are available to the multiversists but that the advocate of universism cannot prove. For example, it is possible to prove results on infinite games in non-well-founded set-theories plus the axiom of determinacy (such as ZF + AFA + PD) that are not available in ZFC + P D. These results, I contend, are philosophically problematic on a strict universist approach to forcing. I suggest that the best way to avoid the difficulty is to adopt a pluralist conception of set theory and embrace a set theoretic multiverse. Consequently, the current practice of set generic forcing better supports a multiverse conception of set theory.

In this article, we want to pin down the discussion of what classes are to the roots of their mathematical use in axiomatic set theories. In order to extrapolate the intended or best suitable interpretation of classes, in the first part we outline most of the set theories that incorporate proper classes and avail ourselves of the attitude the inventors displayed towards their classes. In the second part, we discuss – adopting a more systematic approach – some of the more metamathematical questions about classes and their relation to sets.

We show that Dependent Choice is a sufficient choice principle for developing the basic theory of proper forcing, and for deriving generic absoluteness for the Chang model in the presence of large cardinals, even with respect to DC-preserving symmetric submodels of forcing extensions. Hence, ZF+DC not only provides the right framework for developing classical analysis, but is also the right base theory over which to safeguard truth in analysis from the independence phenomenon in the presence of large cardinals. We also investigate some basic consequences of the Proper Forcing Axiom in ZF, and formulate a natural question about the generic absoluteness of the Proper Forcing Axiom in ZF + DC and ZFC. Our results confirm ZF + DC as a natural foundation for a significant portion of "classical mathematics" and provide support to the idea of this theory being also a natural foundation for a large part of set theory.

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.