Cardinals as Ultrapowers : A Canonical Measure Analysis under the Axiom of Determinacy

This thesis is in the field of Descriptive Set Theory and examines some consequences of the Axiom of Determinacy concerning partition properties that define large cardinals. The Axiom of Determinacy (AD) is a game-theoretic statement expressing that all infinite two-player perfect information games with a countable set of possible moves are determined, i.e., admit a winning strategy for one of the players. By the term "measure analysis'' we understand the following procedure: given a strong partition cardinal κ and some cardinal λ > κ, we assign a measure µ on κ to λ such that the ultrapower with respect to µ equals λ . A canonical measure analysis is a measure assignment for cardinals larger than a strong partition cardinal κ and a binary operation on the measures of this assignment that corresponds to ordinal addition on indices of the cardinals. This thesis provides a canonical measure analysis up to the (ω^ω)th cardinal after an odd projective cardinal. Using this canonical measure analysis we show that all cardinals that are ultrapowers with respect to basic order measures are Jónsson cardinals. With the canonicity results of this thesis we can state that, if κ is an odd projective ordinal, κ^(n), κ^(ωn+1), and κ^(ω^n+1), for n<ω, are Jónsson under AD

Narrow coverings of omega-product spaces

Results of Sierpinski and others have shown that certain finite-dimensional product sets can be written as unions of subsets, each of which is "narrow" in a corresponding direction; that is, each line in that direction intersects the subset in a small set. For example, if the set (omega \times omega) is partitioned into two pieces along the diagonal, then one piece meets every horizontal line in a finite set, and the other piece meets each vertical line in a finite set. Such partitions or coverings can exist only when the sets forming the product are of limited size. This paper considers such coverings for products of infinitely many sets (usually a product of omega copies of the same cardinal kappa). In this case, a covering of the product by narrow sets, one for each coordinate direction, will exist no matter how large the factor sets are. But if one restricts the sets used in the covering (for instance, requiring them to be Borel in a product topology), then the existence of narrow coverings is related to a number of large cardinal properties: partition cardinals, the free subset problem, nonregular ultrafilters, and so on. One result given here is a relative consistency proof for a hypothesis used by S. Mrowka to construct a counterexample in the dimension theory of metric spaces

Laver and set theory

In this commemorative article, the work of Richard Laver is surveyed in its full range and extent.Accepted manuscrip

A strong polarized relation

We prove the consistency of a strong polarized relation for a cardinal and its successor, using pcf and forcingComment: 14 page

Chains, Antichains, and Complements in Infinite Partition Lattices

We consider the partition lattice $\Pi_\kappa$ on any set of transfinite cardinality $\kappa$ and properties of $\Pi_\kappa$ whose analogues do not hold for finite cardinalities. Assuming the Axiom of Choice we prove: (I) the cardinality of any maximal well-ordered chain is always exactly $\kappa$; (II) there are maximal chains in $\Pi_\kappa$ of cardinality $> \kappa$; (III) if, for every cardinal $\lambda < \kappa$, we have $2^{\lambda} < 2^\kappa$, there exists a maximal chain of cardinality $< 2^{\kappa}$ (but $\ge \kappa$) in $\Pi_{2^\kappa}$; (IV) every non-trivial maximal antichain in $\Pi_\kappa$ has cardinality between $\kappa$ and $2^{\kappa}$, and these bounds are realized. Moreover we can construct maximal antichains of cardinality $\max(\kappa, 2^{\lambda})$ for any $\lambda \le \kappa$; (V) all cardinals of the form $\kappa^\lambda$ with $0 \le \lambda \le \kappa$ occur as the number of complements to some partition $\mathcal{P} \in \Pi_\kappa$, and only these cardinalities appear. Moreover, we give a direct formula for the number of complements to a given partition; (VI) Under the Generalized Continuum Hypothesis, the cardinalities of maximal chains, maximal antichains, and numbers of complements are fully determined, and we provide a complete characterization.Comment: 24 pages, 2 figures. Submitted to Algebra Universalis on 27/11/201