Abstract We extend in a natural way the operation of Turing machines to infinite ordinal time, and investigate resulting supertask theory computability decidability on reals. Every set. for example, is decidable by such machines, semi-decidable sets form portion sets. Our oracle concept leads notion relative reals rich degree structure, stratified two jump operators.

Abstract The multiverse view in set theory, introduced and argued for this article, is the that there are many distinct concepts of set, each instantiated a corresponding set-theoretic universe. universe view, contrast, asserts an absolute background concept, with which every question has definite answer. position, I argue, explains our experience enormous range possibilities, phenomenon challenges view. In particular, argue continuum hypothesis settled on by extensive knowledge about how it...

Abstract In this paper, following an idea of Christophe Chalons, I propose a new kind forcing axiom, the Maximality Principle , which asserts that any sentence φ holding in some extension V ℙ and all subsequent extensions ℙ*ℚ holds already . It follows, fact, such sentences must also hold modal terms, therefore, is expressed by scheme (◊ □ ) ⇒ equivalent to theory S 5. article, prove relatively consistent with ZFC. A boldface version Principle, obtained allowing real parameters appear...

If an extension $V\subseteq{\overline{V}}$ satisfies the $\delta$ approximation and cover properties for classes $V$ is a class in ${\overline{V}}$, then every suitably closed embedding $j:{\overline{V}}\to\overline{N}$ ${\overline{V}}$ with criti

We show that the theory , consisting of usual axioms but with power set axiom removed—specifically axiomatized by extensionality, foundation, pairing, union, infinity, separation, replacement and assertion every can be well‐ordered—is weaker than commonly supposed is inadequate to establish several basic facts often desired in its context. For example, there are models which ω 1 singular, reals countable, yet exists, sets size none therefore, collection fails; for Łoś theorem fails, even...

A set theoretical assertion<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="psi"><mml:semantics><mml:mi>ψ</mml:mi><mml:annotation encoding="application/x-tex">\psi</mml:annotation></mml:semantics></mml:math></inline-formula>is<italic>forceable</italic>or<italic>possible</italic>, written<inline-formula alttext="lozenge psi"><mml:semantics><mml:mrow><mml:mi>◊</mml:mi><mml:mi>ψ</mml:mi></mml:mrow><mml:annotation...

Abstract The Lévy-Solovay Theorem[8] limits the kind of large cardinal embeddings that can exist in a small forcing extension. Here I announce generalization this theorem to broad new class notions. One consequence is many iterations most commonly found literature create no weakly compact cardinals, measurable strong Woodin strongly supercompact almost huge and so on.

The dream solution of the continuum hypothesis (CH) would be a by which we settle on basis newly discovered fundamental principle set theory, missing axiom, widely regarded as true. Such indeed solution, since all accept new axiom along with its consequences. In this article, however, I argue that such to CH is unattainable. article adapted from and expands upon material in my "The set-theoretic multiverse", appear Review Symbolic Logic (see arXiv:1108.4223).

The halting problem for Turing machines is decidable on a set of asymptotic probability one.The proof sensitive to the particular computational models.

The notion of computable reducibility between equivalence relations on the natural numbers provides a analogue Borel reducibility.We investigate hierarchy, comparing and contrasting it with hierarchy from descriptive set theory.Meanwhile, appears well suited for an analysis c.e. sets, more specifically, various classes structures.This is rich context many examples, such as isomorphism relation graphs or computably presented groups.Here, our exposition extends earlier work in literature...

We prove that the satisfaction relation $\mathcal{N}\models\varphi[\vec a]$ of first-order logic is not absolute between models set theory having structure $\mathcal{N}$ and formulas $\varphi$ all in common. Two can have same natural numbers, for example, standard model arithmetic $\langle\mathbb{N},{+},{\cdot},0,1,{\lt}\rangle$, yet disagree on their theories truth; two numbers truths, truths-about-truth, at any desired level iterated truth-predicate hierarchy; reals, projective $\langle...

We consider a set-theoretic version of mereology based on the inclusion relation ⊆ and analyze how well it might serve as foundation mathematics. After establishing non-definability ∈ from ⊆, we identify natural axioms for ⊆-based mereology, which constitute finitely axiomatizable, complete, decidable theory. Ultimately, these reasons, conclude that this form cannot by itself Meanwhile, augmented forms such obtained adding singleton operator, are foundationally robust.

After small forcing, almost every strongness embedding is the lift of a in ground model. Consequently, forcing creates neither strong nor Woodin cardinals.

Abstract We analyze the precise modal commitments of several natural varieties set-theoretic potentialism, using tools we develop for a general model-theoretic account building on those Hamkins, Leibman and Löwe [14], including use buttons, switches, dials ratchets. Among potentialist conceptions consider are: rank potentialism (true in all larger $V_\beta $ ), Grothendieck–Zermelo $V_\kappa inaccessible cardinals $\kappa transitive-set transitive sets), forcing extensions),...

If ZFC is consistent, then the collection of countable computably saturated models satisfies all Multiverse Axioms Hamkins.

Boolean ultrapowers extend the classical ultrapower construction to work with ultrafilters on any complete algebra, rather than only a power set algebra. When they are well-founded, associated embeddings exhibit large cardinal nature, and thereby unifies two central themes of theory---forcing cardinals---by revealing them be facets single underlying construction, ultrapower.