Why is forcing the only known method for constructing outer models of set theory?

If V is a standard transitive model of ZFC, then a standard transitive model W of ZFC is an outer model of V if V ⊆ W and V ∩ OR = W ∩ OR.

Is every outer model of a given model a generic extension? At one point Solovay conjectured that if 0# exists, then every real that does not construct 0# lies in L[G], for some G that is generic for some forcing ℙ ∈ L. Famously, this was refuted by Jensen's coding theorem. He produced a real that is generic for an L-definable class forcing property, but does not lie in any set forcing extension of L.

Beller, Jensen, and Welch in Coding the universe [BJW] revived Solovay's conjecture by asking the following question: Let a ⊆ ω be such that L[a] ⊨ “0# does not exist”. Is there ab∈ L[a] such that a ∉ L[b] and a is set generic over L[b].

In [1] and [2] there is a development of a class theory, whose axioms were formulated by Bernays and based on a reflection principle. See [3]. These axioms are formulated in first order logic with ∈:

(A1) Extensionality.

(A2) Class specification. If ϕ is a formula and A is not free in ϕ, then

Note that “x is a set“ can be written as “∃u(x ∈ u)”.

(A3) Subsets.

Note also that “B ⊆ A” can be written as “∀x(x ∈ B → x ∈ A)”.

(A4) Reflection principle. If ϕ(x) is a formula, then

where “u is a transitive set” is the formula “∃v(u ∈ v) ∧ ∀x∀y(x ∈ y ∧ y ∈ u → x ∈ u)” and ϕPu is the formula ϕ relativized to subsets of u.

(A5) Foundation.

(A6) Choice for sets.

We denote by B1 the theory with axioms (A1) to (A6).

The existence of weakly compact and -indescribable cardinals for every n is established in B1 by the method of defining all metamathematical concepts for B1 in a weaker theory of classes where the natural numbers can be defined and using the reflection principle to reflect the satisfaction relation; see [1]. There is a proof of the consistency of B1 assuming the existence of a measurable cardinal; see [4] and [5]. In [6] several set and class theories with reflection principles are developed. In them, the existence of inaccessible cardinals and some kinds of indescribable cardinals can be proved; and also there is a generalization of indescribability for higher-order languages using only class parameters.

The purpose of this work is to develop higher order reflection principles, including higher order parameters, in order to obtain other large cardinals.

We characterize the (κ, λ, < μ)-distributive law in Boolean algebras in terms of cut and choose games , when μ ≤ κ ≤ λ and κ<κ = κ. This builds on previous work to yield game-theoretic characterizations of distributive laws for almost all triples of cardinals κ, λ, μ with μ ≤ λ, under GCH. In the case when μ ≤ κ ≤ λ and κ<κ = κ, we show that it is necessary to consider whether the κ-stationarity of in the ground model is preserved by . In this vein, we develop the theory of κ-club and κ-stationary subsets of . We also construct Boolean algebras in which Player I wins but the (κ, ∞, κ)-d.1. holds, and, assuming GCH, construct Boolean algebras in which many games are undetermined.

In this paper we study with proof-theoretic methods the function(al)s provably recursive relative to Ramsey's theorem for pairs and the cohesive principle (COH).

Our main result on COH is that the type 2 functional provably recursive from are primitive recursive. This also provides a uniform method to extract bounds from proofs that use these principles. As a consequence we obtain a new proof of the fact that is -conservative over PRA.

Recent work of the first author showed that is equivalent to a weak variant of the Bolzano-Weierstraß principle. This makes it possible to use our results to analyze not only combinatorial but also analytical proofs.

For Ramsey's theorem for pairs and two colors we obtain the upper bounded that the type 2 functional provable recursive relative to are in T1. This is the fragment of Gödel's system T containing only type 1 recursion—roughly speaking it consists of functions of Ackermann type. With this we also obtain a uniform method for the extraction of T1-bounds from proofs that use . Moreover, this yields a new proof of the fact that is -conservative over .

The results are obtained in two steps: in the first step a term including Skolem functions for the above principles is extracted from a given proof. This is done using Gödel's functional interpretation. After this the term is normalized, such that only specific instances of the Skolem functions are used. In the second step this term is interpreted using -comprehension. The comprehension is then eliminated in favor of induction using either elimination of monotone Skolem functions (for COH) or Howard's ordinal analysis of bar recursion (for ).

Nous dirons qu'un corps (commutatif) K est un corps de courbe réelle sur le corps k si et seulement si c'est une extension de k, ordonnable, finiment engendrée et de degré de transcendance 1 sur k, telle que k soit relativement algébriquement clos dans K. Un tel corps est le corps d'une courbe (plane) définie sur k. De plus si k est réel-clos, toute courbe définie sur k dont K est le corps, a un point régulier rationnel sur k; en effet soit I ⊂ k[X0,…, Xn] l'idéal d'une telle courbe Γ; on a donc: K = k[x0,…, xn] où x0,…, xn sont les classes d'équivalence de X0,…, Xn; (x0,…, xn) est un point régulier de Γ dans la clôture réelle de K, et par modèle-complétude, Γ a un point régulier rationnel sur k.

Notre but est d'étendre les résultats de Raphael Robinson [6] sur l'indécidabilité des corps de fractions rationnelles sur un corps ordonnable (dans le langage des corps). Comme lui, nous donnerons une méthode qui ne s'applique que lorsque le corps de base k est réel-clos, et une seconde méthode générate.

We provide a short proof that if κ is a regular cardinal with κ < c, then

for any ordinal α < min{, κ}. In particular,

for any ordinal α < . This generalizes an unpublished result of E. Szemerédi that Martin's axiom implies that

for any cardinal κ with κ < c.

We investigate the structure of the Medvedev lattice as a partial order. We prove that every interval in the lattice is either finite, in which case it is isomorphic to a finite Boolean algebra, or contains an antichain of size . the size of the lattice itself. We also prove that it is consistent with ZFC that the lattice has chains of size . and in fact that these big chains occur in every infinite interval. We also study embeddings of lattices and algebras. We show that large Boolean algebras can be embedded into the Medvedev lattice as upper semilattices, but that a Boolean algebra can be embedded as a lattice only if it is countable. Finally we discuss which of these results hold for the closely related Muchnik lattice.

A bracket abstraction algorithm is a means of translating λ-terms into combinators. Broda and Damas, in [1], introduce a new, rather natural set of combinators and a new form of bracket abstraction which introduces at most one combinator for each λ-abstraction. This leads to particularly compact combinatory terms. A disadvantage of their abstraction process is that it includes the whole Schönfinkel [4] algorithm plus two mappings which convert the Schönfinkel abstract into the new abstract. This paper shows how the new abstraction can be done more directly, in fact, using only 2n − 1 algorithm steps if there are n occurrences of the variable to be abstracted in the term. Some properties of the Broda-Damas combinators are also considered.

A set of gödel numbers is invariant if it is closed under automorphisms of (ω, ·), where ω is the set of all gödel numbers of partial recursive functions and · is application (i.e., n · m ≃ φn(m)). The invariant arithmetic sets are investigated, and the invariant recursively enumerable sets and partial recursive functions are partially characterized.

R. K. Meyer once gave precise form to the question of whether relevant implication can be defined in any modal system, and his answer was ‘no’. In the present paper, we extend S4, first with propositional quantifiers, to the system S4π+; and then with definite propositional descriptions, to the system S4π+ip. We show that relevant implication can in some sense be defined in the modal system S4π+ip, although it cannot be defined in S4π+.

The mad spectrum is the set of all cardinalities of infinite maximal almost disjoint families on ω. We treat the problem to characterize those sets ${\rm {\cal A}} $ which, in some forcing extension of the universe, can be the mad spectrum. We give a complete solution to this problem under the assumption $\vartheta ^{ < \vartheta } = \vartheta $, where $\vartheta = {\rm{min}}\left( {\rm {\cal A}} \right) $.

Some 40 years ago, Dana Scott proved that every countable Scott set is the standard system of a model of PA. Two decades later, Knight and Nadel extended his result to Scott sets of size ω1. Here, I show that assuming the Proper Forcing Axiom (PFA), every A-proper Scott set is the standard system of a model of PA. I define that a Scott set is proper if the quotient Boolean algebra /Fin is a proper partial order and A-proper if is additionally arithmetically closed. I also investigate the question of the existence of proper Scott sets.

The Marker-Steinhorn Theorem (cf. [2] and [3]), says the following. If T is an o-minimal theory and M ≺ N is an elementary extension of models of T such that M is Dedekind complete in N, then for every N-definable subset X of Nk, the trace X ∩ Mk is M-definable. The original proof in [2] gives an explicit method how to construct a defining formula of X ∩ Mk out of a defining formula of X. A geometric reformulation of the Marker-Steinhorn Theorem is the definability of Hausdorff limits of families of definable sets. An explicit construction of these Hausdorff limits for expansions of the real field has recently been achieved in [1]. Both proofs and also the treatment [3] are technically involved.

Here we give a short algebraic, but not constructive proof, if T is an expansion of real closed fields. In fact we'll identify the statement of the Theorem with a valuation theoretic property of models of T (namely condition (†) below). Therefore our proof might be applicable to other elementary classes which expand fields, if a notion of dimension and a reasonable valuation theory are available.

From now on, let T be an o-minimal expansion of real closed fields. We have to show the following (cf. [2], Th. 2.1. for this formulation). If M is a model of T and p is a tame n-type over M (i.e., M is Dedekind complete in M ⟨ᾱ⟩ := dcl(Mᾱ) for some realization ᾱ of p), then p is a definable type (cf. [4], 11 .b).

Given a weakly o-minimal structure ${\cal M}$ and its o-minimal completion $\bar{{\cal M}}$, we first associate to $\bar{{\cal M}}$ a canonical language and then prove that Th$\left( {\cal M} \right)$ determines $Th\left( {\bar{{\cal M}}} \right)$. We then investigate the theory of the pair $\left( {\bar{{\cal M}},{\cal M}} \right)$ in the spirit of the theory of dense pairs of o-minimal structures, and prove, among other results, that it is near model complete, and every definable open subset of ${\bar{M}^n}$ is already definable in $\bar{{\cal M}}$.

We give an example of a weakly o-minimal structure interpreting $\bar{{\cal M}}$ and show that it is not elementarily equivalent to any reduct of an o-minimal trace.

Given a cardinal κ and a sequence ${\left( {{\alpha _i}} \right)_{i \in \kappa }}$ of ordinals, we determine the least ordinal β (when one exists) such that the topological partition relation

$$\beta \to \left( {top\,{\alpha _i}} \right)_{i \in \kappa }^1$$

In this paper a theorem will be established which states that a particular axiom of choice is equivalent to complete distributivity of union and intersection. The theorem will be formulated and proved in the system of logic of [4]. In addition to definitions of [4], the following will be used.

In terms of these definitions, the theorem can be formulated as follows.

The dual of this statement, obtained by interchanging I and U, is also a theorem and has a similar proof.