173 research outputs found
Reverse mathematics and well-ordering principles
The paper is concerned with generally Pi^1_2 sentences of the form 'if X is well ordered then f(X) is well ordered', where f is a standard proof theoretic function from ordinals to ordinals. It has turned out that a statement of this form is often equivalent to the existence of countable coded omega-models for a particular theory T_f whose consistency can be proved by means of a cut elimination theorem in infinitary logic which crucially involves the function f. To illustrate this theme, we shall focus on the well-known psi-function which figures prominently in so-called predicative proof theory. However, the approach taken here lends itself to generalization in that the techniques we employ can be applied to many other proof-theoretic functions associated with cut elimination theorems. In this paper we show that the statement 'if X is well ordered then 'X0 is well ordered' is equivalent to ATR0. This was first proved by Friedman, Montalban and Weiermann [7] using recursion-theoretic and combinatorial methods. The proof given here is proof-theoretic, the main techniques being Schuette's method of proof search (deduction chains) [13], generalized to omega logic, and cut elimination for infinitary ramified analysis
A hierarchy of ramified theories below primitive recursive arithmetic
The arithmetical theory EA(I;O) developed by Ăagman, Ostrin and Wainer ([18] and [48]) provides a formal setting for the variable separation of Bellantoni-Cook predicative recursion [6]. As such, EA(I;O) separates variables into outputs, which are quantified over, and inputs, for which induction applies. Inputs remain free throughout giving inductions in EA(I;O) a pointwise character termed predicative induction. The result of this restriction is that the provably recursive functions are the elementary functions. An infinitary analysis brings out a connection to the Slow-Growing Hierarchy yielding Ń0 as the appropriate proof-theoretic ordinal in a pointwise sense. Chapters 1 and 2 are devoted to an exposition of these results. In Chapter 3 a new principle of 1-closure is introduced in constructing a conservative extension of EA(I;O) named EA1. This principle collapses the variable separation in EA(I;O) and allows quantification over inputs by acting as an internalised Ï-rule. EA1 then provides a natural setting to address the problem of input substitution in ramified theories. Chapters 4 and 5 introduce a hierarchy of theories based upon alternate additions of the predicative induction and â1-closure principles. For 0 < k Ń N, the provably recursive functions of the theories EAk are shown to be the Grzegorczyk classes Ek+2. Upper bounds are obtained via embeddings into appropriately layered infinitary systems with carefully controlled bounding functions for existential quantifiers. The theory EA-Ï, defined by closure under finite applications of these two principles, is shown to be equivalent to primitive recursive arithmetic. The hierarchy generated may be considered as an implicit ramification of the sub-system of Peano Arithmetic which restricts induction to â1-formulae.EThOS - Electronic Theses Online ServiceGBUnited Kingdo
Functional interpretation and inductive definitions
Extending G\"odel's \emph{Dialectica} interpretation, we provide a functional
interpretation of classical theories of positive arithmetic inductive
definitions, reducing them to theories of finite-type functionals defined using
transfinite recursion on well-founded trees.Comment: minor corrections and change
Non-contractive logics, paradoxes, and multiplicative quantifiers
The paper investigates from a proof-theoretic perspective various non-contractive logical systems, which circumvent logical and semantic paradoxes. Until recently, such systems only displayed additive quantifiers (GriĆĄin and Cantini). Systems with multiplicative quantifiers were proposed in the 2010s (Zardini), but they turned out to be inconsistent with the naive rules for truth or comprehension. We start by presenting a first-order system for disquotational truth with additive quantifiers and compare it with GriĆĄin set theory. We then analyze the reasons behind the inconsistency phenomenon affecting multiplicative quantifiers. After interpreting the exponentials in affine logic as vacuous quantifiers, we show how such a logic can be simulated within a truth-free fragment of a system with multiplicative quantifiers. Finally, we establish that the logic for these multiplicative quantifiers (but without disquotational truth) is consistent, by proving that an infinitary version of the cut rule can be eliminated. This paves the way to a syntactic approach to the proof theory of infinitary logic with infinite sequents
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
Short Proofs for Slow Consistency
Let denote the finite
consistency statement "there are no proofs of contradiction in with
symbols". For a large class of natural theories , Pudl\'ak
has shown that the lengths of the shortest proofs of
in the theory
itself are bounded by a polynomial in . At the same time he conjectures that
does not have polynomial proofs of the finite consistency
statements . In contrast we show that Peano arithmetic
() has polynomial proofs of
,
where is the slow consistency statement for
Peano arithmetic, introduced by S.-D. Friedman, Rathjen and Weiermann. We also
obtain a new proof of the result that the usual consistency statement
is equivalent to iterations
of slow consistency. Our argument is proof-theoretic, while previous
investigations of slow consistency relied on non-standard models of arithmetic
Recommended from our members
Mathematical Logic: Proof Theory, Constructive Mathematics (hybrid meeting)
The Workshop "Mathematical Logic: Proof Theory,
Constructive Mathematics" focused on
proofs both as formal derivations in deductive systems as well as on
the extraction of explicit computational content from
given proofs in core areas of ordinary mathematics using proof-theoretic
methods. The workshop contributed to the following research strands: interactions between foundations and applications; proof mining; constructivity in classical logic; modal logic and provability logic; proof theory and theoretical computer science; structural proof theory
On Relating Theories: Proof-Theoretical Reduction
The notion of proof-theoretical or finitistic reduction of one theory to another has a long tradition. Feferman and Sieg (Buchholz et al., Iterated inductive definitions and subsystems of analysis. Springer, Berlin, 1981, Chap. 1) and Feferman in (J Symbol Logic 53:364â384, 1988) made first steps to delineate it in more formal terms. The first goal of this paper is to corroborate their view that this notion has the greatest explanatory reach and is superior to others, especially in the context of foundational theories, i.e., theories devised for the purpose of formalizing and presenting various chunks of mathematics.
A second goal is to address a certain puzzlement that was expressed in Fefermanâs title of his Clermont-Ferrand lectures at the Logic Colloquium 1994: âHow is it that finitary proof theory became infinitary?â Hilbertâs aim was to use proof theory as a tool in his finitary consistency program to eliminate the actual infinite in mathematics from proofs of real statements. Beginning in the 1950s, however, proof theory began to employ infinitary methods. Infinitary rules and concepts, such as ordinals, entered the stage.
In general, the more that such infinitary methods were employed, the farther did proof theory depart from its initial aims and methods, and the closer did it come instead to ongoing developments in recursion theory, particularly as generalized to admissible sets; in both one makes use of analogues of regular cardinals, as well as âlargeâ cardinals (inaccessible, Mahlo, etc.). (Feferman 1994).
The current paper aims to explain how these infinitary tools, despite appearances to the contrary, can be formalized in an intuitionistic theory that is finitistically reducible to (actually Î 02
-conservative over) intuitionistic first order arithmetic, also known as Heyting arithmetic. Thus we have a beautiful example of Hilbertâs program at work, exemplifying the Hilbertian goal of moving from the ideal to the real by eliminating ideal elements
- âŠ