from finitary to infinitary second-order logic

Math. Log. Quart. 51, No. 5, 499 – 506 (2005) / DOI 10.1002/malq.200410046 / www.mlq-journal.org

From finitary to infinitary second-order logic

George Weaver∗ and Irena Penev∗∗

Park Science Center, Bryn Mawr College, 101 North Merion Avenue, Bryn Mawr Pa. 19010, USA

Received 2 October 2004, revised 22 April 2005, accepted 17 May 2005Published online 1 August 2005

Key words Second-order logic, infinitary logic, back and forth condition.

MSC (2000) 03B15, 03C85

A back and forth condition on interpretations for those second-order languages without functional variableswhose non-logical vocabulary is finite and excludes functional constants is presented. It is shown that thiscondition is necessary and sufficient for the interpretations to be equivalent in the language. When appliedto second-order languages with an infinite non-logical vocabulary, excluding functional constants, the backand forth condition is sufficient but not necessary. It is shown that there is a class of infinitary second-orderlanguages whose non-logical vocabulary is infinite for which the back and forth condition is both necessary andsufficient. It is also shown that some applications of the back and forth construction for second-order languagescan be extended to the infinitary second-order languages.

c© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

1 Introduction

The Fraısse back and forth characterization of elementary equivalence can be extended to the semantics of severalnon-elementary languages (Mostowski [5, pp. 134 – 135]). In particular, it can be extended to the “standard”semantics for those second-order languages without functional variables whose non-logical vocabulary is finiteand excludes functional constants. This paper presents a back and forth characterization for these languagesnot previously in the literature. However, it fails when the non-logical vocabulary is infinite. Infinitary second-order languages are introduced. These are obtained by augmenting second-order languages in such a way that,when their non-logical vocabularies are infinite, the back and forth condition is both necessary and sufficient forequivalence. Some applications of the back and forth characterization are presented: a distributive normal form,a characterization of finite axiomatizability and a “reduction theorem” due to David Kaplan. These results holdfor both kinds of languages.

Let K be a set (possibly empty) of non-logical constants. Members of K are either individual constants orn-ary predicate constants for n ≥ 1, but not functional constants. Let K0 be a countably infinite well ordered setof individual constants disjoint from K , and for each n ≥ 1 let Kn be a countably infinite well ordered set ofn-ary predicate constants disjoint from K . Kn and Km are disjoint when n �= m. Members of Kn are saidto be of degree n. For each non-negative integer n, m0 . . .mn is a sequence of non-negative integers. ByK[m0 . . .mn] we denote the union of K and the set of the first mt members of Kt for t = 0, . . . , n.

Interpretations of type K[m0 . . .mn] are ordered pairs A = (A, fA), whereA is a non-empty set (the domainof A) and fA is a function on K[m0 . . .mn] defined in the usual way. Given A and B of type K[m0 . . .mn],A is a subinterpretation of B iffA ⊆ B, fA(k) = fB(k) if k is an individual constant, and fA(k) = fB(k) ∩At

if k is a t-ary predicate constant. If A is a non-empty set and n ≥ 0, then D(A, n) is defined by D(A, 0) = Aand D(A, n) = P(An), the power set of An, for n ≥ 1. The n-ary domain of A, D(A, n), is D(A, n). A ≈ Bindicates that A and B are isomorphic.

If A is of type K[m0 . . .mn], 0 ≤ r ≤ n, and a ∈ D(A, r), then (Aa) is the following interpretation oftype K[m0 . . .mr + 1 . . .mn]: the domain of (Aa) is A, f(Aa)(k) = fA(k) if k ∈ K[m0 . . .mn], and

∗ Corresponding author: e-mail: [email protected]∗∗ e-mail: [email protected]


500 G. Weaver and I. Penev: From finitary to infinitary second-order logic

f(Aa)(k) = a if k is the (mr + 1)th member of Kr. If r > n, then (Aa) is the analogous interpretation oftype K[m0 . . .mn . . .mr], where mr is 1 and mt is 0 for n < t < r.

When B is a subset of A, A[B] is the subinterpretation of A generated by B. A[∅] is an interpretation of typeK[m0 . . .mn] provided either K contains individual constants or m0 is non-zero. Since there are no functionalconstants, the cardinality of A[∅] is not greater than the number of individual constants in K[m0 . . .mn].

To simplify the statement of the back and forth condition we use for each t ≥ 1 and s = 0, . . . , t the followingfunction h[t, s] : N

t −→ Nt:

h[t, s](r0, . . . , rs, . . . , rt−1) =

{(r0, . . . , rs − 1, . . . , rt−1) if rs �= 0,

(r0, . . . , rs, . . . , rt−1) otherwise.

Let 0 be that member of Nt all of whose coordinates are zero. Notice that if S ⊆ N

t, then S = Nt if S satisfies

the following conditions: 0 ∈ S, and any ordered t-tuple (r0, . . . , rt−1) of non-negative integers is in S if itsimage under h[t, s] is in S for each s such that rs �= 0.

Given t and n both greater than or equal to 0, r is (r0, . . . , rt) and m is the sequencem0 . . .mn. Let A and Bbe of type K[m]. Then A �r B is defined as follows:

1. If r = 0, i. e. rs = 0 for s = 0, . . . , t, then A �r B iff A[∅] ≈ B[∅].2. If there is an s such that rs �= 0, then A �r B iff for all s such that rs �= 0,

(a) for all a in D(A, s) there is b in D(B, s) such that (Aa) �h[t+1,s](r) (Bb), and

(b) for all b in D(B, s) there is a in D(A, s) such that (Bb) �h[t+1,s](r) (Aa).

2 Finitary second-order languages

In this section we consider the case where K is finite.Let V0 be a countably infinite well ordered set of individual variables, and for n ≥ 1 let Vn be a countably

infinite well ordered set of n-ary predicate variables. Vn and Vm are disjoint when n �= m. Members of Vn aresaid to be of degree n.

Let m be a finite sequence of non-negative integers. Then L[m] is the (finitary) second-order languageoverK[m] (with equality): Individual symbols are individual variables and individual constants. Atomic formulasin L[m] are defined as usual except that the only equations permitted are those between individual symbols.The formulas in L[m] and the distinction between free and bound variables are defined following Hilbert andAckermann [1, p. 66]. Sentences are formulas devoid of free variables. When ϕ is a formula, k is a non-logicalconstant and v is a variable of the same degree as k, ϕ[v/k] is the result of replacing every occurrence of v in ϕby k and ϕ[k/v] is the result of replacing every occurrence of k in ϕ by v.

The back and forth characterization of elementary equivalence is the special case of the above when t = 0: Letr0 be a non-negative integer. Consider that sublanguage of the first-order language over K[m] whose formulascontain no more than r0 distinct individual variables. Then A �r0 B iff A and B agree on all sentences in thissublanguage. In fact, it has been shown in [7] that given A, there is a sentence Ψ[A, r0] in this sublanguage whosemodels are exactly the interpretations B with B �r0 A.

In the second-order case for each t ≥ 0 and r = (r0, . . . , rt) ∈ Nt+1 a sublanguage is also singled out. The

variables that occur in the formulas of this sublanguage are of degree ≤ t and there are no more than rs distinctvariables of degree s = 0, . . . , t that occur in any formula. Analogous to the first-order case, there is a sentenceΨ[A, r] in this sublanguage whose models are exactly those interpretations B with B �r A.

For each n ≥ 0, the n-ary rank of ϕ (rk(n, ϕ)) is the number of distinct variables of degree n occurringin ϕ. The degree of ϕ (D(ϕ)) is the largest n such that the n-ary rank of ϕ is non-zero, when such an n exists,and is 0 otherwise. The rank of ϕ (rk(ϕ)) is (rk(0, ϕ), . . . , rk(D(ϕ), ϕ)). By S[m][t] we denote the set ofsentences in L[m] of degree ≤ t, and S[m][r] is the set of sentences in S[m][t] whose s-ary rank is ≤ rs, foreach s = 0, . . . , t. If r = 0, then S[m][r] is the smallest set of sentences that includes the atomic sentencesin L[m] and is closed under truth-functional combinations. If there is s such that rs �= 0, then S[m][r] is thesmallest set of sentences that includes all sets S[m][h[t + 1, s](r)] for each s such that rs �= 0, that includes allsentences of the forms ∃vϕ and ∀vϕ, where ϕ is a formula in L[m] whose one and only free variable is v andwhere rk(ϕ) = r, and that is closed under forming truth-functional combinations.


Math. Log. Quart. 51, No. 5 (2005) / www.mlq-journal.org 501

We fix a well order of the formulas. When S is a finite set of formulas,∧S is the conjunction of the members

of S in the order in which they occur in the well ordering, and∨S is the corresponding disjunction of the

members of S.

Lemma 2.1 Assume that A and B are of type K[m] and r ∈ Nt+1. If A �r B, then A and B agree on all

sentences in S[m][r].

P r o o f. Let

S = {r : for all m and all A and B of type K[m], if A �r B, then A and B agree on S[m][r]}.

Suppose that r = 0 and suppose that A �r B. Then A[∅] and B[∅] are isomorphic. But A and A[∅] agree onall sentences in S[m][r] and B and B[∅] agree on all sentences in S[m][r]. Thus, r ∈ S.

Suppose now that there is some s such that rs �= 0 and that h[t+ 1, s](r) ∈ S for all such s. Supposethat A �r B. By supposition, A and B agree on S[m][h[t + 1, s](r)]. Let ϕ be a formula in L[m] whoseone and only free variable is v, where rk(ϕ) = r and v is of degree s. It suffices to show that these interpretationsagree on ∃vϕ and ∀vϕ. ∃vϕ is in S[m][r]. Suppose that A � ∃vϕ. Then there is a ∈ D(A, s) that satisfiesϕ in A. Recall that m is of length n + 1. There are two cases to consider. Suppose that s ≤ n. Let k bethe least member of Ks not occuring in K[m]. Let m′ be the sequence of non-negative integers of the samelength as m such that m′

s = ms + 1 and m′i = mi for i �= s. Then ϕ[v/k] is in S[m′][h[t + 1, s](r)],

and (Aa) � ϕ[v/k]. Therefore, there is b ∈ D(B, s) such that (Aa) �h[t+1,s](r) (Bb). Since h[t+ 1, s](r) ∈ S,(Aa) and (Bb) agree on S[m′][h[t+ 1, s](r)]. Therefore, B � ∃vϕ. Suppose now that s > n. Let k be the firstmember of Ks. Let m′ = mmn+1 . . .ms, where ms = 1 and mq = 0 for n + 1 ≤ q < s. Then ϕ[v/k] is inS[m′][h[t+ 1, s](r)]. There is a ∈ D(A, s) such that (Aa) � ϕ[v/k]. Reasoning as above, there is b ∈ D(B, s)such that (Bb) � ϕ[v/k]. Therefore, B � ∃vϕ. In the same way any such existentially quantified sentence trueon B is also true on A. The reasoning for universally quantified sentences is analogous.

The converse of Lemma 2.1 is established by constructing the sentences Ψ[A, r] mentioned above. Thatconstruction parallels the definition of �r.

When r = 0, Ψ[A, r] is the conjunction of the union of the set of atomic sentences true on A and the set ofthe negations of the atomic sentences false on A.

Once there is an s such that rs �= 0, the construction is more complicated. For example, suppose that r0 = 1and that rs = 0, otherwise. Then Ψ[A, r] is the conjunction of the following sentences:∧{∃xΨ[(Aa),0][k/x] : a ∈ A} and ∀x∨{Ψ[(Aa),0][k/x] : a ∈ A},

where k is the individual constant denoting a in (Aa). Note that b satisfies Ψ[A,0][k/x] in B iff (Aa) �0 (Bb).Ψ[Aa,0][k/x] is of rank r, but it contains no bound variables.

To describe such component formulas in general, a second (t + 1)-tuple r′ of non-negative integers is intro-duced. This (t+ 1)-tuple is used to index sequences of variables of order s = 0, . . . , t, and to index sequences ofvalues for these variables: If t ≥ 0, r′ = (r′0, . . . , r

′t) ∈ N

t+1 and s ∈ {0, . . . , t}, then let Vs[r′] be the set of thefirst r′s members of Vs, v(r′s) the sequence of these variables in order of Vs and a(r′s) a sequence of membersof D(A, s) of length r′s; moreover let v(r′) = v(r′0) . . .v(r′t) and a(r′) = a(r′0) . . .a(r′t). Let vr′

s+1 be theleast member of Vs not in Vs[r′]. If a ∈ D(A, s) and s < t, then a(r′)a = a(r′0) . . .a(r′s)a . . .a(r′t), and ifa ∈ D(A, t), then a(r′)a = a(r′0) . . .a(r′t)a; a(r′) � t = a(r′0) . . .a(r′t−1). If ϕ is a formula in free vari-ables V0[r′] ∪ · · · ∪ Vt[r′], then the phrase “ϕ is satisfied by a(r′) in A” is defined in the usual way with theunderstanding that the value of the ith member of v(r′) is the ith member of a(r′).

The formula Ψ[A, r,a(r′)] is then defined as follows:

1. If r = 0, then Ψ[A, r,a(r′)] is the conjunction of the atomic formulas in free variables V0[r′]∪· · ·∪Vt[r′]that are satisfied by a(r′) in A and the negations of such atomic formulas that are not satisfied by a(r′) in A.

2. If there is s such that rs �= 0, then Ψ[A, r,a(r′)] is the conjunction of the union of the following sets:

{∧{∃v′rs+1 Ψ[A, h[t+ 1, s](r),a(r′)a] : a ∈ D(A, s)} : 0 ≤ s ≤ t and rs �= 0},{∀v′rs+1

∨{Ψ[A, h[t+ 1, s](r),a(r′)a] : a ∈ D(A, s)} : 0 ≤ s ≤ t and rs �= 0}.



If rs = 0, then Vs[r′] is the set of members of Vs occurring in Ψ[A, r,a(r′)] and all of these occur free.If rs �= 0, then Vs[r′]∪{vr′

s+1, . . . , vr′s+rs} is the set of variables occurring in Ψ[A, r,a(r′)], where the members

of Vs[r′] occur free and the others occur bound. Hence, if for all s, r′s = 0, then Ψ[A, r,a(r′)] is a sentencein S[m][r]. In this case, Ψ[A, r] is Ψ[A, r,a(r′)].

Given r = (r0, . . . , rt) ∈ Nt+1 and r′ = (r′0, . . . , r

′t′) ∈ N

t′−1, we write r ≤ r′ iff the length t + 1 of r isless than or equal to the length t′ + 1 of r′ and rs ≤ r′s for all s ∈ {0, . . . , t}.

Lemma 2.2 Assume that A and B are of type K[m] and r, r′ ∈ Nt+1. Then the following are equivalent:

1. (Aa(r′)) �r (Bb(r′)).2. b(r′) satisfies Ψ[A, r,a(r′)] in B.

P r o o f. Let

S = {r : r ∈ Nt+1 and for all r′, (Aa(r′)) �r (Bb(r′)) iff b(r′) satisfies Ψ[B, r,a(r′)] in B}.

If r = 0, then it is easily to verify that b(r′) satisfies Ψ[A, r,a(r′)] in B iff (Aa(r′))[∅] ≈ (Bb(r′))[∅].Now suppose that there is s such that rs �= 0. Suppose that, for all such s, h[t+ 1, s](r) ∈ S. Suppose that

b(r′) satisfies Ψ[A, r,a(r′)] in B. Let a ∈ D(A, s). Then there is b ∈ D(B, s) such that b(r′)b satisfiesΨ[A, h[t + 1, s](r),a(r′)a] in B. By supposition, (Bb(r′)b) �h[t+1,s](r) (Aa(r′)a). In the same way, givenb ∈ D(B, s), there is a ∈ D(A, s) such that (Bb(r′)b) �h[t+1,s](r) (Aa(r′)a). Hence, (Aa(r′)) �r (Bb(r′)).If conversely (Aa(r′)) �r (Bb(r′)), then there is m′ such that (Aa(r′)) and (Bb(r′)) are of type K[m′] andΨ[(Aa(r′), r)] is in S[m′][r]. By Lemma 2.1, (Bb(r′)) � Ψ[(Aa(r′), r)]. Thus, b(r′) satisfies Ψ[A, r,a(r′)]in B.

It follows from Lemma 2.2 that A �r B iff B � Ψ[A, r]. Hence, if ϕ ∈ S[m][r], then either ϕ or its negationis a logical consequence of Ψ[A, r]. It is easily verified that b(r) satisfies Ψ[A, r,a(r′)] in B iff Ψ[A, r,a(r′)]and Ψ[B, r, b(r′)] are identical. There are only finitely many formulas of the form Ψ[A, r,a(r′)]. Hence, thenumber of interpretations that are not �r is finite. Given a sentence ϕ in L[m], t ≥ 0 and r ∈ N

t+1, H(ϕ, r) isdefined as follows:

1. If ϕ has no models, then H(ϕ, r) is the conjunction of some sentence in S[m][r] and its negation.

2. If ϕ has models, then H(ϕ, r) is∨{Ψ[A, r] : A � ϕ}.

Then H(ϕ, r) ∈ S[m][r]. Further, B � H(ϕ, r) iff there is A such that A � ϕ and B �r A. Hence,ϕ � H(ϕ, r). Finally, if rk(ϕ) ≤ r′, then ϕ and H(ϕ, r′) have exactly the same models. H(ϕ, rk(ϕ)) is,after Hintikka [2], the (second-order) distributive normal form of ϕ. Below, two generalizations are mentioned.

Let S be a set of sentences in L[m] with models and let H(S, r) be∨{Ψ[A, r] : A � S}. Notice that

B � H(S, r) iff there is A such that A � S and B �r A. Thus, if ϕ ∈ S[m][r], then S � ϕ iff H(S, r) � ϕ. Itfollows that there is a weak form of the consequence formulation of the compactness theorem for sets of sentencesthat have models.

Theorem 2.3 Assume that S has models.

1. S � ϕ iff H(S, rk(ϕ)) � ϕ.

2. S � ϕ iff {H(S, r) : t ≥ 0 and r ∈ Nt+1} � ϕ.

3. If {H(S, r) : t ≥ 0 and r ∈ Nt+1} � ϕ, then H(S, rk(ϕ)) � ϕ.

Let ϕ be a sentence in L[m] and let C(ϕ) be the set of non-logical constants in K[m] occurring in ϕ. If K ′

is a subset of K[m] and A is of type K[m], then A � K ′ is the restriction of A to an interpretation of type K ′.If ϕ has models, then H(ϕ,K ′, r) is defined as

∨{Ψ[A � K ′, r] : A � ϕ}. Then C(H(ϕ,K ′, r)) = K ′,rk(H(ϕ,K ′, r)) = r, and B � H(ϕ,K ′, r) iff there is A such that A � ϕ and A � K ′ �r B � K ′. Thus,ϕ � H(ϕ,K ′, r). The formulasH(ϕ,K[m], r) andH(ϕ, r) have the same models as ϕ andH(ϕ,C(ϕ), rk(ϕ)).

Theorem 2.4 Assume that ϕ ∈ L[m] has models, ψ ∈ S[m][r] and ϕ � ψ. Then

1. ϕ � H(ϕ,C(ψ), r), and

2. H(ϕ,C(ψ), r) � ψ.

Let Σ be a class of interpretations of type K[m]. Σ is a finitary class (in L[m]) iff Σ is the class of models ofa finite set of sentences in L[m]. Lemma 2.1 implies a characterization of finitary classes.



Theorem 2.5 Σ is a finitary class iff there is r such that Σ is closed under �r.

P r o o f. Suppose that Σ is a finitary class. Let ϕ be the conjunction of that finite set of sentences whose classof models is Σ. Let r be the rank of ϕ. Then Σ is closed under �r. Assume in contrary that Σ is closed under �r.Let ϕ be

∨{Ψ[A, r] : A ∈ Σ}. Then Σ is the collection of models of ϕ.

It follows from Theorem 2.5 that a theory in L[m] is finitely axiomatizable iff there is r such that the classof models of the theory are closed under �r. If all of the models of the theory are infinite, then stronger resultsfollow from a lemma of David Kaplan ([4, Lemma 2, p. 257]).

Let �1 and �2 be binary relations on A. Then

�1 + �2 = {(x, y, z) : (z, x) ∈ �1 and (z, y) ∈ �2}.

If � is a subset of A2, then

�1 + �2[�] = {a : there are b and c in A such that (b, c) ∈ � and (b, c, a) ∈ �1 + �2}.

If t ≥ 2 and � a subset of At+1, then

�1 + �2[�] = {(a1, . . . , at−1, a) : there are b and c in A such that

(a1, . . . , at−1, b, c) ∈ � and (b, c, a) ∈ �1 + �2}.

If τ1 . . . τm is a sequence (possibly empty) of n-ary relations on A, then

�1 + �2[τ1 . . . τm] = �1 + �2[τ1] . . . �1 + �2[τm].

�1 +�2 is a pairing function onA iff it is a bijection betweenA2 andA. All and only infinite sets have pairingfunctions. Further, there is a first-order sentence ϕ in five individual variables such that (A�1�2) � ϕ iff �1 + �2

is a pairing function on A.

Lemma 2.6 Assume that A and B are infinite interpretations of type K[m], �1 and �2 are binary relationson A, τ1 and τ2 are binary relations on B, �1 + �2 is a pairing function on A, and τ1 + τ2 is a pairing functionon B. Then for all t ≥ 1 and all r, r′, r′′ ∈ N

t+2,

if (A�1�2)a(r) � (t+ 1)�1 + �2[a(rt+1)] �r′′ (Bτ1τ2)b(r) � (t+ 1)τ1 + τ2[b(rt+1)],r′′0 = r′0 + 1, r′′s = r′s for all s ∈ {1, . . . , t− 1), r′′t = r′t + r′t+1, and r′′t+1 = 0,then (A�1�2)a(r) �r′ (Bτ1τ2)b(r).

P r o o f. We proceed by induction on Nt+2. Suppose r′ ∈ N

t+2 is such that r′s = 0 for all s. Supposethat (A�1�2)a(r) � (t+ 1)�1 + �2[a(rt+1)] �r′′ (Bτ1τ2)b(r) � (t+ 1)τ1 + τ2[b(rt+1)], r′′0 = 1 and r′′s = 0if s �= 0. We want to show that (A�1�2)a(r)[∅] ≈ (Bτ1τ2)b(r)[∅]. By supposition,

(A�1�2)a(r) � (t+ 1)[∅] ≈ (Bτ1τ2)b(r) � (t+ 1)[∅].

Let f be the isomorphism between these interpretations, � a member of a(rt+1) and τ the corresponding memberof b(rt+1). � and τ are both (t + 1)-ary relations. It suffices to show that for all a1, . . . , at+1 in the domain ofthe subinterpretation of (A�1�2)a(r) generated by ∅, (a1, . . . , at+1) ∈ � iff (f(a1), . . . , f(at+1)) ∈ τ . Supposet = 1 and (a1, a2) ∈ �. There is unique a in A such that (a1, a2, a) ∈ �1 + �2 and a ∈ �1 + �2[�]. Since r′′0 = 1,by supposition, there is b in B such that

(A�1�2)a(r) � (t+ 1)a�1 + �2[a(rt+1)] �h[t+2,0](r′′) (Bτ1τ2)b(r) � (t+ 1)bτ1 + τ2[b(rt+1)].

Hence, b ∈ τ1 + τ2[τ ] and (f(a1), f(a2), b) ∈ τ1 + τ2. Therefore, (f(a1), f(a2)) ∈ τ . The reasoning in the otherdirection is analogous. The reasoning for t > 1 is similar to the above.



Now suppose that there is s such that r′s �= 0. Assume that the lemma holds for h[(t+2), s](r′), for all such s.Suppose that (A�1�2)a(r) � (t+ 1)�1 + �2[a(rt+1)] �r′′ (Bτ1τ2)b(r) � (t+ 1)τ1 + τ2[b(rt+1)], r′′0 = r′0 + 1,r′′� = r′� for � ∈ {1, . . . , t− 1}, r′′t = r′t + r′t+1, and r′′t+1 = 0. Suppose that r′s �= 0. It suffices to show that forall a ∈ D(A, s) there is b ∈ D(B, s) such that

(∗) (A�1�2)a(r)a �h[t+2,s](r′) (Bτ1τ2)b(r)b,

and that for all b ∈ D(B, s) there is a ∈ D(A, s) such that a and b satisfy (∗). If s ≤ t, this is immediate fromthe induction hypothesis and the supposition. Suppose that s = t+ 1. Let � ∈ D(A, t+ 1). � is a relation on A.By supposition, there is σ ∈ D(B, t) such that

(A�1�2)a(r) � (t+1)�1+�2[a(rt+1)]�1+�2[�] �h[t+2,t](r′) (Bτ1τ2)b(r) � (t+1)τ1+τ2[b(rt+1)]σ.

Let τ be such that τ1 + τ2[τ ] = σ. Then τ ∈ D(B, t + 1). By induction hypothesis, τ and � satisfy (∗). Bysimilar reasoning, for τ ∈ D(B, t+ 1) there is � ∈ D(A, t+ 1) such that � and τ satisfy (∗). It follows that thelemma holds for r′.

Lemma 2.7 Assume that A and B are interpretations of type K[m], �1 and �2 are binary relations on A,τ1 and τ2 are binary relations on B, �1 + �2 is a pairing function on A and τ1 + τ2 is a pairing functionon B. Then for all t ≥ 1 and all r′ ∈ N

t+2, if (A�1�2) �r′′ (Aτ1τ2) and r′′ = (r′0 + t,∑t+1

i=1 r′i, 0), then

(A�1�2) �r′ (Bτ1τ2).

P r o o f. The proof is direct from Lemma 2.6 by induction on t when rs = 0 for all s.

Theorem 2.8 Assume that A and B are infinite interpretations of type K[m], t ≥ 1 and r ∈ Nt+2. Then, if

A �r′ B and r′ = (max{5, r0 + t},∑t+1i=1 ri, 2), then A �r B.

P r o o f. Suppose that A �(max{5,(r0+t)},Pt+1

i=1 ri,2)B. Given binary relations �1 and �2 onA such that �1+�2

is a pairing function on A, there are binary relations τ1 and τ2 on B such that

(A�1�2) �(max{5,r0+t},Pt+1

i=1 ri,0)(Bτ1τ2).

Since there is a first-order sentence in five variables that is true on (Bτ1τ2) iff τ1 + τ2 is a pairing function on B,by Lemma 2.2 and by the fact that this sentence is true on (A�1�2), τ1 + τ2 is a pairing function on B. Thereasoning is then immediate from Lemma 2.7.

It follows from Theorem 2.8 that if the theory of an infinite interpretation is finitely axiomatizable by a sentenceof rank (r0, . . . , rt+1), where t ≥ 1, then the theory has an axiomatization of rank (max{5, r0 + t},∑t+1

i=1 ri, 2).As a corollary to Theorem 2.8 we obtain a version of Kaplan’s lemma.

Corollary 2.9 Assume that A and B are infinite interpretations of type K[m]. If A �(n0,n1,2) B forall n0, n1, then for all t ≥ 3 and all r ∈ N

t, A �r B.

3 Infinitary second-order languages

In this section we consider the case where K is infinite. K[m], L[m] and S[m][r] are understood as above. Thereasoning for Lemma 2.1 is easily extended to establish the following.

Lemma 3.1 Assume that K is infinite and that A and B are of type K[m]. If A �r B, then A and B agreeon all sentences in S[m][r].

The converse of Lemma 3.1 fails when K is infinite. For example, let K consist of ℵ0 individual constantsand let A and B be of type K , where B = A ∪ {d}, d /∈ A, fA is one-one, every member of A is denotedin A by a member of K , and fA and fB agree on K . Then A and B agree on all sentences in S[0], butnot A �(1,1) B. In the following, second-order languages are augmented so that Lemma 3.1 and its conversehold for these languages. More specifically, the languages are augmented in such a way that infinitary analoguesof the formulas Ψ[A,a(r), r′] satisfy the appropriate extension of Lemma 2.2. These formulas contain infiniteconjunctions and disjunctions but only finitely many variables (free or bound).



A set T of formulas is bound iff there is t ≥ 0 and r ∈ Nt+1 such that no member of T contains a variable of

degree > t and for all s ∈ {0, . . . , t}, no more than the first rs members of Vs occur in any member of T . Thenr is called a variable bound for T .

Given m ∈ Nn+1, the set of formulas of the augmented second-order language La[m] over K[m] is closed

under conjunctions and disjunctions of bound sets of formulas. More precisely, the set of formulas in La[m] isthe union of the sets {Fn[m] : n ≥ 0}, where Fn[m] is defined as follows:

1. F0[m] is the set of all formulas in L[m] and

2. Fn+1[m] is the smallest set including Fn[m] and closed under forming truth-functional combinations,universal and existential quantification and conjunctions and disjunctions of bound subsets of Fn[m].

Every formulaϕ in La[m] has a degree D(ϕ), an n-ary rank rk(n, ϕ), and a rank rk(ϕ), where D(ϕ), rk(n, ϕ)and rk(ϕ) are defined as in the finite case. By Sa[m][t] we denote the set of sentences in La[m] of degree ≤ t,and Sa[m][r] is the set of sentences in Sa[m][t] whose s-ary rank is ≤ rs for each s ∈ {0, . . . , t}. As in thefinitary case, if r �= 0, then Sa[m][r] is the smallest set that includes all sets Sa[m][h[t + 1, s](r)] for each ssuch that rs �= 0 and all sentences of the forms ∃vϕ and ∀vϕ, where ϕ is a formula in La[m] whose one andonly free variable is v and rk(ϕ) = r, and that is closed under forming finitary truth functional combinations andconjunctions and disjunctions of bound sets of sentences. The formula Ψ[A, r,a(r′)] is defined as in the finitecase. To distinguish between the finite and infinite cases we write Ψa[A, r,a(r′)] in the infinite case. As in thefinite case, if r′ = 0, then Ψa[A, r,a(r′)] is a sentence in Sa[m][r], and this sentence is Ψa[A, r].

Lemma 3.1 holds for augmented sentences:

Lemma 3.2 Assume that K is infinite and that A and B are of type K[m]. If A �r B, then A and B agreeon all sentences in Sa[m][r].

The reasoning for Lemma 2.1 is easily extended to establish Lemma 3.2. In the same way, the reasoning forLemma 2.2 can be extended to prove the following.

Lemma 3.3 Assume thatK is infinite and that A and B are of typeK[m]. Then the following are equivalent:1. (Aa(r′)) �r (Bb(r′)).2. b(r′) satisfies Ψa[A, r,a(r′)] in B.

It follows from Lemma 3.3 that A �r B iff B � Ψa[A, r]. Therefore, if ϕ ∈ Sa[m][r], then either ϕ or itsnegation is a logical consequence of Ψa[A, r]. If ϕ is a sentence in La[m], then {Ψa[A, r] : A � ϕ} is bound.Thus, the disjunction of this set is in Sa[m][r]. Therefore,H(ϕ, r), defined as in Section 2, is a sentence. Similarremarks apply to H(S, r) and H(ϕ,K ′, r). Hence, Theorem 2.3 and Theorem 2.4 hold for La[m]. Assume thatΣ is a class of interpretations, Σ is a bound class (in La[m]) provided Σ is the class of models of a bound setof sentences in La[m]. A theory in an augmented language has a bound base provided there is a bound set ofsentences whose logical consequences are exactly the logical consequences of the theory. The reasoning forTheorem 2.5 is easily extended to establish the following.

Theorem 3.4 Σ is a bound class iff there is r such that Σ is closed under �r.

Theorem 3.4 implies that a theory in an augmented language has a bound base iff there is r such that the classof models of the theory are closed under �r. Theories with a bound base in augmented second-order logic arethe natural analogues of finitely axiomatizable theories in (finitary) second-order logic, and many questions aboutfinitely axiomatizable theories can be reformulated for theories with a bound base. For example, the Fraenkel-Carnap question (is every finitely axiomatizable, semantically complete theory categorical?) can be reformulatedin augmented second-order logic as the following: is every semantically complete theory with a bound basecategorical? For other related examples see [8]. Lemma 2.6 and Lemma 2.7 can easily be extended to augmentedsecond-order languages. The reasoning in Section 2 establishes the following.

Theorem 3.5 Assume that K is infinite, A and B are infinite interpretations of type K[m], t ≥ 1and r ∈ N

t+2. Then, if A �r′ B and r′ = (max{5, r0 + t},∑t+1i=1 ri, 2), then A �r B.

Corollary 3.6 Assume that K is infinite and A and B are infinite interpretations of type K[m]. If forall n0, n1, A �(n0,n1,2) B, then for all t ≥ 3 and all r ∈ N

t, A �r B.

The logic composed of La[m] and its standard semantics is stronger than standard second-order logic whenKis infinite. Recall that the two interpretations exhibited after the proof of Lemma 3.1 are second-order equivalentbut are distinguished by an augmented second-order sentence of quantifier rank (1, 1). Other infinitary higher



order languages have appeared in the literature. For example, [3] contains higher order languages obtained fromfinitary higher order languages by allowing conjunctions and disjunctions of sets of cardinality less than theinfinite cardinal κ, but only finite sequences of variables. Let L[m]κω be the language obtained from L[m] inthis way. These languages include La[m]. However, there are sentences in the L[m]ω1ω that are not sentencesin La[m]. For example, conjunctions and disjunctions of countably infinite sets of second-order sentences hav-ing no variable bound are sentences in L[m]ω1ω, but not in La[m]. L[m]ω1ω is expressively more powerfulthan La[m]. It follows from Lemma 4.1 that the least cardinal not characterizable by a pure sentence in La[m](in the sense of [8, p. 93]) is characterizable by a pure sentence in L[m]ω1ω .La[m] has an infinitary first-order sublanguage. This sublanguage is a subset of L∞ω. When K is infinite

it is the language of an infinitary extension of first-order logic in which the Fraısse back and forth condition(see [7]) is both a necessary and sufficient condition for equivalence of interpretations of type K[m0]. As above,the syntax of this particular infinitary first-order language is different from that of Lω1ω. To the knowledge of theauthors, neither this infinitary first-order logic nor the corresponding infinitary monadic second-order logic havebeen studied.

4 Augmented finitary second-order languages

In this section we consider augmenting those second-order languages whose non-logical vocabulary is finite.Among the languages considered are the pure languages, i. e. those whose non-logical vocabulary is empty. It iseasily shown that if K is finite, and A and B are of type K[m], then A and B agree on all sentences in L[m] iffthey agree on all sentences in La[m]. This result is immediate from the following lemma.

Lemma 4.1 Assume thatK is finite and that A is of type K[m]. Then Ψa[A, r,a(r′)] and Ψ[A, r,a(r′)] areidentical.

It follows from Lemma 4.1 that non-zero cardinals are indistinguishable in L[m] (in the sense of [8, p. 96]) iffthey are indistinguishable in La[m].

Acknowledgements The first author wishes to thank those of his students who have participated in the discussions of themodel theory of second-order languages and the back and forth characterization of equivalence for interpretations for theselanguages: Kenneth J. Woodring, Ben George, and Jennifer Vaughan. The latter two and the second author were supportedby grants from Bryn Mawr College.

References

[1] D. Hilbert and W. Ackermann, Principles of Mathematical Logic (Chelsea Publishing, New York 1950).[2] J. Hintikka, Distributive normal forms in first-order logic. In: Formal Systems and Recursive Functions (J. N. Crossley

and M. A. E. Dummett, eds.), pp. 48 – 91 (North-Holland, Amsterdam 1965).[3] M. Magidor, On the role of supercompact and extendible cardinals in logic. Israel J. Math. 10, 147 – 157 (1971).[4] R. Montague, Reductions of higher-order logic. In: The Theory of Models (J. W. Addision, L. Henkin, and

A. Tarski, eds.), pp. 251 – 264 (North-Holland, Amsterdam 1965).[5] A. Mostowski, Thirty Years of Foundational Studies (Barnes and Noble, New York 1966).[6] S. Shapiro, Foundations without Foundationalism (Oxford University Press, Oxford 1991).[7] G. Weaver, Finite partitions and their generators. Z. Math. Logik Grundlagen Math. 20, 255 – 260 (1974).[8] G. Weaver and B. George, The Fraenkel-Carnap question for Dedekind algebras. Math. Logic Quarterly 49, 92 – 96

(2003).


from finitary to infinitary second-order logic

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