This paper reconstructs Steel’s multiverse project in his ‘Gödel’s program’ (Steel, 2014), first by comparing it to those of Hamkins (2012) and Woodin (2011), then by detailed analysis what’s presented in Steel’s brief text. In particular, we reconstruct his notion of a ‘natural’ theory, describe his multiverse axioms and his translation function, and assess the resulting status of the Continuum Hypothesis. In the end, we reconceptualize the defect that Steel thinks might suffer from and isolate what it would take to remove it while working within his framework. As our goal is to present as coherent and compelling a philosophical and mathematical story as we can, we allow ourselves to augment Steel’s story in places (e.g., in the treatment of Amalgamation) and to depart from it in others (e.g., the removal of ‘meaning’ from the account). The relevant mathematics is laid out in the appendices.

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Abstract. We provide infinitary proof theories for three common semantic theories of truth: strong Kleene, van Fraassen supervaluation and Cantini supervaluation. The value of these systems is that they provide an easy method of proving simple facts about semantic theories. Moreover we shall show that they also give us a simpler understanding of the computational complexity of these definitions and provide a direct proof that the closure ordinal for Kripke's definition is ω C K 1 . This work can be understood as an effort to provide a proof-theoretic counterpart to Welch's gametheoretic (Welch, 2009). §1. Introduction. In Kripke (1975), Kripke introduced fixed point theories of truth to the philosophy of language. The definitions of these truth predicates are conducted using transfinite recursion and the theory of inductive definitions. While such theories are well understood in mathematical logic (Moschovakis, 1974), the resultant definitions are complicated both in terms of heuristics and computation. Proofs about membership in these fixed points are conducted informally in the metalanguage and are often contingent on a series of lemmas establishing various properties about the fixed point in question. This sort of reasoning is analogous to the way one may reason about a modal logic using its semantics. With modal logic, however, we usually also have a proof theory which, having established soundness and completeness, allows us to establish claims in a simple and transparent manner.The aim of this paper is to provide a similarly simple and transparent means of verifying simple claims about the extension of the truth predicate. To do this we shall make use of infinitary tableau systems. We shall then establish that each of the systems provided is sound and complete with respect to their associated fixed points. The paper is broken into a section for each of the tableau systems developed. After this, we prove that each system gives a 1 1 -complete set as its intended extension and provide a direct proof showing that the height of the fixed point is ω C K 1 . 1.1. Semantic theories of truth. In this section, we define each of the fixed point truth definitions used in this paper, although we shall assume some familiarity with the basic construction (Kripke, 1975). We restrict ourselves to providing a truth definition for the standard model of arithmetic, N and we assume that we are in the language L T of arithmetic expanded with a predicate T intended to represent truth. The language of arithmetic will be denoted as L. Let Sent L and Sent L T denote the sentences of L and L T respectively. We assume that we have a recursive bijection · : Sent L T ∼ = ω. We use ϕ, ψ, χ as variables for sentences from Sent L T in the metalanguage.

AbstractfThe purpose of this paper is to investigate categoricity arguments conducted in second order logic and the philosophical conclusions that can be drawn from them. We provide a way of seeing this result, so to speak, through a first order lens divested of its second order garb. Our purpose is to draw into sharper relief exactly what is involved in this kind of categoricity proof and to highlight the fact that we should be reserved before drawing powerful philosophical conclusions from it.

It is a commonplace of set theory to say that there is no set of all wellorderings nor a set of all sets. We are implored to accept this due to the threat of paradox and the ensuing descent into unintelligibility. In the absence of promising alternatives, we tend to take up a conservative stance and tow the line: there is no universe (Halmos, in: Naive set theory, 1960). In this paper, I am going to challenge this claim by taking seriously the idea that we can talk about the collection of all the sets and many more collections beyond that. A method of articulating this idea is offered through an indefinitely extending hierarchy of set theories. It is argued that this approach provides a natural extension to ordinary set theory and leaves ordinary mathematical practice untouched.Keywords Philosophy of mathematics · Philosophy of set theory · Formal theories of truth · Absolute generality · Indefinite extensibility You're gonna need a bigger boat.(Jaws)The idea that there ought to be a set containing all of the sets is the focus of this paper. It is not a new idea. Moreover, I suspect that my way of addressing this issue will have occurred to many readers, although usually they they will have set it aside. I do not want to suggest that the techniques explored in this paper are of themselves particularly original; we shall see that related approaches have been explored from a technical angle for many years. Rather, what I am aiming to do is give this group of ideas a more philosophical twist by drawing out their underlying motivations and B Toby Meadows