THE IMPLICITLY CONSTRUCTIBLE UNIVERSE

The Constructible Universe

Assuming the axiom of constructibility, points in closed discrete subspaces of certain normal spaces can be simultaneously separated. This is a partial result towards the normal Moore space conjecture. The normal Moore space conjecture states that every normal Moore space is metrizable. This is known to be not provable from the usual axioms of set theory, since Silver [4] shows that Martin's ax...

Generalisations of Gödel’s universe of constructible sets

The cumulative hierarchy and the constructible universe of ZFA

We present two results which shed some more light on the deep connection between ZFA and the standard ZF set theory: First of all we refine a result of Forti and Honsell (see [5]) in order to prove that the universe of ZFA can also be obtained (without appealing to choice) as the least fixed point of a continuous operator and not only as the greatest fixed point of the powerset operator. Next w...

Recognizable Sets and Woodin Cardinals: Computation beyond the Constructible Universe

We call a subset of an ordinal λ recognizable if it is the unique subset x of λ for which some Turing machine with ordinal time and tape, which halts for all subsets of λ as input, halts with the final state 0. Equivalently, such a set is the unique subset x which satisfies a given Σ1 formula in L[x]. We prove several results about sets of ordinals recognizable from ordinal parameters by ordina...

Recognizable sets and Woodin cardinals: computation beyond the constructible universe

We call a subset of an ordinal recognizable if it is the unique subset x of for which some Turing machine with ordinal time and tape and an ordinal parameter, that halts for all subsets of as input, halts with the final state 0. Equivalently, such a set is the unique subset x which satisfies a given ⌃1 formula in L[x]. We further define the recognizable closure for subsets of by closing under r...