Incomparable $\omega_1$-like models of set theory

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 .

Proceedings of the American Mathematical Society, 1980

We determine when a model M \mathfrak {M} of ZF can be expanded to a model ⟨ M , X ⟩ \langle \mathfrak {M},\mathfrak {X}\rangle of a weak extension of Gödel Bernays: GB + {\text {GB}} + the Δ 1 1 \Delta _1^1 comprehension axiom. For nonstandard M \mathfrak {M} , the ordinal of the standard part of M \mathfrak {M} must equal the inductive closure ordinal of M \mathfrak {M} , and M \mathfrak {M} must satisfy the axioms of ZF with replacement and separation for formulas involving predicates for all hyperelementary relations on M \mathfrak {M} . We also consider expansions to models of GB + Σ 1 1 {\text {GB}} + \Sigma _1^1 choice, observe that the results actually apply to more general theories of well-founded relations, and observe relationships to expansibility to models of other second order theories.

Bulletin of Symbolic Logic, 1995

In this paper, we sketch the development of two important themes of modern set theory, both of which can be regarded as growing out of work of Kurt Gödel. We begin with a review of some basic concepts and conventions of set theory. §0. The ordinal numbers were Georg Cantor's deepest contribution to mathematics. After the natural numbers 0, 1, …, n, … comes the first infinite ordinal number ω, followed by ω + 1, ω + 2, …, ω + ω, … and so forth. ω is the first limit ordinal as it is neither 0 nor a successor ordinal. We follow the von Neumann convention, according to which each ordinal number α is identified with the set {ν ∣ ν α} of its predecessors. The ∈ relation on ordinals thus coincides with <. We have 0 = ∅ and α + 1 = α ∪ {α}. According to the usual set-theoretic conventions, ω is identified with the first infinite cardinal ℵ0, similarly for the first uncountable ordinal number ω1 and the first uncountable cardinal number ℵ1, etc. We thus arrive at the following picture...

Fundamenta Mathematicae, 2003

In the early 1970's S.Tennenbaum proved that all countable models of P A − + ∀ 1 − T h(N) are embeddable into the reduced product N ω /F, where F is the cofinite filter. In this paper we show that if M is a model of P A − + ∀ 1 − T h(N), and |M | = ℵ 1 , then M is embeddable into N ω /D, where D is any regular filter on ω.

This paper explores the undecidability of the existence of weakly inaccessible cardinals in the von Neumann universe under Zermelo-Fraenkel set theory with the Axiom of Choice (ZFC). Through a novel logical framework inspired by Gödel's incompleteness theorems, we demonstrate that the statement "there exists a weakly inaccessible cardinal" is neither provable nor disprovable in ZFC, assuming ZFC is consistent. The framework introduces intuitive concepts such as the intrinsic distinctness of "super-large" objects, a "functional escape" from the properties of smaller objects, and the limits of formal systems. These ideas are formalized to show that weakly inaccessible cardinals transcend ZFC's constructive capabilities, aligning with Gödelian incompleteness. The paper provides technical details on forcing and the constructible universe L, clarifies the original contribution of the logical approach, and contextualizes weakly inaccessible cardinals among other large cardinals. Implications for set theory and future research directions are discussed.

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.

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.

2016

A pointwise definable model is one in which every object is definable without parameters. In a model of set theory, this property strengthens V = HOD, but is not first-order expressible. Nevertheless, if ZFC is consistent, then there are continuum many pointwise definable models of ZFC. If there is a transitive model of ZFC, then there are continuum many pointwise definable transitive models of ZFC. What is more, every countable model of ZFC has a class forcing extension that is pointwise definable. Indeed, for the main contribution of this article, every countable model of Gödel-Bernays set theory has a pointwise definable extension, in which every set and class is first-order definable without parameters. 2000 Mathematics Subject Classification. 03E55. Key words and phrases. set theory, forcing. 1 See [Ani] for an instance of the argument at MathOverflow, which surely serves a brisk cup of math tea online. We leave aside the remark of Horatio, eight-year-old son of the first author, who announced, "Sure, papa, I can describe any number. Let me show you: tell me any number, and I'll tell you a description of it!"

British Journal of Mathematics & Computer Science, 2015

In this article we derived an importent example of the inconsistentcountable set in second order ZF C (ZF C2) with the full second-order semantic.Main results is: (i) ¬Con(ZF C2), (ii) let k be an inaccessible cardinal and H k is a set of all sets having hereditary size less then k, then ¬Con(ZF C + (V = H k )).

In this article we derived an importent example of the inconsistent countable set in second order ZFC ZFC 2  with the full second-order semantic. Main results is:(i) ConZFC 2 , (ii) let k be an inaccessible cardinal and H k is a set of all sets having hereditary size less then k, then ConZFC  V  H k .