8 research outputs found
Laver and set theory
In this commemorative article, the work of Richard Laver is surveyed in its full range and extent.Accepted manuscrip
Joint Laver diamonds and grounded forcing axioms
I explore two separate topics: the concept of jointness for set-theoretic
guessing principles, and the notion of grounded forcing axioms. A family of
guessing sequences is said to be joint if all of its members can guess any
given family of targets independently and simultaneously. I primarily
investigate jointness in the case of various kinds of Laver diamonds. In the
case of measurable cardinals I show that, while the assertions that there are
joint families of Laver diamonds of a given length get strictly stronger with
increasing length, they are all equiconsistent. This is contrasted with the
case of partially strong cardinals, where we can derive additional consistency
strength, and ordinary diamond sequences, where large joint families exist
whenever even one diamond sequence does. Grounded forcing axioms modify the
usual forcing axioms by restricting the posets considered to a suitable ground
model. I focus on the grounded Martin's axiom which states that Martin's axioms
holds for posets coming from some ccc ground model. I examine the new axiom's
effects on the cardinal characteristics of the continuum and show that it is
quite a bit more robust under mild forcing than Martin's axiom itself.Comment: This is my PhD dissertatio
The power function
The axioms of ZFC provide very little information about the possible values of the power function (i.e. the map K---->2ᴷ). In this dissertation, we examine various theorems concerning the behaviour of the power function inside the formal system ZFC , and we :;hall be p:trticul:trly interested in results which provide eonstraints on the possible values of the power function. Thus most of the results presented here will be consistency results. A theorem of Easton (Theorem 2.3.1) shows that, when restricted to regular cardinals, the power function may take on any reasonable value, and thus a considerable part of this thesis is concerned with the power function on singular cardinals. We also examine the influence of various strong axioms of infinity, and their generalization to smaller cardinals, on the possible behaviour of the power function
The power function
The axioms of ZFC provide very little information about the possible values of the power function (i.e. the map K---->2ᴷ). In this dissertation, we examine various theorems concerning the behaviour of the power function inside the formal system ZFC , and we :;hall be p:trticul:trly interested in results which provide eonstraints on the possible values of the power function. Thus most of the results presented here will be consistency results. A theorem of Easton (Theorem 2.3.1) shows that, when restricted to regular cardinals, the power function may take on any reasonable value, and thus a considerable part of this thesis is concerned with the power function on singular cardinals. We also examine the influence of various strong axioms of infinity, and their generalization to smaller cardinals, on the possible behaviour of the power function