Let . Consider the class of all Borel with null vertical sections , x ∈ X. We show that if for all such F and all null Z ⊆ X, is null, then for all such F, . The theorem generalizes the fact that every Sierpiński set is strongly meager and was announced in [P].
Assuming the continuum hypothesis, we construct a pure subgroup G of the Baer-Specker group with the following properties. Every endomorphism of G differs from a scalar multiplication by an endomorphism of finite rank. Yet G has uncountably many homomorphisms to ℤ.
For a t-norm T on a bounded lattice , a partial order was recently defined and studied. In [11], it was pointed out that the binary relation is a partial order on , but may not be a lattice in general. In this paper, several sufficient conditions under which is a lattice are given, as an answer to an open problem posed by the authors of [11]. Furthermore, some examples of t-norms on such that is a lattice are presented.
We formulate, within the frame-theory for the foundations of Mathematics outlined in [2], a list of axioms which state that almost all "interesting" collections and almost all "interesting" operations are elements of the universe. The resulting theory would thus have the important foundational feature of being completely self-contained. Unfortunately, the whole list is inconsistent, and we are led to formulate the following problem, which we call the problem of self-reference: "Find out...
We introduce the sum of observables in fuzzy quantum spaces which generalize the Kolmogorov probability space using the ideas of fuzzy set theory.
It is shown that for every Darboux function F there is a non-constant continuous function f such that F + f is still Darboux. It is shown to be consistent - the model used is iterated Sacks forcing - that for every Darboux function F there is a nowhere constant continuous function f such that F + f is still Darboux. This answers questions raised in [5] where it is shown that in various models of set theory there are universally bad Darboux functions, Darboux functions whose sum with any nowhere...
Let κ < λ be regular cardinals. We say that an embedding j: V → M with critical point κ is λ-tall if λ < j(κ) and M is closed under κ-sequences in V. Silver showed that GCH can fail at a measurable cardinal κ, starting with κ being κ⁺⁺-supercompact. Later, Woodin improved this result, starting from the optimal hypothesis of a κ⁺⁺-tall measurable cardinal κ. Now more generally, suppose that κ ≤ λ are regular and one wishes the GCH to fail at λ with κ being λ-supercompact. Silver’s methods show...
We force and construct a model containing supercompact cardinals in which, for any measurable cardinal δ and any ordinal α below the least beth fixed point above δ, if is regular, δ is strongly compact iff δ is δ + α + 1 strong, except possibly if δ is a limit of cardinals γ which are strongly compact. The choice of the least beth fixed point above δ as our bound on α is arbitrary, and other bounds are possible.