Sentence examples for model of arithmetic from inspiring English sources

In 1934 the Norwegian Thoralf Skolem gave an explicit construction of what is now called a nonstandard model of arithmetic, containing "infinite numbers" and infinitesimals, each of which is a certain class of infinite sequences.

The intuitive idea in this method is to establish that a sentence is true in the ultraproduct if and only if it is true in "almost all" of the given structures (i.e., "almost everywhere"—an idea that was present in a different form in Skolem's construction of a nonstandard model of arithmetic in 1933).

The software is based on Triplet Structure Model of arithmetic word problem.

In the standard semantics, the only model of the Peano postulates, up to isomorphism, is the usual model of arithmetic.

Indeed, as Gödel's arithmetisation of syntax showed, the elements and inter-relationships of standard formal syntax can modelled as an infinite substructure inside the standard model of arithmetic.

For example, the L ω1,ω -sentence characterizing the standard model of arithmetic has a model of cardinality ℵ0 but no models of any other cardinality.

Nonetheless, just like the classical non-standard models of arithmetic, there is a class of inconsistent models of arithmetic (or more accurately models of inconsistent arithmetic) which have an interesting and important mathematical structure.

Such interpretations are among those that are sometimes called unintended, or non-standard models of arithmetic.

But, Q can be shown to have models of finite size too by referring to the inconsistent models of arithmetic.

One interesting implication of the existence of inconsistent models of arithmetic is that some of them are finite (unlike the classical non-standard models).

can be seen to be true in all and only non-standard models of arithmetic, by letting xx be all and only the non-standard elements of the model.