Set-Theoretic Geology

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.

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...

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.

The Journal of Symbolic Logic, 1995

We prove that a number of axioms, each a consequence of PFA (the Proper Forcing Axiom) are equivalent. In particular we show that TOP (the Thinning-out Principle as introduced by Baumgartner in the Handbook of set-theoretic topology), is equivalent to the following statement: If I is an ideal on co, with co, generators, then there exists an uncountable X C co,, such that either [X]w n I = 0 or [X]w C I. ?1. Introduction. In this paper we study relations between some consequences of the Proper Forcing Axiom (PFA). Among them we consider the Thinning-out Principle (TOP) introduced by Baumgartner in [B], and the partition calculus axiom co, ) (co, (co,; fin cw1))2 proposed by Todorcevic in [T]. We show that each of these two axioms can be restated in a simpler way, and then we easily deduce that Todorcevic's axiom (which we call Axiom S in this paper) is a consequence of TOP. We will then show how our versions of these axioms give simplified proofs of the applications of these axioms in [B] and [T]. We will show that the following axiom is equivalent to TOP: AXIOM 0. Let S = {S,: a < co } be a collection of (countable) subsets of coi, such that for every uncountable X C co,, there exists a countable set Q C X which cannot be covered by a finite subfamily of S. Then there exists an uncountable subset of co1 which meets each S, in a finite set. We will show that the following (weaker) version of Axiom 0 is equivalent to Axiom S: AXIOM 1. Let S = {S,: a < co, } be a collection of (countable) subsets of W1, such that for every uncountable X C co,, there exists a countable set Q C X which cannot be covered by a finite subfamily of S. Then there exists uncountable set X C co,, such that for each a C X, S, n X is a finite set.

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.

Proceedings of the American Mathematical Society, 1989

Let M M be a countable transitive model of ZFC and A A be a countable M M -generic family of Cohen reals. We prove that there is no smallest transitive model N N of ZFC that either M ∪ A ⊆ N M \cup A \subseteq N or M ∪ { A } ⊆ N M \cup \{ A\} \subseteq N . It is also proved that there is no smallest transitive model N N of ZFC − ^{-} (ZFC theory without the power set axiom) such that M ∪ { A } ⊆ N M \cup \{ A\} \subseteq N . It is also proved that certain classes of extensions of M M obtained by Cohen generic reals have no minimal model.

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.

A central area of current philosophical debate in the foundations of mathematics concerns whether or not there is a single, maximal, universe of set theory. Universists maintain that there is such a universe, while Multiversists argue that there are many universes, no one of which is ontologically privileged. Often model-theoretic constructions that add subsets to models are cited as evidence in favour of the latter. This paper informs this debate by analysing ways the Uni-versist might interpret talk that seems to necessitate the addition of subsets to V. We argue that, despite the prima facie incoherence of such talk for the Universist, she nonetheless has reason to try and provide interpretation of this discourse. We analyse extant interpretations of such talk, and find some of them lacking. Finally, we argue that the Universist should accept the existence of a countable transitive model elementarily equivalent to V , and can interpret constructions over models of this form.

Acta Mathematica, 2013

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 .