axiomatisation , also: axiomatization

But unlike these traditional accounts, we provide an axiomatisation from which the similarities follow.

The same is true of any consistent first order axiomatisation of set theory.

With the axiomatisation of maths came logical foundations for both old and new subjects.

Thus one chooses the axiomatisation suited for the application.

It was taken as an axiom in his first axiomatisation of propositional logic.

Mally was the first logician ever to attempt an axiomatisation of ethics.

Their objective was the axiomatisation of branches of mathematics like geometry, arithmetic, analysis and set theory.

Theorem: This axiomatisation of deontic logic implies that !

The axiomatisation of other LOTOS relations is also considered.

Yet Andrey Kolmogorov, whose rival axiomatisation was better received, was less severe:

The axiom of empty set is generally considered uncontroversial, and it or an equivalent appears in just about any alternative axiomatisation of set theory.

Applications of measures include the Lebesgue integral, Kolmogorov's axiomatisation of probability theory, and ergodic theory.

The introduction of the concepts of abstractions and axiomatisation is attributable to the Greek mathematician/philosophers of the period 600-300 BC.

The hyperreals can be constructed in the framework of Zermelo-Fraenkel set theory, the standard axiomatisation of set theory used elsewhere in mathematics.

The axiom schema of replacement was not part of Ernst Zermelo's 1908 axiomatisation of set theory (Z).

John von Neumann in his 1925 An axiomatisation of set theory wrestled with the same issues as did Russell, Zermelo, Skolem, and Fraenkel.

Skolem's paradox is that every countable axiomatisation of set theory in first-order logic, if it is consistent, has a model that is countable.

While he mentions the efforts of David Hilbert to prove the consistency of his axiomatisation of mathematics von Neumann placed him in the same group as Russell.

Universal quantification is often given an alternative axiomatisation using an extra rule of generalisation (see the section on Metatheorems), in which case the rules Q6 and Q7 are redundant.

The study of various kinds of axiomatisation may be interesting to enlighten the link between experience and the birth of mathematical concepts (and the very nature of mathematics itself, too).

We describe here a Hilbert system with nine axioms and just the rule modus ponens, which we call the one-rule axiomatisation and which describes classical equational logic.

Euclid's Elements being the earliest extant documentation of the axioms of plane geometry- though Proclus tells of an earlier axiomatisation by Hippocrates of Chios.

The downward form of this theorem shows that if a countable first-order axiomatisation is satisfied by any infinite structure, then the same axioms are satisfied by some countable structure.

A central goal of early research into set theory was to find a first order axiomatisation for set theory which was categorical, meaning that the axioms would have exactly one model, consisting of all sets.

Or, one can use a different axiomatisation of set theory, such as W. V. Quine's New Foundations, which allows for a set of all sets and avoids Russell's paradox in another way.