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.
We show that the notions of "strongly unfoldable cardinals", introduced by Villaveces in his model-theoretic studies of models of set theory, and "shrewd cardinals", introduced by Rathjen in a proof-theoretic context, coincide. We then proceed by using ideas from the proof of this equivalence to establish the existence of "ordinal anticipating Laver functions" for strong unfoldability. With the help of these functions, we show that the principle $\Diamond_κ(\mathrm{Reg})$ holds at every strongly unfoldable cardinal $κ$ with the property that there exists a subset $z$ of $κ$ such that every sub…
Since being isolated by Viale and Weiss in 2009, the Guessing Model Property has emerged as a particularly prominent and powerful consequence of the Proper Forcing Axiom. In this paper, we investigate connections between variations of the Guessing Model Property and cardinal arithmetic, broadly construed. We improve upon results of Viale and Krueger by proving that a weakening of the Guessing Model Property implies Shelah's Strong Hypothesis. We also prove that, though the Guessing Model Property is known not to put an upper bound on the size of the continuum, it does imply that $2^{ω_1}$ is a…
Let $\mathsf{MM}^{++}(κ)$ state that the forcing axiom $\mathsf{MM}^{++}$ can be instantiated only for stationary set preserving posets of size at most $κ$. We give a detailed account of Asperò and Schindler's proof that $\mathsf{MM}^{++}(κ)+$there are class many Woodin cardinals implies Woodin's axiom $(*)$ if $\Diamond_κ$ holds and $κ>\aleph_2$. Our presentation takes advantage of the notion of consistency property: specifically we rephrase Asperò and Schindler's forcing as a specific instantiation of the notion of ``consistency property'' used by Makkai, Keisler, Mansfield and others in the…
We prove two general results about the preservation of extendible and $C^{(n)}$-extendible cardinals under a wide class of forcing iterations (Theorems 5.4 and 7.5). As applications we give new proofs of the preservation of Vopěnka's Principle and $C^{(n)}$-extendible cardinals under Jensen's iteration for forcing the GCH, previously obtained by Brooke-Taylor and Tsaprounis, res\-pectively. We prove that $C^{(n)}$-extendible cardinals are preserved by forcing with standard Easton-support iterations for any possible $Δ_2$-definable behaviour of the power-set function on regular cardinals. We sh…
