We provide a model theoretical and tree property-like characterization of $\lambda $ - $\Pi ^1_1$ -subcompactness and supercompactness. We explore the behavior of these combinatorial principles at accessible cardinals.

Computable analysis provides ways of representing points in a topological space, and therefore of defining a notion of computable points of the space. In this article, we investigate when two topologies on the same space induce different sets of computable points. We first study a purely topological version of the problem, which is to understand when two topologies are not $\sigma $-homeomorphic. We obtain a characterization leading to an effective version, and we prove that two topologies satisfying this condition induce different sets of computable points. Along the way, we propose an effective version of the Baire category theorem which captures the construction technique, and enables one to build points satisfying properties that are co-meager with respect to a topology, and are computable with respect to another topology. Finally, we generalize the result to three topologies and give an application to prove that certain sets do not have computable type, which means that they have a homeomorphic copy that is semicomputable but not computable.

This paper looks at the concept of neighborhood canonicity introduced by Brian Chellas [2]. We follow the lead of the author's paper [9] where it was shown that every non-iterative logic is neighborhood canonical and here we will show that all logics whose axioms have a simple syntactic form—no intensional operator is in boolean combination with a propositional letter—and which have the finite model property are neighborhood canonical. One consequence of this is that KMcK, the McKinsey logic, is neighborhood canonical, an interesting counterpoint to the results of Robert Goldblatt and Xiaoping Wang who showed, respectively, that KMcK is not relational canonical [5] and that KMcK is not relationally strongly complete [11].

In Part I of this paper no official significance was given to free occurrences of restricted-variables. Indeed doubt has been expressed as to the desirability or feasibility of such usage. Thus, to quote from Rosser [4], p. 146:

“We now raise the question of the significance of F (α), in which the occurrences of α are free [α a variable subject to the restriction K(α)]. In deciding to take (x).K(x) ⊃ F(x) and (Ex).K(x).F(x) as the meanings of (α)F(α) and (Eα)F(α), we were guided by the intuitive meanings. In the case of F(α), the intuitive meaning does not furnish a satisfactory guide. In everyday mathematics, if it has been agreed that α stands for a quantity satisfying the restriction K(α), it is commonly the case that, if one is assuming F (α), then K(α)&F(α) is understood, but if one is trying to prove F (α), then K(α) ⊃ F (α) is understood. It seems that in symbolic logic perhaps it is best not to give any especial significance to α in F(α) when it occurs free.”

Despite this anomalous behaviour we shall, in this Part, show how one can have unhampered use of free restricted-variables in appropriate contexts.

We present a characterization of weakly compact cardinals in terms of generalized stationarity. We apply this characterization to construct a model with no partial square sequences.

A hundred years ago, Frege proposed a logical definition of the natural numbers based on the following idea:

He replaced this circular definition by the following one:

He tried afterwards to found his theory over a notion of class satisfying a general comprehension principle:

Russell quickly derived a contradiction from this principle (the famous Russell's paradox) but saved Frege's arithmetic with his theory of types based on the following comprehension principle:

In 1979, talking at the Claude Bernard University in Lyon, I remarked that 3 types suffice to provide Frege's arithmetic, showing in fact that PA2 (second order Peano arithmetic) holds in TT3 + AI (theory of types 0, 1, 2 plus a suitable axiom of infinity). I asked whether TT3 + AI was a conservative extension of PA2. Pabion [3] gave a positive answer by a subtle use of the Fraenkel-Moskowski method. This result will be improved in the present paper, with a view to getting models of NF3 + AI in which Frege's arithmetic forms a model isomorphic to a given countable model of PA2.

We exhibit a Horn sentence expressing the statement of the title; the construction generalizes to arbitrary primes in place of five.

For the reader's convenience, we begin Part II by restating the rules for constructing Sq derivations given in §2:

We let R be the set of 14 rules for Sq, R* the set of pure *-rules in R, R≃ the set of pure ≃-rules in R, and R− the set R ∩ − {Exs≃}. The closure of any F ⊆ Sq under R will be denoted by csR F, and the direct image of F under Cut*, for example, by Cut*(F).

In this paper it is shown that the global statement that the dominating number for κ is less than 2κ for all regular κ, is internally consistent, given the existence of 0#. The possible range of values for the dominating number for κ and 2κ which may be simultaneously true in an inner model is also explored.

We introduce the class PIO of functions computable in time that is polynomial in max {the length of input, the length of output}, observe that there is no notation system for total PIO functions but there are notation systems for partial PIO functions, and give an algebra of partial PIO functions from binary strings to binary strings.

We consider an infinite two-person game. The second player has a winning perfect information strategy; we show that this player has a winning strategy which depends on substantially less information. The game studied here is a variation on a game of Gale.

A definition of nonstandard universe which gets over the limitation to the finite levels of the cumulative hierarchy is proposed. Though necessarily nonwellfounded, nonstandard universes are arranged in strata in the likeness of superstructures and allow a rank function taking linearly ordered values. Nonstandard universes are also constructed which model the whole ZFC theory without regularity and satisfy the κ-saturation property.

M. Beiglböck, V. Bergelson, and A. Fish proved that if G is a countable amenable group and A and B are subsets of G with positive Banach density, then the product set AB is piecewise syndetic. This means that there is a finite subset E of G such that EAB is thick, that is, EAB contains translates of any finite subset of G. When G = ℤ, this was first proven by R. Jin. We prove a quantitative version of the aforementioned result by providing a lower bound on the density (with respect to a Følner sequence) of the set of witnesses to the thickness of EAB. When G = ℤd , this result was first proven by the current set of authors using completely different techniques.

There is a proper countable support iteration of length ω adding no new reals at finite stages and adding a Sacks real in the limit.

We present a solution to the problem of denning a counterpart in Algebraic Set Theory of the construction of internal sheaves in Topos Theory. Our approach is general in that we consider sheaves as determined by Lawvere-Tierney coverages, rather than by Grothendieck coverages, and assume only a weakening of the axioms for small maps originally introduced by Joyal and Moerdijk, thus subsuming the existing topos-theoretic results.