VON NEUMANN’S CONSISTENCY PROOF
- MathematicsThe Review of Symbolic Logic
Von Neumann’s proof is the first rigorous proof of the consistency of an axiomatization of the first-order theory of a successor function.
Simplified Lower Bounds for Propositional Proofs
- MathematicsNotre Dame J. Formal Log.
This article presents a simpliﬁed proof of the result that bounded depth propositional proofs of the pigeonhole principle are exponentially large. The proof uses the new techniques for proving…
NOMINALISTIC ORDINALS, RECURSION ON HIGHER TYPES, AND FINITISM
- PhilosophyThe Bulletin of Symbolic Logic
A historical account of the idea of nominalistic ordinals in the context of the Hilbert Programme as well as Gentzen and Bernays’ finitary interpretation of transfinite induction are presented.
The Practice of Finitism: Epsilon Calculus and Consistency Proofs in Hilbert's Program
The paper traces the development of the ``simultaneous development of logic and mathematics'' through the ∈-notation and provides an analysis of Ackermann's consistency proofs for primitive recursive arithmetic and for the first comprehensive mathematical system, the latter using thesubstitution method.
03 A 05 Philosophical and critical aspects of logic and foundations 03 D 10 Turing machines and related notions
In his famous paper, “An unsolvable problem of elementary number theory” [Am. J. Math. 58, 345–363 (1936; JFM 62.0046.01)], A. Church identified the intuitive notion of effective calculability with…
The consistency of arithmetic from a point of view of constructive tableau method with strong negation, Part I: the system without complete induction
In this Part I, we shall prove the consistency of arithmetic without complete induction from a point of view of strong negation, using its embedding to the tableau system $\bf SN$ of constructive…
A propósito del formalismo de Johann von Neumann
In 1930, Johann von Neumann, together with Rudolf Carnap and Arend Heyting, participated in a conference held in Konigsberg, called “Second Seminar on the Epistemology of Exact Sciences”. The idea…
We must know -- We shall know
In this article, I focus on the resiliency of the P=?NP problem. The main point to deal with is the change of the underlying logic from first to second-order logic. In this manner, after developing…
The Epsilon Calculus with Equality and Herbrand Complexity
The contribution of this paper is the upper and lower bounds analysis of the length of Herbrand disjunctions in extended first ePSilon theorem for epsilon calculus with equality and shows that the complexity analysis for Herbrand's theorem with equality is a straightforward consequence of the one for extended firstepsilon theorem without equality.
From Solvability to Formal Decidability: Revisiting Hilbert’s “Non-Ignorabimus”
- MathematicsJournal of Humanistic Mathematics
The topic of this article is Hilbert’s axiom of solvability, that is, his conviction of the solvability of every mathematical problem by means of a finite number of operations. The question of…