The theory is defined as a placed of sentences which is uniform; typically these are besides defined to exist as closed under logical consequence. For instance, a placed of everthing sentences confessedly inside a few particular model (e.g. a reals) occurs as theory.

Gödel's completeness theorem says that a theory has a model if and only if it is consistent, i.e. there are no contradiction is proved per theory. This is the heart of exemplary theory when it lets u.s. guide questions all about theories by shopping at system & vice-versa. 1 should non confuse the completeness theorem sustaining a notion of a complete theory. The complete theory occurs as theory which contains each sentence or its negation. Importantly, of these potty locate the complete uniform theory extending any uniform theory.

A compactness theorem states that a placed of sentences S is satiable, i.e., has the model, whenever each finite subset of S is satiable. In the context of proof theory the correspondent statement is trivial, since each proof could develop just a finite total of antecedents utilized in the proof; in the context of exemplary theory, nonetheless, this proof is somewhat additional hard. There are 2 swell known proofs, a single by Gödel (which goes via proofs) and of these by Malcev (which is more directly & allows u.s. to limit a cardinality of the ensuant model).

Exemplary theory is ordinarily caring by having first order logic, and numbers of crucial resolutions (like a completeness and compactness theorems) fail in second order logic or other alternatives. Within number one the correct sequence logic totally infinite cardinals look the equivalent to a language which is countable. This is expressed in the Löwenheim-Skolem theorems, which state that any theory with an infinite model The has system of tons infinite cardinalities (at least that of the language) which agree by using The in everthing sentences, i.e. it is 'elementarily equivalent'.

Thus particularly, set theory (whose language is countable) has a denumerable model; this is referred to as Skolem's paradox, even though it's true (providing you accept the axioms of set theory)! To understand how come it was thought paradoxical, consider that there are sentences around placed theory which require a being of uncountable sets—& these sentences come confessedly within my enumerable model. Particularly a proof of the independence of the continuum hypothesis requires considering sets around system which come out to become uncountable once viewed from either inside a model, however come enumerable to individual outside a model.