elementarily

Since the theory is complete, all of its models are elementarily equivalent.

F is elementarily equivalent to the real numbers.

It is easy to prove that if Duplicator wins this game for all n, then and are elementarily equivalent.

A first-order theory is complete if and only if any two of its models are elementarily equivalent.

These models are elementarily equivalent; their theory admits the following axiomatization (verbally):

In his thesis, Fraïssé used the back-and-forth method to determine whether two model-theoretic structures were elementarily equivalent.

Thus, one can define p-adically closed fields as those whose first-order theory is elementarily equivalent to that of .

If M and N are elementarily equivalent, one writes M N.

The back-and-forth method can be used to show that any two countable atomic models of a theory that are elementarily equivalent are isomorphic.

In particular, his work implies that if a finitely generated group G is elementarily equivalent to a word-hyperbolic group then G is word-hyperbolic as well.

In model theory, a branch of mathematical logic, two structures M and N of the same signature σ are called elementarily equivalent if they satisfy the same first-order σ-sentences.

Specifically, a model is prime if it admits an elementary embedding into any model to which it is elementarily equivalent (that is, into any model satisfying the same complete theory as ).

Nevertheless, purely real expressions of the solutions may be obtained using hypergeometric functions, or more elementarily in terms of trigonometric functions, specifically in terms of the cosine and arccosine functions.

The whole world will surely have a common language, that is quite elementarily Utopian, and since we are free of the trammels of convincing story-telling, we may suppose that language to be sufficiently our own to understand.

Both residue fields are given by an ultraproduct over the fields F, so are isomorphic and have characteristic 0, and both value groups are the same, so the ultraproducts are elementarily equivalent.

When applied to first-order logic, the first incompleteness theorem implies that any sufficiently strong, consistent, effective first-order theory has models that are not elementarily equivalent, a stronger limitation than the one established by the Löwenheim-Skolem theorem.

A theory is defined to be model complete if whenever M and N are models of the theory and M is a submodel of N, then M and N are elementarily equivalent.

This is expressed in the Löwenheim-Skolem theorems, which state that any countable theory with an infinite model has models of all infinite cardinalities (at least that of the language) which agree with on all sentences, i.e. they are 'elementarily equivalent'.

Then one shows that if two Henselian valued fields have equivalent valuation groups and residue fields, and the residue fields have characteristic 0, then they are elementarily equivalent (which means that a first order sentence is true for one if and only if it is true for the other).