Realizability algebras: a program to well order R

The theory of classical realizability is a framework in which we can develop the proof-program correspondence. Using this framework, we show how to transform into programs the proofs in classical analysis with dependent choice and the existence of a well ordering of the real line. The principal tools are: The notion of realizability algebra, which is a three-sorted variant of the well known combinatory algebra of Curry. An adaptation of the method of forcing used in set theory to prove consistency results. Here, it is used in another way, to obtain programs associated with a well ordering of R and the existence of a non trivial ultrafilter on N

Bar recursion in classical realisability : dependent choice and continuum hypothesis

This paper is about the bar recursion operator in the context of classical realizability. After the pioneering work of Berardi, Bezem & Coquand [1], T. Streicher has shown [10], by means of their bar recursion operator, that the realizability models of ZF, obtained from usual models of $\lambda$-calculus (Scott domains, coherent spaces, . . .), satisfy the axiom of dependent choice. We give a proof of this result, using the tools of classical realizability. Moreover, we show that these realizability models satisfy the well ordering of $\mathbb{R}$ and the continuum hypothesis These formulas are therefore realized by closed $\lambda_c$-terms. This allows to obtain programs from proofs of arithmetical formulas using all these axioms.Comment: 11 page

On the structure of classical realizability models of ZF

The technique of "classical realizability" is an extension of the method of "forcing"; it permits to extend the Curry-Howard correspondence between proofs and programs, to Zermelo-Fraenkel set theory and to build new models of ZF, called "realizability models". The structure of these models is, in general, much more complicated than that of the particular case of "forcing models". We show here that the class of constructible sets of any realizability model is an elementary extension of the constructibles of the ground model (a trivial fact in the case of forcing, since these classes are identical). It follows that Shoenfield absoluteness theorem applies to realizability models.Comment: 17 page

Valid formulas, games and network protocols

We describe a remarkable relation between the notion of valid formula of predicate logic and the specification of network protocols. We give several examples such as the acknowledgement of one packet or of a sequence of packets. We show how to specify the composition of protocols.Comment: 12 page

Realizability algebras II : new models of ZF + DC

Using the proof-program (Curry-Howard) correspondence, we give a new method to obtain models of ZF and relative consistency results in set theory. We show the relative consistency of ZF + DC + there exists a sequence of subsets of R the cardinals of which are strictly decreasing + other similar properties of R. These results seem not to have been previously obtained by forcing.Comment: 28

Analyzing Fragmentation of Simple Fluids with Percolation Theory

We show that the size distributions of fragments created by high energy nuclear collisions are remarkably well reproduced within the framework of a parameter free percolation model. We discuss two possible scenarios to explain this agreement and suggest that percolation could be an universal mechanism to explain the fragmentation of simple fluids.Comment: 12 pages, 11 figure

Partial energies fluctuations and negative heat capacities

We proceed to a critical examination of the method used in nuclear fragmentation to exhibit signals of negative heat capacity. We show that this method leads to unsatisfactory results when applied to a simple and well controlled model. Discrepancies are due to incomplete evaluation of potential energies.Comment: Modified figures 3 and

A program for the full axiom of choice

The theory of classical realizability is a framework for the Curry-Howard correspondence which enables to associate a program with each proof in Zermelo-Fraenkel set theory. But, almost all the applications of mathematics in physics, probability, statistics, etc. use Analysis i.e. the axiom of dependent choice (DC) or even the (full) axiom of choice (AC). It is therefore important to find explicit programs for these axioms. Various solutions have been found for DC, for instance the lambda-term called "bar recursion" or the instruction "quote" of LISP. We present here the first program for AC.Comment: 22 pages - The paper has been formatted for publication in Log. Meth. Comp. S