The aim of this paper is to provide an addendum to a paper by Rose with the same title which has appeared in an earlier issue of this Journal [2]. Our new result is:

Theorem. A ring of characteristic zero which admits elimination of quantifiers in the language (0, 1, +, ·) is an algebraically closed field.

An (m + n)-valued propositional calculus2 may happen to be a subsystem of an m-valued propositional calculus , though the converse is never true. This fact may give us the impression that, as m grows, the content of becomes meagre. The present treatment is intended to remove this impression by constructing a complete, m-valued sub-system of any (m + n)-valued propositional calculus.

In the following we adopt the customary, autonymous mode of speech according to which symbols belonging to the object calculi or languages are used in the syntactic language as names for themselves, and juxtaposition serves to denote juxtaposition.

2.1 = df stands for definational identity in the syntactic language.

2.11 ≡ stands for definational identity in the object calculi.

2.2 ∊, ⊂, ∩, {x1, …, xn} are used in their meanings as customarily employed in the theory of sets—∊ for class membership, ⊂ for proper inclusion, ∩ for the product operation of classes, {x1, …, xn} for the class with x1, …, xn as its only elements.

2.3 x, y, z are used as unspecified natural numbers including 0. m, n, i, j are used as unspecified natural numbers other than 0.

2.401 Definition. δ = df as the function of two variables defined for any x, y such that

2.41 Definition. For m ≧ 2, ιm = df the function of two variables denned on the set {0, …, m − 1} such that ιm (x, y) = y − x for x ≦ y and ιn(x, y) = 0 for x > y.

1. In this paper we construct a model for part of the system NF of [4]. Specifically, we define a relation R of natural numbers such that the R-relativiseds of all the axioms except P9 of Hailperin's finitization [2] of NF become theorems of say Zermelo set theory. We start with an informal explanation of the model.

2. Scrutiny of P1-P8 of [2] suggests that a model for these axioms might be constructed by so to speak starting with a universe that contained a “universe set” and a “cardinal 1”, and passing to its closure under the operations implicit in P1-P7, viz., the Boolean, the domain, the direct product, the converse, and the mixtures of product and inverse operations represented by P3 and P4. To obtain such closure we must find a way of representing the operations that involve ordered pairs and triples.

We take as universe of the model the set of natural numbers ω; we let 0 represent the “universe set” and 1 represent “cardinal 1”. Then, in order to be able to refer in the model to the unordered pair of two sets, we determine all representatives of unordered pairs in advance by assigning them the even numbers in unique fashion (see d3 and d25); we can now define the operations that involve ordered pairs and triples, and obtain closure under them using the odd numbers. It remains to weed out, as in d26, the unnecessary sets so as to satisfy the axiom of extensionality.

Type two cuts, bad cuts and very bad cuts are introduced in [10] for studying the relationship between Loeb measure and U-topology of a hyperfinite time line in an ω1-saturated nonstandard universe. The questions concerning the existence of those cuts are asked there. In this paper we answer, fully or partially, some of those questions by showing that: (1) type two cuts exist, (2) the ℵ1-isomorphism property implies that bad cuts exist, but no bad cuts are very bad.

Suppose that α = γ + δ where γ ≥ δ > 0. Then there is a graph which has no independent set of order type and has no pentagram (a pentagram is a set of five points with all pairs joined by edges). In the notation of Erdős and Rado. who generalized Ramsey's Theorem to this setting.

We prove that the class of trees with no branches of cardinality ≤ κ is not RPC definable in L∞κ when κ is regular. Earlier such a result was known for under the assumption κ<κ = κ. Our main result is actually proved in a stronger form which covers also L∞κ (and makes sense there) for every strong limit cardinal λ < κ of cofinality κ.

In [12] we proved the following isotropy-reflection principle:

Theorem. Let F be a formally real field and let Fp denote its Pythagorean closure. The natural embedding of reduced special groups from Gred(F) into Gred(Fp) = G(FP) induced by the inclusion of fields, reflects isotropy.

Here Gred(F) denotes the reduced special group (with underlying group Ḟ/ΣḞ2) associated to the field F, henceforth assumed formally real; cf. [11], Chapter 1, §3, for details.

The result proved in [12] is, in fact, more general. For example, the Pythagorean closure Fp can be replaced in the statement above by the intersection of all real closures of F (inside a fixed algebraic closure). Similar statements hold, more generally, for all relative Pythagorean closures of F in the sense of Becker [3], Chapter II, §3.

Since the notion of isotropy of a quadratic form can be expressed by a first-order formula in the natural language LSG for special groups (with the coefficients as parameters), this result raises the question whether the embedding ιFFp: Gred(F) ↪ G (Fp) is elementary. Further, since the LSG-formula expressing isotropy is positive-existential, one may also ask whether ιFFp reflects all (closed) formulas ofthat kind with parameters in Gred(F).

In this paper we give a negative answer to the first of these questions, for a vast class of formally real (non-Pythagorean) fields F (Prop. 5.1). This follows from rather general preservation results concerning the “Boolean hull” and the “reduced quotient” operations on special groups.

Recently, the Dimension Problem for effective Hausdorff dimension was solved by J. Miller in [14], where the author constructs a Turing degree of non-integral Hausdorff dimension. In this article we settle the Dimension Problem for effective packing dimension by constructing a real of strictly positive effective packing dimension that does not compute a real of effective packing dimension one (on the other hand, it is known via [10. 3. 7] that every real of strictly positive effective Hausdorff dimension computes reals whose effective packing dimensions are arbitrarily close to, but not necessarily equal to, one).

We consider the structure $({\Bbb Z}, + ,0,|_{p_1 } , \ldots ,|_{p_n } )$, where $x|_p y$ means $v_p \left( x \right) \leqslant v_p \left( y \right)$ and vp is the p-adic valuation. We prove that this structure has quantifier elimination in a natural expansion of the language of abelian groups, and that it has dp-rank n. In addition, we prove that a first order structure with universe ${\Bbb Z}$ which is an expansion of $({\Bbb Z}, + ,0)$ and a reduct of $({\Bbb Z}, + ,0,|_p )$ must be interdefinable with one of them. We also give an alternative proof for Conant’s analogous result about $({\Bbb Z}, + ,0, < )$.

Several attempts have been done to distinguish “positive” information in an arbitrary first order theory, i.e., to find a well behaved class of closed sets among the definable sets. In many cases, a definable set is said to be closed if its conjugates are sufficiently distinct from each other. Each such definition yields a class of theories, namely those where all definable sets are constructible, i.e., boolean combinations of closed sets. Here are some examples, ordered by strength:

Weak normality describes a rather small class of theories which are well understood by now (see, for example, [P]). On the other hand, normalization is so weak that all theories, in a suitable context, are normalizable (see [HH]). Thus equational theories form an interesting intermediate class of theories. Little work has been done so far. The original work of Srour [S1, S2, S3] adopts a point of view that is closer to universal algebra than to stability theory. The fundamental definitions and model theoretic properties can be found in [PS], though some easy observations are missing there. Hrushovski's example of a stable non-equational theory, the first and only one so far, is described in the unfortunately unpublished manuscript [HS]. In fact, it is an expansion of the free pseudospace constructed independently by Baudisch and Pillay in [BP] as an example of a strictly 2-ample theory. Strong equationality, defined in [Hr], is also investigated in [HS].

For G a group definable in a saturated model of a NIP theory T, we prove that there is a smallest type-definable subgroup H of G such that the quotient G / H is stable. This generalizes the existence of G00, the smallest type-definable subgroup of G of bounded index.

There is an ergodic, measure preserving, countable Borel equivalence relation E on a standard Borel probability space (X, μ) such that E∣c is not essentially free on any conull C ⊂ X.

We show that, assuming the existence of the canonical inner model with one Woodin cardinal $M_1 $ , there is a model of $ZFC$ in which the nonstationary ideal on $\omega _1 $ is $\aleph _2 $-saturated and whose reals admit a ${\rm{\Sigma }}_4^1 $-wellorder.

We analyze the set-theoretic strength of determinacy for levels of the Borel hierarchy of the form $\Sigma _{1 + \alpha + 3}^0 $, for α < ω1. Well-known results of H. Friedman and D.A. Martin have shown this determinacy to require α + 1 iterations of the Power Set Axiom, but we ask what additional ambient set theory is strictly necessary. To this end, we isolate a family of weak reflection principles, Π1-RAPα, whose consistency strength corresponds exactly to the logical strength of ${\rm{\Sigma }}_{1 + \alpha + 3}^0 $ determinacy, for $\alpha < \omega _1^{CK} $. This yields a characterization of the levels of L by or at which winning strategies in these games must be constructed. When α = 0, we have the following concise result: The least θ so that all winning strategies in ${\rm{\Sigma }}_4^0 $ games belong to Lθ+1 is the least so that $L_\theta \models {\rm{``}}{\cal P}\left( \omega \right)$ exists, and all wellfounded trees are ranked”.

We introduce a notion of complexity for interpretations, which is used to prove some new results about interpretations of sequential theories. In particular, we give a new, elementary proof of Pudlák's theorem that sequential theories are connected. We also demonstrate a counterexample to the infinitary distributive law

in the lattice of chapters, in which the chapters a and bi are compact. (Counterexamples in which a is not compact have been found previously.)

This paper is an appendix to Keisler's papers [3], [4]. We assume that the reader is fully acquainted with these papers and use their terminology without further explanation. One of the main tools in a number of proofs in [4] is the following result which is all but stated explicitly in [4].

We generalise the main theorems from the paper “The Borel cardinality of Lascar strong types” by I. Kaplan, B. Miller and P. Simon to a wider class of bounded invariant equivalence relations. We apply them to describe relationships between fundamental properties of bounded invariant equivalence relations (such as smoothness or type-definability) which also requires finding a series of counterexamples. Finally, we apply the generalisation mentioned above to prove a conjecture from a paper by the first author and J. Gismatullin, showing that the key technical assumption of the main theorem (concerning connected components in definable group extensions) from that paper is not only sufficient but also necessary to obtain the conclusion.

This note shows that the inconsistency proof of the system of illative combinatory logic given in [1] can be simplified as well as extended to the absolute inconsistency of a more general system.

One extension of the result in [1] lies in the fact that the following weakened form of the deduction theorem for implication will lead to the inconsistency:

Also the inconsistency follows almost as easily for

as it does for ⊢ H2X for arbitrary X, so we will consider the more general case.

The only properties we require other than (DT), (1) and Rule Eq for equality are modus ponens,

and

Let G = [x] Hn−1x⊃: . … H2x⊃ :Hx⊃ . x ⊃ A, where A is arbitrary. Then if Y is the paradoxical combinator and X = YG, X = GX.

Now X ⊂ X, i.e.,

A Boolean algebra B is said to be openly generated if {A : A ≤rcB, ∣A∣ = ℵ0} includes a club subset of . We show:

(V = L). For any cardinal κ there exists an L∞κ-free Boolean algebra which is not openly generated (Proposition 4.1).

(MA+(σ-closed)). Every -free Boolean algebra is openly generated (Theorem 4.2).

The last assertion follows from a characterization of openly generated Boolean algebras under MA+(σ-closed) (Theorem 3.1). Using this characterization we also prove the independence of problem 7 in Ščepin [15] (Proposition 4.3 and Theorem 4.4).