In set theory, a maximality principle is a principle that asserts some maximality property of the universe of sets or some part thereof. Set theorists have formulated a variety of maximality principles in order to settle statements left undecided by current standard set theory. In addition, philosophers of mathematics have explored maximality principles whilst attempting to prove categoricity theorems for set theory or providing criteria for selecting foundational theories. This article reviews recent work concerned with the formulation, investigation and justification (...) of maximality principles. (shrink)

We will give a simple philosophical "proof" of the negation of Cantor's continuum hypothesis (CH). (A formal proof for or against CH from the axioms of ZFC is impossible; see Cohen [1].) We will assume the axioms of ZFC together with intuitively clear axioms which are based on some intuition of Stuart Davidson and an old theorem of Sierpinski and are justified by the symmetry in a thought experiment throwing darts at the real number line. We will in fact show (...) why there must be an infinity of cardinalities between the integers and the reals. We will also show why Martin's Axiom must be false, and we will prove the extension of Fubini's Theorem for Lebesgue measure where joint measurability is not assumed. Following the philosophy--if you reject CH you are only two steps away from rejecting the axiom of choice (AC)--we will point out along the way some extensions of our intuition which contradict AC. (shrink)

If we replace first-order logic by second-order logic in the original definition of Gödel’s inner model L, we obtain the inner model of hereditarily ordinal definable sets [33]. In this paper...

In this work we give a complete answer as to the possible implications between some natural properties of Lebesgue measure and the Baire property. For this we prove general preservation theorems for forcing notions. Thus we answer a decade-old problem of J. Baumgartner and answer the last three open questions of the Kunen-Miller chart about measure and category. Explicitly, in \S1: (i) We prove that if we add a Laver real, then the old reals have outer measure one. (ii) We (...) prove a preservation theorem for countable-support forcing notions, and using this theorem we prove (iii) If we add ω 2 Laver reals, then the old reals have outer measure one. From this we obtain (iv) $\operatorname{Cons}(\mathrm{ZF}) \Rightarrow \operatorname{Cons}(\mathrm{ZFC} + \neg B(m) + \neg U(m) + U(c))$ . In \S2: (i) We prove a preservation theorem, for the finite support forcing notion, of the property " $F \subseteq ^\omega\omega$ is an unbounded family." (ii) We introduce a new forcing notion making the old reals a meager set but the old members of ω ω remain an unbounded family. Using this we prove (iii) $\operatorname{Cons}(\mathrm{ZF}) \Rightarrow \operatorname{Cons}(\mathrm{ZFC} + U(m) + \neg B(c) + \neg U(c) + C(c))$ . In \S3: (i) We prove a preservation theorem, for the finite support forcing notion, of a property which implies "the union of the old measure zero sets is not a measure zero set," and using this theorem we prove (ii) $\operatorname{Cons}(\mathrm{ZF}) \Rightarrow \operatorname{Cons}(\mathrm{ZFC} + \neg U(m) + C(m) + \neg C(c))$. (shrink)

This article addresses the question of fundamental entities in set theory. It takes up J. Hamkins’ claim that models of set theory are such fundamental entities and investigates it using the methodology of P. Maddy’s naturalism, Second Philosophy. In accordance with this methodology, I investigate the historical case study of the use of models in the introduction of forcing, compare this case to contemporary practice and give a systematic account of how set-theoretic practice can be said to introduce models as (...) new entities. In conclusion, I argue for a view that takes both sets and models to be fundamental entities in set theory. (shrink)

We define the notion of Souslin forcing, and we prove that some properties are preserved under iteration. We define a weaker form of Martin's axiom, namely MA(Γ + ℵ 0 ), and using the results on Souslin forcing we show that MA(Γ + ℵ 0 ) is consistent with the existence of a Souslin tree and with the splitting number s = ℵ 1 . We prove that MA(Γ + ℵ 0 ) proves the additivity of measure. Also we introduce (...) the notion of proper Souslin forcing, and we prove that this property is preserved under countable support iterated forcing. We use these results to show that ZFC + there is an inaccessible cardinal is equiconsistent with ZFC + the Borel conjecture + Σ 1 2 -measurability. (shrink)

We consider a classM of Boolean algebras with strictly positive, finitely additive measures. It is shown thatM is closed under iterations with finite support and that the forcing via such an algebra does not destroy the Lebesgue measure structure from the ground model. Also, we deduce a simple characterization of Martin's Axiom reduced to the classM.

Let J be any proper ideal of subsets of the real line R which contains all finite subsets of R. We define an ideal J * ∣B as follows: X ∈ J * ∣B if there exists a Borel set $B \subset R \times R$ such that $X \subset B$ and for any x ∈ R we have $\{y \in R: \langle x,y\rangle \in B\} \in \mathscr{J}$ . We show that there exists a family $\mathscr{A} \subset \mathscr{J}^\ast\mid\mathscr{B}$ of power ω (...) 1 such that $\bigcup\mathscr{A} \not\in \mathscr{J}^\ast\mid\mathscr{B}$ . In the last section we investigate properties of ideals of Lebesgue measure zero sets and meager sets in Cohen extensions of models of set theory. (shrink)

We strengthen a result of Harrington and Shelah by showing that, unless ω1 is an inaccessible cardinal in L, a relatively weak fragment of Martin's axiom implies that there exists a δ13 set of reals without the property of Baire.

In this paper, we explore some of the consequences of Martin’s Conjecture on degree invariant Borel maps. These include the strongest conceivable ergodicity result for the Turing equivalence relation with respect to the filter on the degrees generated by the cones, as well as the statement that the complexity of a weakly universal countable Borel equivalence relation always concentrates on a null set.

We deal with the consistency strength of ZFC + variants of MA + suitable sets of reals are measurable (and/or Baire, and/or Ramsey). We improve the theorem of Harrington and Shelah [2] repairing the asymmetry between measure and category, obtaining also the same result for Ramsey. We then prove parallel theorems with weaker versions of Martin's axiom (MA(σ-centered), (MA(σ-linked)), MA(Γ + ℵ 0 ), MA(K)), getting Mahlo, inaccessible and weakly compact cardinals respectively. We prove that if there exists r ∈ (...) R such that ω L[ r] 1 = ω 1 and MA holds, then there exists a ▵ 1 3 -selective filter on ω, and from the consistency of ZFC we build a model for ZFC + MA(I) + every ▵ 1 3 -set of reals is Lebesgue measurable, has the property of Baire and is Ramsey. (shrink)

On the occasion of his 70th birthday, the work of Adrian Mathias in set theory is surveyed in its full range and extent.

We introduce a property of forcing notions, called the anti-, which comes from Aronszajn trees. This property canonically defines a new chain condition stronger than the countable chain condition, which is called the property . In this paper, we investigate the property . For example, we show that a forcing notion with the property does not add random reals. We prove that it is consistent that every forcing notion with the property has precaliber 1 and for forcing notions with the (...) property fails. This negatively answers a part of one of the classical problems about implications between fragments of. (shrink)

We study the projective posets and their properties as forcing notions. We also define Martin's axiom restricted to projective sets, MA, and show that this axiom is weaker than full Martin's axiom by proving the consistency of ZFC + ¬lCH + MA with “there exists a Suslin tree”, “there exists a non-strong gap”, “there exists an entangled set of reals” and “there exists κ < 20 such that 20 < 2k”.

We show that if there is a nonconstructible real, then every perfect set has a nonconstructible element, answering a question of K. Prikry. This is a specific instance of a more general theorem giving a sufficient condition on a pair $M\subset N$ of models of set theory implying that every perfect set in N has an element in N which is not in M.

Moore introduced the Mapping Reflection Principle and proved that the Bounded Proper Forcing Axiom implies that the size of the continuum is ℵ2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\aleph _2$$\end{document}. The Mapping Reflection Principle follows from the Proper Forcing Axiom. To show this, Moore utilized forcing notions whose conditions are countable objects. Chodounský–Zapletal introduced the Y-Proper Forcing Axiom that is a weak fragments of the Proper Forcing Axiom but implies some important conclusions from the Proper Forcing Axiom, (...) for example, the P-ideal Dichotomy. In this article, it is proved that the Y-Proper Forcing Axiom implies the Mapping Reflection Principle by introducing forcing notions whose conditions are finite objects. (shrink)

Let K 0 λ be the class of structures $\langle\lambda, , where $A \subseteq \lambda$ is disjoint from a club, and let K 1 λ be the class of structures $\langle\lambda, , where $A \subseteq \lambda$ contains a club. We prove that if $\lambda = \lambda^{ is regular, then no sentence of L λ+κ separates K 0 λ and K 1 λ . On the other hand, we prove that if $\lambda = \mu^+,\mu = \mu^{ , and a forcing axiom (...) holds (and ℵ L 1 = ℵ 1 if μ = ℵ 0 ), then there is a sentence of L λλ which separates K 0 λ and K 1 λ. (shrink)

If the bounded proper forcing axiom BPFA holds and ω 1 = ${\mathrm{\omega }}_{1}^{\mathrm{L}}$ , then there is a lightface ${\mathrm{\Sigma }}_{3}^{1}$ well-ordering of the reals. The argument combines a well-ordering due to Caicedo-Veličković with an absoluteness result for models of MA in the spirit of "David's trick." We also present a general coding scheme that allows us to show that BPFA is equiconsistent with R being lightface ${\mathrm{\Sigma }}_{4}^{1}$ , for many "consistently locally certified" relations R on $\mathrm{\mathbb{R}}$ . (...) This is accomplished through a use of David's trick and a coding through the ∑ 2 stable ordinals of L. (shrink)

Azriel Levy did fundamental work in set theory when it was transmuting into a modern, sophisticated field of mathematics, a formative period of over a decade straddling Cohen’s 1963 founding of forcing. The terms “Levy collapse”, “Levy hierarchy”, and “Levy absoluteness” will live on in set theory, and his technique of relative constructibility and connections established between forcing and definability will continue to be basic to the subject. What follows is a detailed account and analysis of Levy’s work and contributions (...) to set theory. (shrink)

We improve a theorem of Raisonnier by showing that Cons(ZFC+every Σ 2 1 -set of reals in Lebesgue measurable+every Π 2 1 -set of reals isK σ-regular) implies Cons(ZFC+there exists an inaccessible cardinal). We construct, fromL, a model where every Δ 3 1 -sets of reals is Lebesgue measurable, has the property of Baire, and every Σ 2 1 -set of reals isK σ-regular. We prove that if there exists a Σ n+1 1 unbounded filter on ω, then there exists (...) a nonK σ-regular Π 2 1 -subset. (shrink)

We define a generic Vopěnka cardinal to be an inaccessible cardinal \ such that for every first-order language \ of cardinality less than \ and every set \ of \-structures, if \ and every structure in \ has cardinality less than \, then an elementary embedding between two structures in \ exists in some generic extension of V. We investigate connections between generic Vopěnka cardinals in models of ZFC and the number and complexity of \-Suslin sets of reals in models (...) of ZF. In particular, we show that ZFC + is equiconsistent with ZF + \\) where \ is the pointclass of all \-Suslin sets of reals, and also with ZF + \\) + \\) where \ is the least ordinal that is not a surjective image of the reals. (shrink)

The basic purpose of this essay, the first of an intended pair, is to interpret standard von Neumann quantum theory in a framework of iterated measure algebraic truth for mathematical (and thus mathematical-physical) assertions — a framework, that is, in which the truth-values for such assertions are elements of iterated boolean measure-algebras (cf. Sections 2.2.9, 5.2.1–5.2.6 and 5.3 below).The essay itself employs constructions of Takeuti's boolean-valued analysis (whose origins lay in work of Scott, Solovay, Krauss and others) to provide a (...) metamathematical interpretation of ideas sometimes considered disparate, heuristic, or simply ill-defined: the collapse of the wave function, for example; Everett's many worlds'-construal of quantum measurement; and a natural product space of contextual (nonlocal) hidden variables. (shrink)

We analyse here the monadic theory of the rational order, the monadic theory of the real line with quantification over "small" subsets and models of these theories. We prove that the results are in some sense the best possible.

Let κ B be the least cardinal for which the Baire category theorem fails for the real line R. Thus κ B is the least κ such that the real line can be covered by κ many nowhere dense sets. It is shown that κ B cannot have countable cofinality. On the other hand it is consistent that the corresponding cardinal for 2 ω 1 be ℵ ω . Similar questions are considered for the ideal of measure zero sets, other (...) ω 1 saturated ideals, and the ideal of zero-dimensional subsets of R ω 1. (shrink)

We consider the possible complexity of the set of reals belonging to an inner model M of set theory. We show that if this set is analytic then either 1M is countable or else all reals are in M. We also show that if an inner model contains a superperfect set of reals as a subset then it contains all reals. On the other hand, it is possible to have an inner model M whose reals are an uncountable Fσ set (...) and which does not have all reals. A similar construction shows that there can be an inner model M which computes correctly 1, contains a perfect set of reals as a subset and yet not all reals are in M. These results were motivated by questions of H. Friedman and K. Prikry. (shrink)

It is consistent that for many cardinals λ there is a family of at least λ + unbounded subsets of λ which have uniformization properties. In particular if it is consistent that a supercompact cardinal exists, then it is consistent that ℵ ω has such a family. We have applications to point set topology, Whitehead groups and reconstructing separable abelian p-groups from their socles.

In the absence of Woodin cardinals, fine structural inner models for mild large cardinal hypotheses admit forcing extensions where bounded forcing axioms hold and yet the reals are projectively well-ordered.

The class of stable posets is defined and investigated. We give a forcing construction of a universe of set theory which satisfies a weak form of Martin's Axiom and $2^{\aleph_0} > \aleph_1$ and yet some propositions which follow from CH hold in this universe.