Axiomatic Thinking II

First, we consider Hilbert’s program, focusing on the three different aspect of mathematics called actual mathematics, formal mathematics, and metamathematics. Then, we investigate the relationship between metamathematics and actual mathematics, describe what shall be achieved with metamathematics, and propose a framework for metamathematics.

The purpose of this article is to present a new and simplified cut elimination procedure for \(\textsf{KP}\). We start off from the basic language of set theory and add constants for all elements of the constructible hierarchy up to the Bachmann-Howard ordinal \(\psi (\varepsilon _{\Omega {+}1})\). This enriched language is then used to set up an infinitary proof system \(\textsf{IP}\) whose ordinal-theoretic part is based on a specific notation system \(C(\varepsilon _{\Omega +1},0)\) due to Buchholz and his idea of operator controlled derivations. \(\textsf{KP}\) is embedded into \(\textsf{IP}\) and complete cut elimination for \(\textsf{IP}\) is proved.

Several theorems about the equivalence of familiar theories of reverse mathematics with certain well-ordering principles have been proved by recursion-theoretic and combinatorial methods (Friedman, Marcone, Montalbán et al.) and with far-reaching results by proof-theoretic technology (Afshari, Freund, Girard, Rathjen, Thomson, Valencia Vizcaíno, Weiermann et al.), employing deduction search trees and cut elimination theorems in infinitary logics with ordinal bounds in the latter case. At type level 1, the well-ordering principles are of the form where f is a standard proof-theoretic function from ordinals to ordinals (such f’s are always dilators). One aim of the paper is to present a general methodology underlying these results that enables one to construct omega-models of particular theories from \((*)\) and even \(\beta \)-models from the type 2 version of \((*)\). As \((*)\) is of complexity \(\Pi ^1_2\) such a principle cannot characterize stronger comprehensions at the level of \(\Pi ^1_1\)-comprehension. This requires a higher order version of \((*)\) that employs ideas from ordinal representation systems with collapsing functions used in impredicative proof theory. The simplest one is the Bachmann construction. Relativizing the latter construction to any dilator f and asserting that this always yields a well-ordering turns out to be equivalent to \(\Pi ^1_1\)-comprehension. This result has been conjectured and a proof strategy adumbrated roughly 10 years ago, but a detailed proof has only been worked out in recent years.

In Hilbert’s paper “Axiomatic Thinking”—the published version of his 1917 Zürich talk - he touches on the axiomatic treatment of continuity and, as he puts it, “the dependence of the propositions of a field of knowledge on the axiom of continuity”. By the “axiom of continuity”, Hilbert seems to mean a number of things. In this paper I speculate on the various meanings Hilbert may have ascribed to the term. I focus in particular on interpreting the “axiom of continuity” as the central principal of Synthetic Differential Geometry that all real functions are smooth.

The paper contrasts two ways of generalizing and gives examples: probably most people think of examples like generalizing Cartesian coordinate geometry to differential manifolds. One kind of structure is replaced by another more complicated but more flexible kind. Call this articulating generalization as it articulates some general assumptions behind an earlier concept. On the other hand, by unifying generalization, I mean simply dropping some assumptions from an earlier concept or theorem. Hilbert, Noether, and Grothendieck were all known for highly non-trivial unifying generalizations.

By providing quantifier-free axioms systems, without any form of induction, for a slight variation of Euclid’s proof and for the Goldbach proof for the existence of infinitely many primes, we highlight the fact that there are two distinct and very likely incompatible concepts of infiniteness that are part of the theorems proved. One of them is the concept of cofinality, the other is the concept of equinumerosity with the universe.

My purpose is to comment some claims of André Weil (1906–1998) in his letter of March 26, 1940 to his sister Simone, in particular, the following quotation: “it is essential, if mathematics is to stay as a whole, to provide a unification, which absorbs in some simple and general theories all the common substrata of the diverse branches of the science, suppressing what is not so useful and necessary, and leaving intact what is truly the specific detail of each big problem. This is the good one can achieve with axiomatics”. For Weil (and Bourbaki), the main problem was to find “strategies” for finding complex proofs of “big problems”. For that, the dialectic balance between general structures and specific details is crucial. I will focus on the fact that, for these creative mathematicians, the concept of structure is a functional concept, which has always a “strategic” creative function. The “big problem” here is Riemann Hypothesis (RH). Artin, Schmidt, Hasse, and Weil introduced an intermediary third world between, on the one hand, Riemann original hypothesis on the non-trivial zeroes of the zeta function in analytic theory of numbers, and, on the other hand, the algebraic theory of compact Riemman surfaces. The intermediary world is that of projective curves over finite fields of characteristic \(p\ge 2\). RH can be translated in this context and can be proved using sophisticated tools of algebraic geometry (divisors, Riemann-Roch theorem, intersection theory, Severi-Castelnuovo inequality) coupled with the action of Frobenius maps in characteristic \(p\ge 2\). Recently, Alain Connes proposed a new strategy and constructed a new topos theoretic framework à la Grothendieck where Weil’s proof could be transferred by analogy back to the original RH.

We aim to put some order to the multiple interpretations of the Church-Turing Thesis and to the different approaches taken to prove or disprove it.

In the first part of this contribution, I will present aspects and attitudes towards “axiomatic thinking” in various branches of theoretical physics. In the second and more technical part, which is approximately of the same size, I will focus on mathematical results that are relevant for axiomatic schemes of space-time in connection with attempts to axiomatize Special and General Relativity.

The purpose of the paper is to show that axiomatic thinking can also be applied to religion provided a part of the language used in religion (here called: Religious Discourse) consists of propositions or norms. Although David Hilbert was not concerned with religion when he gave his famous talk “Axiomatisches Denken” in 1917, his published essay (in 1918) treats this topic in such a broad sense that such an application seems appropriate.

This application is done in the following way: The first part discusses the possibility of applying axiomatic thinking to religion by considering the necessary preconditions to be satisfied for a successful application. The second part discusses the specific logical language that will be used in the application. The third part offers two concrete examples of such an application: a short and preliminary axiomatic theory of omniscience and omnipotence.