We consider combinatorial principles based on playing several two person games simultaneously. We call strategies for playing two or more games simultaneously parallel. The principles are easy consequences of the determinacy of games, in particular they are true for all finite games. We shall show that the principles fail for infinite games. The statements of these principles are of lower logical complexity than the sentence expressing the determinacy of games, therefore, they can be studied in weak axiomatic systems for arithmetic (Bounded Arithmetic). We pose several open problems about the provability of these statements in Bounded Arithmetic and related computational problems.

A version of Herbrand's theorem tells us that a universal sentence of a first-order language with at least one constant is satisfiable if and only if the conjunction of all its ground instances is. In general the set of such instances is infinite, and arbitrarily large finite subsets may have to be inspected in order to detect inconsistency. Essentially, the reason that every member of such an infinite set may potentially matter, can be traced back to sentences like

(1)

Loosely put, such sentences effectively sabotage any attempt to build a model from below in a finite number of steps, since new members of the Herbrand universe are constantly brought to attention. Since they cause an indefinite expansion of the relevant part of the Herbrand universe, such sentences could quite appropriately be called expanding.

When such sentences are banned, stronger versions of Herbrand's theorem can be stated. Define a clause (disjunction of literals) to be non-expanding if every non-ground term occurring in a positive literal also occurs (possibly as an embedded subterm) in a negative literal of the same clause. Written as a disjunction of literals, the matrix of (1) clearly fails this criterion. Moreover, say that a sentence is non-expanding if it is a universal sentence with a quantifier-free matrix that is a conjunction of non-expanding clauses. Such sentences do in a sense never reach out beyond themselves, and the relevant part of the Herbrand universe is therefore drastically reduced.

In [1] Ludwik Borkowski takes a quantifier symbol ‘Q1’ (e.g., the familiar ‘∀’) to permit definition of another quantifier symbol ‘Q1’ if, where ‘f’ is a singulary predicate variable, there exists a formula A of QC1—a first-order quantificational calculus (without identity and individual constants) having ‘Q1’ as its one primitive quantifier symbol—such that: (1) under the intended interpretations of ‘Q1’ and ‘Q1’ the biconditional (Q1X)f(X) = A is valid, (2) no individual variable occurs free in A, and (3) A contains no propositional variable, and no predicate variable other than ‘f.’

In [7] it is shown that if Σ is a type omitted in the structure = ω, +, ·, < and complete with respect to Th() then Σ is omitted in models of Th() of all infinite powers. The proof given there extends readily to other models of P. In this paper the result is extended to models of ZFC. For pre-tidy models of ZFC, the proof is a straightforward combination of the methods in [7] and in Keisler and Morley ([9], [6]). For other models, the proof involves forcing. In particular, it uses Solovay and Cohen's original forcing proof that GB is a conservative extension of ZFC (see [2, p. 105] and [5, p. 77]).

The method of proof used for pre-tidy models of set theory can be used to obtain an alternate proof of the result for This new proof yields more information. First of all, a condition is obtained which resembles the hypothesis of the “Omitting Types” theorem, and which is sufficient for a theory T to have a model omitting a type Σ and containing an infinite set of indiscernibles. The proof that this condition is sufficient is essentially contained in Morley's proof [9] that the Hanf number for omitting types is so the condition will be called Morley's condition.

If T is a pre-tidy theory, Morley's condition guarantees that T will have models omitting Σ in all infinite powers.

Let a be an admissible ordinal and let ∧ ≤ α be a limit ordinal. A sequence of a-r.e. degrees is said to be ascending, simultaneous and of length ∧ if (i) there is an α-recursive function t: α × ∧ → α such that, for all ϒ < ∧, A ϒ = {t(σ, ϒ)∣ σ < α} is of degree a ϒ; (ii) if ϒ < ⊤ < ∧, then a ϒ ≤α a τ and (iii) for all ϒ < ∧, there is a ⊤ > ϒ with a ϒ, >α a ϒ. Lerman [4] showed that such an exists for every ∧ ≤ α. An upper bound a of is an α-r.e. degree in which every element of is α-recursive. a is minimal if there is no α-r.e. degree b <α a which is also an upper bound of . Sacks [6] proved that every ascending sequence of simultaneously ω-r.e. degrees of length ω cannot have 0ω′, the complete ω-r.e. degree, as a minimal upper bound. In contrast, Cooper [2] showed that there exists an ascending sequence of simultaneously ω-r.e. degrees of length to having a minimal upper bound which is an ω-r.e. degree. In this paper we investigate the behavior of ascending sequences of simultaneously α-r.e. degrees for admissible ordinals α > ω. Call α Σ∞-admissibIe if it is Σ n -nadmissible for all n. Let Φ(∧) say: No ascending sequence of simultaneously α-r.e. degrees of length ∧ can have 0α′, the complete α-r.e. degree, as a minimal upper bound. Our main result in this paper is:

Let α be either a constructible cardinal with σ2ci(α) < α or Σ∞-admissible. Then σ2cf(α) is the least ordinal ν for which every ∧ ≤ α of cofinality ν (over L α) can satisfy Φ(∧).

In [3] we have associated to a structure an ordinal which gives us information about elementary substructures of the structure. For example a structure whose ascending chain number (as we call the ordinal) is ω could be called Noetherian since all ascending elementary chains inside it are finite (and there are arbitrarily large finite chains). Theorem 2 shows that such structures exist. In fact we prove that for any α < ω1 there is a structure whose ascending chain number is α. The construction is based on the existence of a certain group of permutations of ω (see Theorem 1). The second part of this paper deals with the relevance of the chain number to the study of Jonsson algebras.

The universe of sets, V, is usually seen as an entity structured in successive levels, each level being made up of objects and collections of objects belonging to the previous levels. This process of obtaining sets and axioms for set theory can be seen in Scott [74] and Shoenfield [77].

The approach we want to take differs from the previous one very strongly: the seeds from which we want to generate our universe of classes are to be the one-variable predicates (given by one-free-variable formulas) of the formal language we shall be using. In other words, any one-variable predicate of the language is to be represented as a class in our universe. In this sense, we view our theory as being about a self-referential language, a language whose predicates refer to objects which are predicates of the language itself.

We want, in short, a system such that: (i) any predicate may be represented by an object to be studied by the theory itself; (ii) the axioms for the theory may be derived from the general principle that we are dealing with a language that aims at describing its own predicates; and (iii) the theory should be strong enough to derive ZFC and suggest answers to the existence of large cardinals and to the continuum hypothesis.

An objection to such a project arises immediately: in view of the Russell-Zermelo paradox, how is it possible to have all predicates of the language as elements of the universe? This objection will be easy to deal with: we shall provide our language with a type structure to avoid paradox.