elementarily's Usage Examples:

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.

expressive enough to support most everyday mathematics (via the addition of elementarily-expressible axioms such as Zermelo–Fraenkel set theory (ZFC)).

back-and-forth games) is a technique for determining whether two structures are elementarily equivalent.

elementary embedding into any model M {\displaystyle M} to which it is elementarily equivalent (that is, into any model M {\displaystyle M} satisfying the.

the universe of sets start to look similar, in the sense that each is elementarily embeddable into a later one.

back-and-forth method to determine whether two model-theoretic structures were elementarily equivalent.

structure need not be a strongly minimal theory, that is, there may be an elementarily equivalent structure which is not minimal.

generating a meta-level, set-theoretic falsehood, namely in relation to other elementarily-equivalent set-theories not identical to the one chosen.

Such groups are elementarily equivalent to the integers ( Z , + , < ) {\displaystyle (\mathbb {Z}.

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.

Benacerraf noted that there are elementarily equivalent, set-theoretic ways of relating natural numbers to pure sets.

used to show that any two countable atomic models of a theory that are elementarily equivalent are isomorphic.

complexity of cut-elimination, Herbrand disjunctions with cuts can be non-elementarily smaller than a standard Herbrand disjunction.

torsion-free word-hyperbolic groups and to characterize all groups that are elementarily equivalent to (that is, have the same first-order theory as) a given.

Artin-Schreier theorem, the degree of this extension is necessarily 2, and F is elementarily equivalent to R.

class of binary relations (each with set-sized domain), there is one elementarily embeddable into another.

The Bijou was the first theatre in the United States to be elementarily lighted by electricity, which Thomas Edison personally installed and.

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

the field of real numbers and the field of real algebraic numbers are elementarily equivalent structures.

notion for saturated models is that any "reasonably small" model of T is elementarily embedded in a saturated model, where "reasonably small" means cardinality.