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Extra info for Model Theory: Third Edition (Dover Books on Mathematics)
THEOREM three. 2. three. A concept T is preserved below unions of chains if and provided that T has a suite of universal–existential (i. e. , ) axioms. evidence. (). For the simple path, allow Γ be a collection of axioms for T and permit , be a series of types of T with union . think about a sentence in Γ the place ψ has no quantifiers. every one is a version of γ. enable a1, …, am A. For a few β < α, we've a1, …, am Aβ. Then there exist b1, …, bn Aβ such that It follows that whence satisfies γ. as a result is a version of Γ, and hence of T. (). Now think T is preserved below unions of chains. enable Δ be the set of all sentences that are logically reminiscent of sentences. Then Δ is closed less than finite disjunctions. consider and are versions such that and each sentence precise in is correct in . Then each sentence precise in is right in . we will end up the subsequent: To end up (1), we upload a brand new consistent cb for every b B, forming the version and language . enable T1 be the full conception of in , and T2 the set of all common sentences of which carry in . Then T1 ∪ T2 is constant, simply because as much as equivalence T2 is closed less than conjunction, and for any ψ(cb1 … cbn) in T2, the sentence (∃y1 … yn)ψ holds in , and for this reason in . enable be a version of T1 ∪ T2. Then and . in addition, each common sentence of real in is correct in , whence each existential sentence precise in is right in . Now extend the language extra through including a brand new consistent ca for every . Then the idea the diagram of plus the user-friendly diagram of , is constant. It therefore has a version . Then and , so (1) is proved. Iterating (1) we receive a series such that for every n, allow be the union of this chain. on the grounds that every one is a version of T, is a version of T. in spite of the fact that, can also be the union of the effortless chain accordingly by means of the ordinary chain theorem, , so is additionally a version of T. by way of Lemma three. 2. 1, we finish that T has a suite of axioms. A formulation φ is expounded to be confident iff it's outfitted up from atomic formulation utilizing in basic terms the connectives , and the quantifiers ∀, ∃. THEOREM three. 2. four. A constant conception T is preserved less than homomorphisms if and provided that T has a suite of confident axioms. facts. ( ). allow us to first money the straightforward course. A formulation φ(x1 … xn) is preserved below homomorphisms iff for any homomorphism f of a version onto a version and all a1, …, an A, if , then . It suffices to end up that every one optimistic formulation φ are preserved below homomorphisms. this is often performed by way of induction at the complexity of φ. It evidently holds while φ is atomic and passes over and . suppose φ(xx1 … xn) is preserved lower than homomorphisms. First, believe . Then for a few a A, we now have , and accordingly . Now think . permit b be any component to B. Then for a few a A we've b = fa. hence and . for that reason (∃x)φ and (∀x)φ are preserved less than homomorphisms. (). We now turn out the difficult course. allow pos suggest that each confident sentence real in is correct in . We express first that: To end up (1), we upload new constants ca, a A, and db, b B. allow T1 be the set of all confident sentences of precise in . allow T2 be the set of all sentences of real in . Then T1 ∪ T2 is constant simply because pos . permit be a version of T1 ∪ T2; T2 indicates that .