232 research outputs found
Hilbert's "Verunglueckter Beweis," the first epsilon theorem, and consistency proofs
In the 1920s, Ackermann and von Neumann, in pursuit of Hilbert's Programme,
were working on consistency proofs for arithmetical systems. One proposed
method of giving such proofs is Hilbert's epsilon-substitution method. There
was, however, a second approach which was not reflected in the publications of
the Hilbert school in the 1920s, and which is a direct precursor of Hilbert's
first epsilon theorem and a certain 'general consistency result' due to
Bernays. An analysis of the form of this so-called 'failed proof' sheds further
light on an interpretation of Hilbert's Programme as an instrumentalist
enterprise with the aim of showing that whenever a `real' proposition can be
proved by 'ideal' means, it can also be proved by 'real', finitary means.Comment: 18 pages, final versio
Hilbert's Program Then and Now
Hilbert's program was an ambitious and wide-ranging project in the philosophy
and foundations of mathematics. In order to "dispose of the foundational
questions in mathematics once and for all, "Hilbert proposed a two-pronged
approach in 1921: first, classical mathematics should be formalized in
axiomatic systems; second, using only restricted, "finitary" means, one should
give proofs of the consistency of these axiomatic systems. Although Godel's
incompleteness theorems show that the program as originally conceived cannot be
carried out, it had many partial successes, and generated important advances in
logical theory and meta-theory, both at the time and since. The article
discusses the historical background and development of Hilbert's program, its
philosophical underpinnings and consequences, and its subsequent development
and influences since the 1930s.Comment: 43 page
Transfinite Update Procedures for Predicative Systems of Analysis
We present a simple-to-state, abstract computational problem, whose solution implies the 1-consistency of various systems of predicative Analysis and offers a way of extracting witnesses from classical proofs. In order to state the problem, we formulate the concept of transfinite update procedure, which extends Avigad\u27s notion of update procedure to the transfinite and can be seen as an axiomatization of learning as it implicitly appears in various computational interpretations of predicative Analysis. We give iterative and bar recursive solutions to the problem
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
Cut elimination for a simple formulation of epsilon calculus
AbstractA simple cut elimination proof for arithmetic with the epsilon symbol is used to establish the termination of a modified epsilon substitution process. This opens a possibility of extension to much stronger systems
The Quantum Strategy of Completeness: On the Self-Foundation of Mathematics
Gentzen’s approach by transfinite induction and that of intuitionist Heyting arithmetic to completeness and the self-foundation of mathematics are compared and opposed to the Gödel incompleteness results as to Peano arithmetic. Quantum mechanics involves infinity by Hilbert space, but it is finitist as any experimental science. The absence of hidden variables in it interpretable as its completeness should resurrect Hilbert’s finitism at the cost of relevant modification of the latter already hinted by intuitionism and Gentzen’s approaches for completeness. This paper investigates both conditions and philosophical background necessary for that modification. The main conclusion is that the concept of infinity as underlying contemporary mathematics cannot be reduced to a single Peano arithmetic, but to at least two ones independent of each other. Intuitionism, quantum mechanics, and Gentzen’s approaches to completeness an even Hilbert’s finitism can be unified from that viewpoint. Mathematics may found itself by a way of finitism complemented by choice. The concept of information as the quantity of choices underlies that viewpoint. Quantum mechanics interpretable in terms of information and quantum information is inseparable from mathematics and its foundation
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