Sentence examples for system of axioms from inspiring English sources

In Zürich he delved into the realm of set theory, trying to streamline the Zermelo-Fraenkel system of axioms (see logic, history of: 20th-century set theory).

In 1940 Kurt Gödel had shown that, if one accepts the Zermelo-Fraenkel system of axioms for set theory, then the continuum hypothesis is not disprovable.

On the other hand, since the axiom schema of replacement of ZFC provides an axiom for each formula, ZFC has infinitely many axioms which is unavoidable because it is known that no finite subset yields the full system of axioms.

Lindley said his aim in his early years as an academic was to make statistics a respectable branch of mathematics: all the other courses he had studied had the classic mathematical form, of a system of axioms from which the consequences were deduced, but statistics lacked this.

We'll discuss some of these applications, among them: Church's theorem that there is no algorithm for deciding when a formula is valid in the predicate calculus; Tarski's theorem that the set of true sentence of a language isn't definable within that language; and Gödel's second incompleteness theorem, which says that no consistent system of axioms can prove its own consistency.

It is well known that there is an alternative system of axioms for a hyperbolic plane.

Various systems of axioms, or basic rules and assumptions, have been formulated as a basis for measurement theory.

Ontologies are complex systems of axioms in which unanticipated consequences of changes are both frequent, and difficult for ontology authors to apprehend.

Even fairly small programs and systems of axioms are difficult to comprehend and can be highly unpredicable, yielding unexpected consequences and unanticipated interactions.

For all of these thinkers, what is crucial in, and sufficient for, mathematics is that the completeness and consistency of certain basic concepts, or of the corresponding systems of axioms, are established.

And there is an – often only implicit – reference to the basic ideas of MVL in independence proofs (e.g. for systems of axioms) which often refer to logical matrices with more than two truth degrees.