number-theoretic

The strongest results as of today are, in a very brief outline, the following: let X range over number-theoretic predicates.

Chores 2009

Analysis was to be formulated as a system of second-order arithmetic, which means that quantification is extended over number-theoretic predicates or, equivalently, over sets of natural numbers.

Chores 2009

We should begin here by reviewing two related aspects: the roots of Dedekind's number-theoretic investigations in the works of Gauss, Dirichlet, and Ernst Eduard Kummer; and the contrast between Dedekind's approach in this area and that of Leopold Kronecker.

Dedekind's Contributions to the Foundations of Mathematics Reck, Erich 2008

As a third spin-off, Dedekind's number-theoretic investigations led to the introduction of the notion of a “lattice” (under the name “Dualgruppe”), at first implicitly and then explicitly.

Dedekind's Contributions to the Foundations of Mathematics Reck, Erich 2008

Given that one cannot quantify over an infinite mathematical domain, the question arises: What, if anything, does any number-theoretic proof by mathematical induction actually prove?

Wittgenstein's Philosophy of Mathematics Rodych, Victor 2007

First, number-theoretic expressions that quantify over an infinite domain are not algorithmically decidable, and hence are not meaningful mathematical propositions.

Wittgenstein's Philosophy of Mathematics Rodych, Victor 2007

However, self-referential constructions attained an adequate degree of mathematical rigor and became genuine mathematical tools only when non trivial number-theoretic techniques were put to work (see the entry recursive functions), for instance in the analysis of syntactical substitution and in providing arithmetical models of formal provability

Paradoxes and Contemporary Logic Cantini, Andrea 2007

According to Gödel's incompleteness theorems, the statement that Peano Arithmetic (PA) is consistent, in its guise as a number-theoretic statement (given the technique of Gödel numbering), cannot be derived in PA itself.

Axiomatic Theories of Truth Halbach, Volker 2007

According to him, number-theoretic induction and the axiom of choice constitute independent intuitions, truly synthetic a priori judgements.

Paradoxes and Contemporary Logic Cantini, Andrea 2007

A number-theoretic formula φ (n1, ¦, nk) is numeralwise expressible in

Kurt Gödel Kennedy, Juliette 2007