36 research outputs found
A new model construction by making a detour via intuitionistic theories II: Interpretability lower bound of Feferman's explicit mathematics T0
We partially solve a long-standing problem in the proof theory of explicit mathematics or the proof theory in general. Namely, we give a lower bound of Feferman’s system T0 of explicit mathematics (but only when formulated on classical logic) with a concrete interpretat ion of the subsystem Σ12-AC+ (BI) of second order arithmetic inside T0. Whereas a lower bound proof in the sense of proof-theoretic reducibility or of ordinalanalysis was already given in 80s, the lower bound in the sense of interpretability we give here is new. We apply the new interpretation method developed by the author and Zumbrunnen (2015), which can be seen as the third kind of model construction method for classical theories, after Cohen’s forcing and Krivine’s classical realizability. It gives us an interpretation between classical theories, by composing interpretations between intuitionistic theories
On Power Set Axiom: Internal Classes vs. External Sets
Usual math sets have special types: countable, compact, open, occasionally
Borel, rarely projective, etc. Generic sets dependent on Power Set axiom appear
mostly in esoteric areas, logic of Set Theory (ST), etc. Recognizing internal
to math (formula-specified) and external (based on parameters in those
formulas) aspects of math objects greatly simplifies the foundations.
I postulate external sets (not internally specified, treated as the domain of
variables) to be hereditarily countable and independent of formula-defined
classes, i.e. with only finite Kolmogorov Information about them. This allows
elimination of all non-integer quantifiers in ST formulas.Comment: 5 page
Changing a semantics: opportunism or courage?
The generalized models for higher-order logics introduced by Leon Henkin, and
their multiple offspring over the years, have become a standard tool in many
areas of logic. Even so, discussion has persisted about their technical status,
and perhaps even their conceptual legitimacy. This paper gives a systematic
view of generalized model techniques, discusses what they mean in mathematical
and philosophical terms, and presents a few technical themes and results about
their role in algebraic representation, calibrating provability, lowering
complexity, understanding fixed-point logics, and achieving set-theoretic
absoluteness. We also show how thinking about Henkin's approach to semantics of
logical systems in this generality can yield new results, dispelling the
impression of adhocness. This paper is dedicated to Leon Henkin, a deep
logician who has changed the way we all work, while also being an always open,
modest, and encouraging colleague and friend.Comment: 27 pages. To appear in: The life and work of Leon Henkin: Essays on
his contributions (Studies in Universal Logic) eds: Manzano, M., Sain, I. and
Alonso, E., 201
Type-Two Well-Ordering Principles, Admissible Sets, and Pi^1_1-Comprehension
This thesis introduces a well-ordering principle of type two, which we call the Bachmann-Howard principle. The main result states that the Bachmann-Howard principle is equivalent to the existence of admissible sets and thus to Pi^1_1-comprehension. This solves a conjecture of Rathjen and Montalbán. The equivalence is interesting because it relates "concrete" notions from ordinal analysis to "abstract" notions from reverse mathematics and set theory.
A type-one well-ordering principle is a map T which transforms each well-order X into another well-order T[X]. If T is particularly uniform then it is called a dilator (due to Girard). Our Bachmann-Howard principle transforms each dilator T into a well-order BH(T). The latter is a certain kind of fixed-point: It comes with an "almost" monotone collapse theta:T[BH(T)]->BH(T) (we cannot expect
full monotonicity, since the order-type of T[X] may always exceed the order-type of X). The Bachmann-Howard principle asserts that such a collapsing structure exists. In fact we define three variants of this principle: They are equivalent but differ in the sense in which the order BH(T) is "computed".
On a technical level, our investigation involves the following achievements: a detailed discussion of primitive recursive set theory as a basis for set-theoretic reverse
mathematics; a formalization of dilators in weak set theories and second-order arithmetic; a functorial version of the constructible hierarchy; an approach to deduction chains (Schütte) and beta-completeness (Girard) in a set-theoretic context; and a beta-consistency proof for Kripke-Platek set theory.
Independently of the Bachmann-Howard principle, the thesis contains a series of results connected to slow consistency (introduced by S.-D. Friedman, Rathjen and Weiermann): We present a slow reflection statement and investigate its consistency strength, as well as its computational properties. Exploiting the latter, we show that instances of the Paris-Harrington principle can only have extremely long proofs in certain fragments of arithmetic
Adding an Abstraction Barrier to ZF Set Theory
Much mathematical writing exists that is, explicitly or implicitly, based on
set theory, often Zermelo-Fraenkel set theory (ZF) or one of its variants. In
ZF, the domain of discourse contains only sets, and hence every mathematical
object must be a set. Consequently, in ZF, with the usual encoding of an
ordered pair , formulas like have truth values, and operations like have results that are sets. Such 'accidental theorems' do not match
how people think about the mathematics and also cause practical difficulties
when using set theory in machine-assisted theorem proving. In contrast, in a
number of proof assistants, mathematical objects and concepts can be built of
type-theoretic stuff so that many mathematical objects can be, in essence,
terms of an extended typed -calculus. However, dilemmas and
frustration arise when formalizing mathematics in type theory.
Motivated by problems of formalizing mathematics with (1) purely
set-theoretic and (2) type-theoretic approaches, we explore an option with much
of the flexibility of set theory and some of the useful features of type
theory. We present ZFP: a modification of ZF that has ordered pairs as
primitive, non-set objects. ZFP has a more natural and abstract axiomatic
definition of ordered pairs free of any notion of representation. This paper
presents axioms for ZFP, and a proof in ZF (machine-checked in Isabelle/ZF) of
the existence of a model for ZFP, which implies that ZFP is consistent if ZF
is. We discuss the approach used to add this abstraction barrier to ZF
Type-free truth
This book is a contribution to the flourishing field of formal and philosophical work on truth and the semantic paradoxes. Our aim is to present several theories of truth, to investigate some of their model-theoretic, recursion-theoretic and proof-theoretic aspects, and to evaluate their philosophical significance.
In Part I we first outline some motivations for studying formal theories of truth, fix some terminology, provide some background on Tarski’s and Kripke’s theories of truth, and then discuss the prospects of classical type-free truth. In Chapter 4 we discuss some minimal adequacy conditions on a satisfactory theory of truth based on the function that the truth predicate is intended to fulfil on the deflationist account. We cast doubt on the adequacy of some non-classical theories of truth and argue in favor of classical theories of truth.
Part II is devoted to grounded truth. In chapter 5 we introduce a game-theoretic semantics for Kripke’s theory of truth. Strategies in these games can be interpreted as reference-graphs (or dependency-graphs) of the sentences in question. Using that framework, we give a graph-theoretic analysis of the Kripke-paradoxical sentences. In chapter 6 we provide simultaneous axiomatizations of groundedness and truth, and analyze the proof-theoretic strength of the resulting theories. These range from conservative extensions of Peano arithmetic to theories that have the full strength of the impredicative system ID1.
Part III investigates the relationship between truth and set-theoretic comprehen- sion. In chapter 7 we canonically associate extensions of the truth predicate with Henkin-models of second-order arithmetic. This relationship will be employed to determine the recursion-theoretic complexity of several theories of grounded truth and to show the consistency of the latter with principles of generalized induction. In chapter 8 it is shown that the sets definable over the standard model of the Tarskian hierarchy are exactly the hyperarithmetical sets. Finally, we try to apply a certain solution to the set-theoretic paradoxes to the case of truth, namely Quine’s idea of stratification. This will yield classical disquotational theories that interpret full second-order arithmetic without set parameters, Z2- (chapter 9). We also indicate a method to recover the parameters.
An appendix provides some background on ordinal notations, recursion theory and graph theory
Universism and extensions of V
A central area of current philosophical debate in the foundations of mathematics concerns whether or not there is a single, maximal, universe of set theory. Universists maintain that there is such a universe, while Multiversists argue that there are many universes, no one of which is ontologically privileged. Often model-theoretic constructions that add sets to models are cited as evidence in favour of the latter. This paper informs this debate by developing a way for a Universist to interpret talk that seems to necessitate the addition of sets to V. We argue that, despite the prima facie incoherence of such talk for the Universist, she nonetheless has reason to try and provide interpretation of this discourse. We present a method of interpreting extension-talk (V-logic), and show how it captures satisfaction in `ideal' outer models and relates to impredicative class theories. We provide some reasons to regard the technique as philosophically virtuous, and argue that it opens new doors to philosophical and mathematical discussions for the Universist