Pure Σ2\Sigma_2-Elementarity beyond the Core

We display the entire structure ${\cal R}_2$ coding $\Sigma_1$- and $\Sigma_2$-elementarity on the ordinals. This will enable the analysis of pure $\Sigma_3$-elementary substructures.Comment: Extensive rewrite of the introduction. Mathematical content of sections 2 and 3 unchanged, extended introduction to section 2. Removed section 4. Theorem 4.3 to appear elsewhere with corrected proo

A predicative variant of a realizability tripos for the Minimalist Foundation.

open2noHere we present a predicative variant of a realizability tripos validating the intensional level of the Minimalist Foundation extended with Formal Church thesis.the file attached contains the whole number of the journal including the mentioned pubblicationopenMaietti, Maria Emilia; Maschio, SamueleMaietti, MARIA EMILIA; Maschio, Samuel

The Theory of Countable Analytical Sets

The purpose of this paper is the study of the structure of countable sets in the various levels of the analytical hierarchy of sets of reals. It is first shown that, assuming projective determinacy, there is for each odd n a largest countable ∏_n^1 set of reals, C_n (this is also true for n even, replacing ∏_n^1 by Σ_n^1 and has been established earlier by Solovay for n = 2 and by Moschovakis and the author for all even n > 2). The internal structure of the sets C_n is then investigated in detail, the point of departure being the fact that each C_n is a set of Δ_n^1-degrees, wellordered under their usual partial ordering. Finally, a number of applications of the preceding theory is presented, covering a variety of topics such as specification of bases, ω-models of analysis, higher-level analogs of the constructible universe, inductive definability, etc

Finite Sets And Symmetric Simplicial Sets

The category of finite cardinals (or, equivalently, of finite sets) is the symmetric analogue of the category of finite ordinals, and the ground category of a relevant category of presheaves, the augmented symmetric simplicial sets. We prove here that this ground category has characterisations similar to the classical ones for the category of finite ordinals, by the existence of a universal symmetric monoid, or by generators and relations. The latter provides a definition of symmetric simplicial sets by faces, degeneracies and transpositions, under suitable relations

Executing Gödel’s programme in set theory

The study of set theory (a mathematical theory of infinite collections) has garnered a great deal of philosophical interest since its development. There are several reasons for this, not least because it has a deep foundational role in mathematics; any mathematical statement (with the possible exception of a few controversial examples) can be rendered in set-theoretic terms. However, the fruitfulness of set theory has been tempered by two difficult yet intriguing philosophical problems: (1.) the susceptibility of naive formulations of sets to contradiction, and (2.) the inability of widely accepted set-theoretic axiomatisations to settle many natural questions. Both difficulties have lead scholars to question whether there is a single, maximal Universe of sets in which all set-theoretic statements are determinately true or false (often denoted by ‘V ’). This thesis illuminates this discussion by showing just what is possible on the ‘one Universe’ view. In particular, we show that there are deep relationships between responses to (1.) and the possible tools that can be used in resolving (2.). We argue that an interpretation of extensions of V is desirable for addressing (2.) in a fruitful manner. We then provide critical appraisal of extant philosophical views concerning (1.) and (2.), before motivating a strong mathematical system (known as‘Morse-Kelley’ class theory or ‘MK’). Finally we use MK to provide a coding of discourse involving extensions of V , and argue that it is philosophically virtuous. In more detail, our strategy is as follows: Chapter I (‘Introduction’) outlines some reasons to be interested in set theory from both a philosophical and mathematical perspective. In particular, we describe the current widely accepted conception of set (the ‘Iterative Conception’) on which sets are formed successively in stages, and remark that set-theoretic questions can be resolved on the basis of two dimensions: (i) how ‘high’ V is (i.e. how far we go in forming stages), and (ii) how ‘wide’ V is (i.e. what sets are formed at successor stages). We also provide a very coarse-grained characterisation of the set-theoretic paradoxes and remark that extensions of universes in both height and width are relevant for our understanding of (1.) and (2.). We then present the different motivations for holding either a ‘one Universe’ or ‘many universes’ view of the subject matter of set theory, and argue that there is a stalemate in the dialectic. Instead we advocate filling out each view in its own terms, and adopt the ‘one Universe’ view for the thesis. Chapter II (‘G¨odel’s Programme’) then explains the Universist project for formulating and justifying new axioms concerning V . We argue that extensions of V are relevant to both aspects of G¨odel’s Programme for resolving independence. We also identify a ‘Hilbertian Challenge’ to explain how we should interpret extensions of V , given that we wish to use discourse that makes apparent reference to such nonexistent objects. Chapter III (‘Problematic Principles’) then lends some mathematical precision to the coarse-grained outline of Chapter I, examining mathematical discourse that seems to require talk of extensions of V . Chapter IV (‘Climbing above V ?’), examines some possible interpretations of height extensions of V . We argue that several such accounts are philosophically problematic. However, we point out that these difficulties highlight two constraints on resolution of the Hilbertian Challenge: (i) a Foundational Constraint that we do not appeal to entities not representable using sets from V , and (ii) an Ontological Constraint to interpret extensions of V in such a way that they are clearly different from ordinary sets. 5 Chapter V (‘Broadening V ’s Horizons?’), considers interpretations of width extensions. Again, we argue that many of the extant methods for interpreting this kind of extension face difficulties. Again, however, we point out that a constraint is highlighted; a Methodological Constraint to interpret extensions of V in a manner that makes sense of our naive thinking concerning extensions, and links this thought to truth in V . We also note that there is an apparent tension between the three constraints. Chapter VI (‘A Theory of Classes’) changes tack, and provides a positive characterisation of apparently problematic ‘proper classes’ through the use of plural quantification. It is argued that such a characterisation of proper class discourse performs well with respect to the three constraints, and motivates the use of a relatively strong class theory (namely MK). Chapter VII (‘V -logic and Resolution’) then puts MK to work in interpreting extensions of V . We first expand our logical resources to a system called V -logic, and show how discourse concerning extensions can be thereby represented. We then show how to code the required amount of V -logic usingMK. Finally, we argue that such an interpretation performs well with respect to the three constraints. Chapter VIII (‘Conclusions’) reviews the thesis and makes some points regarding the exact dialectical situation. We argue that there are many different philosophical lessons that one might take from the thesis, and are clear that we do not commit ourselves to any one such conclusion. We finally provide some open questions and indicate directions for future research, remarking that the thesis opens the way for new and exciting philosophical and mathematical discussion

Gap-minimal systems of notations and the constructible hierarchy

If a constructibly countable ordinal alpha is a gap ordinal, then the order type of the set of index ordinals smaller than alpha is exactly alpha. The gap ordinals are the only points of discontinuity of a certain ordinal-valued function. The notion of gap minimality for well ordered systems of notations is defined, and the existence of gap-minimal systems of notations of arbitrarily large constructibly countable length is established