19 research outputs found
Interactive Realizability and the elimination of Skolem functions in Peano Arithmetic
We present a new syntactical proof that first-order Peano Arithmetic with
Skolem axioms is conservative over Peano Arithmetic alone for arithmetical
formulas. This result - which shows that the Excluded Middle principle can be
used to eliminate Skolem functions - has been previously proved by other
techniques, among them the epsilon substitution method and forcing. In our
proof, we employ Interactive Realizability, a computational semantics for Peano
Arithmetic which extends Kreisel's modified realizability to the classical
case.Comment: In Proceedings CL&C 2012, arXiv:1210.289
Proof theory of weak compactness
We show that the existence of a weakly compact cardinal over the
Zermelo-Fraenkel's set theory is proof-theoretically reducible to iterations of
Mostowski collapsings and Mahlo operations
Learning, realizability and games in classical arithmetic
PhDAbstract. In this dissertation we provide mathematical evidence that the concept of
learning can be used to give a new and intuitive computational semantics of classical
proofs in various fragments of Predicative Arithmetic.
First, we extend Kreisel modi ed realizability to a classical fragment of rst order
Arithmetic, Heyting Arithmetic plus EM1 (Excluded middle axiom restricted to 0
1 formulas).
We introduce a new realizability semantics we call \Interactive Learning-Based
Realizability". Our realizers are self-correcting programs, which learn from their errors
and evolve through time, thanks to their ability of perpetually questioning, testing and
extending their knowledge. Remarkably, that capability is entirely due to classical principles
when they are applied on top of intuitionistic logic.
Secondly, we extend the class of learning based realizers to a classical version PCFClass
of PCF and, then, compare the resulting notion of realizability with Coquand game semantics
and prove a full soundness and completeness result. In particular, we show there
is a one-to-one correspondence between realizers and recursive winning strategies in the
1-Backtracking version of Tarski games.
Third, we provide a complete and fully detailed constructive analysis of learning as it
arises in learning based realizability for HA+EM1, Avigad's update procedures and epsilon
substitution method for Peano Arithmetic PA. We present new constructive techniques to
bound the length of learning processes and we apply them to reprove - by means of our
theory - the classic result of G odel that provably total functions of PA can be represented
in G odel's system T.
Last, we give an axiomatization of the kind of learning that is needed to computationally
interpret Predicative classical second order Arithmetic. Our work is an extension of
Avigad's and generalizes the concept of update procedure to the trans nite case. Trans-
nite update procedures have to learn values of trans nite sequences of non computable
functions in order to extract witnesses from classical proofs
Unprovability and phase transitions in Ramsey theory
The first mathematically interesting, first-order arithmetical example of incompleteness was given in the late seventies and is know as the Paris-Harrington principle. It is a strengthened form of the finite Ramsey theorem which can not be proved, nor refuted in Peano Arithmetic. In this dissertation we investigate several other unprovable statements of Ramseyan nature and determine the threshold functions for the related phase transitions.
Chapter 1 sketches out the historical development of unprovability and phase transitions, and offers a little information on Ramsey theory. In addition, it introduces the necessary mathematical background by giving definitions and some useful lemmas.
Chapter 2 deals with the pigeonhole principle, presumably the most well-known, finite instance of the Ramsey theorem. Although straightforward in itself, the principle gives rise to unprovable statements. We investigate the related phase transitions and determine the threshold functions.
Chapter 3 explores a phase transition related to the so-called infinite subsequence principle, which is another instance of Ramseyâs theorem.
Chapter 4 considers the Ramsey theorem without restrictions on the dimensions and colours. First, generalisations of results on partitioning α-large sets are proved, as they are needed later. Second, we show that an iteration of a finite version of the Ramsey theorem leads to unprovability.
Chapter 5 investigates the template âthin implies Ramseyâ, of which one of the theorems of Nash-Williams is an example. After proving a more universal instance, we study the strength of the original Nash-Williams theorem. We conclude this chapter by presenting an unprovable statement related to Schreier families.
Chapter 6 is intended as a vast introduction to the Atlas of prefixed polynomial equations. We begin with the necessary definitions, present some specific members of the Atlas, discuss several issues and give technical details
Achieving while maintaining:A logic of knowing how with intermediate constraints
In this paper, we propose a ternary knowing how operator to express that the
agent knows how to achieve given while maintaining
in-between. It generalizes the logic of goal-directed knowing how proposed by
Yanjing Wang 2015 'A logic of knowing how'. We give a sound and complete
axiomatization of this logic.Comment: appear in Proceedings of ICLA 201