Given a Mahlo cardinal k and a regular ε such that ω1 < ε < k we show that ◇k(cf = ε) holds in V provided that there are only non-stationarily many β < k with o(β) ≥ ε in K.

We show, under the assumption that factoring is hard, that a model of PV exists in which the polynomial hierarchy does not collapse to the linear hierarchy; that a model of exists in which NP is not in the second level of the linear hierarchy; and that a model of exists in which the polynomial hierarchy collapses to the linear hierarchy.

Our methods are model-theoretic. We use the assumption about factoring to get a model in which the weak pigeonhole principle fails in a certain way, and then work with this failure to obtain our results.

As a corollary of one of the proofs, we also show that in the failure of WPHP (for definable relations) implies that the strict version of PH does not collapse to a finite level.

These systems are roughly natural number theory in, respectively, nth order function calculus, for all positive integers n. Each of these systems is expressed in the notation of the theory of types, having variables with type subscripts from 1 to n. Variables of type 1 stand for natural numbers, variables of type 2 stand for classes of natural numbers, etc. Primitive atomic wff's (well-formed formulas) of Tn are those of number theory in variables of type 1, and of the following kind for n > 1: xi ϵ yi+1. Other wff's are formed by truth functions and quantifiers in the usual manner. Quantification theory holds for all the variables of Tn. Tn has the axioms Z1 to Z9, which are, respectively, the nine axioms and axiom schemata for the system Z (natural number theory) on p. 371 of [1]. These axioms and axiom schemata contain only variables of type 1, except for the schemata Z2 and Z9, which are as follows:

where ‘F(x1)’ can be any wff of Tn. Identity is primitive for variables of type 1; for variables of other types it is defined as follows:

This paper reviews various symbolisms for the two-valued propositional calculus, and introduces a new set of signs which embodies the principles of Leśniewski's symbolism, yet resembles better known signs. This type of symbolism, serving as a diagram, may be used either in place of or as auxiliary to the usual symbolisms.

We give a sharper version of a theorem of Rosický, Trnková and Adámek [13], and a new proof of a theorem of Rosický [12], both about colimits in categories of structures. Unlike the original proofs, which use category-theoretic methods, we use set-theoretic arguments involving elementary embeddings given by large cardinals such as α-strongly compact and C(n)-extendible cardinals.

A nonstandard set theory *ZFC is proposed that axiomatizes the nonstandard embedding *. Besides the usual principles of nonstandard analysis, all axioms of ZFC except regularity are assumed. A strong form of saturation is also postulated. *ZFC is a conservative extension of ZFC.

It is well-known that natural axiomatic theories are well-ordered by consistency strength. However, it is possible to construct descending chains of artificial theories with respect to consistency strength. We provide an explanation of this well-orderedness phenomenon by studying a coarsening of the consistency strength order, namely, the $\Pi ^1_1$ reflection strength order. We prove that there are no descending sequences of $\Pi ^1_1$ sound extensions of $\mathsf {ACA}_0$ in this ordering. Accordingly, we can attach a rank in this order, which we call reflection rank, to any $\Pi ^1_1$ sound extension of $\mathsf {ACA}_0$ . We prove that for any $\Pi ^1_1$ sound theory T extending $\mathsf {ACA}_0^+$ , the reflection rank of T equals the $\Pi ^1_1$ proof-theoretic ordinal of T. We also prove that the $\Pi ^1_1$ proof-theoretic ordinal of $\alpha $ iterated $\Pi ^1_1$ reflection is $\varepsilon _\alpha $ . Finally, we use our results to provide straightforward well-foundedness proofs of ordinal notation systems based on reflection principles.

Let T be a model-complete theory that eliminates the quantifier ∃∞x For T we construct a theory T+ such that any element in a model of T+ determines a model of T. We show that T+ has a model companion T1. We can iterate the construction. The produced theories are investigated.

We present a cut-elimination proof for simple type theory with an axiom of choice formulated in the language with an epsilon-symbol. The proof is modeled after Takahashi's proof of cut-elimination for simple type theory with extensionality. The same proof works when types are restricted, for example for second-order classical logic with an axiom of choice.

A sequence of graphs is FO-convergent if the probability of satisfaction of every first-order formula converges. A graph modeling is a graph, whose domain is a standard probability space, with the property that every definable set is Borel. It was known that FO-convergent sequence of graphs do not always admit a modeling limit, but it was conjectured that FO-convergent sequences of sufficiently sparse graphs have a modeling limits. Precisely, two conjectures were proposed:

1. 1. If a FO-convergent sequence of graphs is residual, that is if for every integer d the maximum relative size of a ball of radius d in the graphs of the sequence tends to zero, then the sequence has a modeling limit.

2. 2. A monotone class of graphs ${\cal C}$ has the property that every FO-convergent sequence of graphs from ${\cal C}$ has a modeling limit if and only if ${\cal C}$ is nowhere dense, that is if and only if for each integer p there is $N\left( p \right)$ such that no graph in ${\cal C}$ contains the pth subdivision of a complete graph on $N\left( p \right)$ vertices as a subgraph.

In this article we prove both conjectures. This solves some of the main problems in the area and among others provides an analytic characterization of the nowhere dense–somewhere dense dichotomy.

Various theorems for the preservation of set-theoretic axioms under forcing are proved, regarding both forcing axioms and axioms true in the Lévy collapse. These show in particular that certain applications of forcing axioms require to add generic countable sequences high up in the set-theoretic hierarchy even before collapsing everything down to ℵ1. Later we give applications, among them the consistency of MM with ℵω not being Jónsson which answers a question raised in the set theory meeting at Oberwolfach in 2005.

We use a new construction of an o-minimal structure, due to Lipshitz and Robinson, to answer a question of van den Dries regarding the relationship between arbitrary o-minimal expansions of real closed fields and structures over the real numbers. We write a first order sentence which is true in the Lipshitz-Robinson structure but fails in any possible interpretation over the field of real numbers.

In [1] it is shown that for every sequence x = 〈xn : n ∈ ω〉 ∈ Xω Λ there is an isol xω (essentially an immunized product) such that

Here we have used the notation: Λ = the isols, ω = the nonnegative integers, pn is the nth prime rational integer starting with p0 = 2, ∣ denotes divisibility and ∤ its negation. If p is an arbitrary prime, py ∣ x, pz ∣ x, and y < z then py+1 ∣ x. In particular since y ∈ ω is comparable with every element of Λ, the conditions py ∣ x and py+1 ∤ x uniquely determine y. Thus every sequence x ∈ Xωω is uniquely determined by an xω satisfying (1) and consequently may be used as a “code” for that sequence. In Theorem 1 it is shown that (1) does not uniquely determine the values of an arbitrary sequence x ∈ XωΛ, however in Theorem 3 we find a different scheme which does. At the very end of the paper we give some reasons why coding theorems are useful. It should also be mentioned that for a coding theorem to be meaningful it is necessary to restrict the operations by which a sequence can be recaptured from its code. Otherwise a triviality results. Our coding theorem will allow all operations which are first order definable in Λ with respect to addition, multiplication, and exponentiation. We conjecture that the latter operation is really necessary.

Theorem. The Jacobson radical of an ℵ0-categorical associative ring is nilpotent.

We calculate the complexity of Scott sentences of scattered linear orders. Given a countable scattered linear order L of Hausdorff rank $\alpha $ we show that it has a ${d\text {-}\Sigma _{2\alpha +1}}$ Scott sentence. It follows from results of Ash [2] that for every countable $\alpha $ there is a linear order whose optimal Scott sentence has this complexity. Therefore, our bounds are tight. We furthermore show that every Hausdorff rank 1 linear order has an optimal ${\Pi ^{\mathrm {c}}_{3}}$ or ${d\text {-}\Sigma ^{\mathrm {c}}_{3}}$ Scott sentence and give a characterization of those linear orders of rank $1$ with ${\Pi ^{\mathrm {c}}_{3}}$ optimal Scott sentences. At last we show that for all countable $\alpha $ the class of Hausdorff rank $\alpha $ linear orders is $\boldsymbol {\Sigma }_{2\alpha +2}$ complete and obtain analogous results for index sets of computable linear orders.

We present two theorems whose applications are to eliminate diagonalization arguments from a variety of constructions of degrees of unsolvability.

All definitions and notations come from [1, Chapter 1]. We give a brief resumé of them here.

We identify a set with its characteristic function. (A(x)= 1 if x ∈ A and A (x) = 0 if x ∉ A.) A string σ is the restriction of a characteristic function to a finite initial segment of natural numbers, lh(σ) = length of σ = n + 1 if σ = A[n] for some set A. (A [n] is the restriction of A to {m: m ≤ n}.) If i = 0 or 1, σ * i is defined as the string of length lh(σ) + 1 such that σ * i ⊇ σ and σ * i(lh(σ)) = i. We write σ ∣ τ if σ ⊉ τ and τ ⊉ σ.

{Φn} is a listing of the partial recursive functionals. We write “A ≤TB” (“A is Turing reducible to B”) if ∃n∀xΦn(B)(x) = A(x).

A partial function, T, from strings to strings is a tree if T is order preserving and for all strings, σ, if one of T(σ * 0), T(σ * 1) is defined then T(σ), T(σ * 0), T(σ * 1) are all defined and T(σ * 0)∣(σ * 1).

Dynamic Topological Logic is a modal framework for reasoning about dynamical systems, that is, pairs 〈X, f〉 where X is a topological space and f: X → X a continuous function.

In this paper we consider the case where X is a metric space. We first show that any formula which can be satisfied on an arbitrary dynamic topological system can be satisfied on one based on a metric space; in fact, this space can be taken to be countable and have no isolated points. Since any metric space with these properties is homeomorphic to the set of rational numbers, it follows that any satisfiable formula can be satisfied on a system based on ℚ.

We then show that the situation changes when considering complete metric spaces, by exhibiting a formula which is not valid in general but is valid on the class of systems based on a complete metric space. While we do not attempt to give a full characterization of the set of valid formulas on this class we do give a relative completeness result; any formula which is satisfiable on a dynamical system based on a complete metric space is also satisfied on one based on the Cantor space.