Distributive Aronszajn trees

Ben-David and Shelah proved that if $\lambda$ is a singular strong-limit cardinal and $2^\lambda=\lambda^+$, then $\square^*_\lambda$ entails the existence of a normal $\lambda$-distributive $\lambda^+$-Aronszajn tree. Here, it is proved that the same conclusion remains valid after replacing the hypothesis $\square^*_\lambda$ by $\square(\lambda^+,{<}\lambda)$. As $\square(\lambda^+,{<}\lambda)$ does not impose a bound on the order-type of the witnessing clubs, our construction is necessarily different from that of Ben-David and Shelah, and instead uses walks on ordinals augmented with club guessing. A major component of this work is the study of postprocessing functions and their effect on square sequences. A byproduct of this study is the finding that for $\kappa$ regular uncountable, $\square(\kappa)$ entails the existence of a partition of $\kappa$ into $\kappa$ many fat sets. When contrasted with a classic model of Magidor, this shows that it is equiconsistent with the existence of a weakly compact cardinal that $\omega_2$ cannot be split into two fat sets.Comment: 45 pages; improved and generalized some results, and streamlined the presentatio

Change blindness: eradication of gestalt strategies

Arrays of eight, texture-defined rectangles were used as stimuli in a one-shot change blindness (CB) task where there was a 50% chance that one rectangle would change orientation between two successive presentations separated by an interval. CB was eliminated by cueing the target rectangle in the first stimulus, reduced by cueing in the interval and unaffected by cueing in the second presentation. This supports the idea that a representation was formed that persisted through the interval before being 'overwritten' by the second presentation (Landman et al, 2003 Vision Research 43149–164]. Another possibility is that participants used some kind of grouping or Gestalt strategy. To test this we changed the spatial position of the rectangles in the second presentation by shifting them along imaginary spokes (by ±1 degree) emanating from the central fixation point. There was no significant difference seen in performance between this and the standard task [F(1,4)=2.565, p=0.185]. This may suggest two things: (i) Gestalt grouping is not used as a strategy in these tasks, and (ii) it gives further weight to the argument that objects may be stored and retrieved from a pre-attentional store during this task

Guessing axioms, invariance and suslin trees

In this thesis we investigate the properties of a group of axioms known as 'Guessing Axioms,' which extend the standard axiomatisation of set theory, ZFC. In particular, we focus on the axioms called 'diamond' and 'club,' and ask to what extent properties of the former hold of the latter. A question of 1. Juhasz, of whether club implies the existence of a Suslin tree, remains unanswered at the time of writing and motivates a large part of our in- vestigation into diamond and club. We give a positive partial answer to Juhasz's question by defining the principle Superclub and proving that it implies the exis- tence of a Suslin tree, and that it is weaker than diamond and stronger than club (though these implications are not necessarily strict). Conversely, we specify some conditions that a forcing would have to meet if it were to be used to provide a negative answer, or partial answer, to Juhasz's question, and prove several results related to this. We also investigate the extent to which club shares the invariance property of diamond: the property of being formally equivalent to many of its natural strength- enings and weakenings. We show that when certain cardinal arithmetic statements hold, we can always find different variations on club t.hat will be provably equiv- alent. Some of these hold in ZFC. But, in the absence of the required cardinal arithmetic, we develop a general method for proving that most variants of club are pairwise inequivalent in ZFC.EThOS - Electronic Theses Online ServiceGBUnited Kingdo