Basic Logic, Fall 2017Midterm 2 Review Sheet

Clinton T. Conley(Please contact me if you find any errors!)

In this handout we briefly summarize the major concepts and results thatfall within the portion of the course covered by the second midterm exam.

1 Completeness of first-order logic

• Henkin completeness: usual consistency and completeness PLUS if S `∃xα then S ` α[x/d] for some constant symbol d (Henkin witnesses).

• If S is a (countable) consistent L-theory, we may enlarge L to L′ byadding new constants and also enlarge S to a (L′-)Henkin-completetheory S ′.

• (Countable) Henkin-complete theories have (countable) models.

• Putting this all together, we obtain that (countable) consistent theorieshave (countable) models, which implies the completeness theorem (ifS |= σ then S ` σ) by the usual argument.

• As a bonus, we obtain the compactness theorem: if S is finitely satis-fiable then S is satisfiable.

2 Embeddings, isomorphisms, and substruc-

tures

• An embedding from an L-structure M to another L structure N is amap between their underlying sets that respects interpretations of all

1

symbols in L. For example, if ϕ is an embedding from M to N and cis some constant symbol, then ϕ(cM) = cN .

• An isomorphism is a bijective embedding. Note that when the lan-guage has equality, all embeddings are injective so it suffices to checksurjectivity. We write M∼= N if there exists an isomorphism betweenthem.

• M is a substructure of N if M ⊆ N andM interprets the symbols thesame way asN does. More formally, the inclusion map is an embeddingof L-structures.

3 Elementary equivalence, elementary embed-

dings, and elementary substructures

• Two L-structures are elementarily equivalent if they satisfy the sameL-sentences. In other words, Th(M) = Th(N ). Note that this is highlydependent upon the choice of language L. We write M≡ N .

• M ∼= N implies M ≡ N but the converse fails drastically by aLowenheim-Skolem argument.

• A function ϕ from M to N is an elementary embedding if wheneverα(~x) is an L-formula, and ~a ∈ Mk, then M |= α(~a) iff N |= α(ϕ(~a)).Isomorphisms are always elementary embeddings, but general embed-dings need not be elementary.

• A substructureM of N is an elementary substructure if whenever α(~x)is an L-formula, and ~a ∈Mk, thenM |= α(~a) iff N |= α(~a). This is thesame as saying that M and N satisfy the same LM-sentences, wherenow we have expanded L by adding constants for each element of M .We write M≺ N .

• If we put Diag(M) = {σ an LM-sentence :M |= σ}, then a substruc-ture M of N is elementary exactly when N |= Diag(M). Note thatDiag(M) is really just ThLM(M), the theory of M computed in theenlarged language LM.

• M ≺ N implies M ≡ N , but again a Lowenheim-Skolem argumentrules out the converse.

2

4 Tarski-Vaught and Lowenheim-Skolem

• Suppose that M is a substructure of N . The Tarski-Vaught Criterion(TVC) says that to check whether M ≺ N , it is equivalent to checkfor all L-formulas α(~x, y) and ~a ∈Mk with N |= ∃y α(~a, y) that thereis some b ∈M such that N |= α(~a, b).

• From this we obtain the (countable) downward Lowenheim-Skolem the-orem. If L is a countable language and N is an L-structure containingsome particular countable set X ⊆ N , then we may find a countableelementary substructure M≺ N such that X ⊆M .

• To prove this we enlarged our language to add Skolem functions forexistential L-formulas (analogous to Henkin witnesses) before findingthe substructure M. These helped us verify the TVC, guaranteeingM≺ N .

• We also have the upward Lowenheim-Skolem theorem. If M is aninfinite L-structure we can find arbitrarily large (in the sense of cardi-nality) L-structures N with M≺ N .

• To prove this we used compactness. Add a ton of new constant symbols,check finite satisfiability of Diag(M) along with sentences saying thenew constant symbols are unequal, then build a model of this theoryand forget about the new constants.

5 Quantifier elimination

• Given some L-theory S, we say two L-terms s and t are S-equivalentif S ` s ≡ t. Similary, we say two L-formulas α and β are S-equivalentif S ` α↔ β.

• An L-formula is quantifier-free if it contains no quantifiers (an unusu-ally accurately named notion in logic!).

• We showed that any quantifier free formula is ∅-equivalent to one inDNF: a finite disjunction of BCs. And a BC is a finite conjunction ofatomic formulas and negations of atomic formulas.

3

• An L-theory S has quantifier elimination (QE) if every L-formula isS-equivalent to a quantifier-free formula. Typically we want the twoformulas have the same free variables, but in general this requires thelanguage to have a constant symbol (so we can encode “always true”as c ≡ c, etc.).

• To check whether S has QE it’s enough to check that whenever θ(~x, y)is a BC then ∃y θ(~x, y) is equivalent to a quantifier-free formula.

6 Linear orders

While perhaps not officially part of logic, linear orders are useful examplesof various model-theoretic phenomena so we spent some time on them.

• The language of linear orders, LLO has a binary relation ≤ and equality.

• LO is the LLO-theory axiomatizing reflexivity, transitivity, and antisym-metry of ≤.

• DLOWOE is the LLO-theory containing LO and additional axioms of density(between any two distinct points is a third point) and no endpoints (nomax or min points in the order).

• We showed Cantor’s theorem that any two countable models of DLOWOEare isomorphic by the “back and forth” argument. From this it followsthat DLOWOE is a complete theory.

• We also showed that DLOWOE admits quantifier elimination, and usedthis to conclude that (Q,≤) is an elementary substructure of (R,≤).

4