Increasing the second uniform indiscernible by strongly ssp forcing

We introduce a new and natural stationary set preserving forcing $\mathbb P^{c-c}({\lambda},{\mu})$ that (under ${\rm NS}$ precipitous + existence of H_{\theta}^# for a sufficiently large regular ${\theta}$) increases the second uniform indiscernible $u_2$ beyond some given ordinal ${\lambda}$. The forcing $\mathbb P^{c-c}$ shares this property with forcings defined in [9] and [2]. As a main tool we use certain natural open two player games which are of independent interest, viz. the capturing games $G^{cap}(X)$ and the catching-capturing games $G^{c-c}(X)$. In particular, these games are used to isolate a special family of countable elementary submodels $M \prec H_{\theta}$ that occur as side conditions in $\mathbb P^{c-c}$ and thus allow to control the forcing in a strong way

Incompatible bounded category forcing axioms

We introduce bounded category forcing axioms for well-behaved classes $\Gamma$. These are strong forms of bounded forcing axioms which completely decide the theory of some initial segment of the universe $H_{\lambda_\Gamma^+}$ modulo forcing in $\Gamma$, for some cardinal $\lambda_\Gamma$ naturally associated to $\Gamma$. These axioms naturally extend projective absoluteness for arbitrary set-forcing--in this situation $\lambda_\Gamma=\omega$--to classes $\Gamma$ with $\lambda_\Gamma>\omega$. Unlike projective absoluteness, these higher bounded category forcing axioms do not follow from large cardinal axioms, but can be forced under mild large cardinal assumptions on $V$. We also show the existence of many classes $\Gamma$ with $\lambda_\Gamma=\omega_1$, and giving rise to pairwise incompatible theories for $H_{\omega_2}$.Comment: arXiv admin note: substantial text overlap with arXiv:1805.0873

Set theory and the analyst

This survey is motivated by specific questions arising in the similarities and contrasts between (Baire) category and (Lebesgue) measure - category-measure duality and non-duality, as it were. The bulk of the text is devoted to a summary, intended for the working analyst, of the extensive background in set theory and logic needed to discuss such matters: to quote from the Preface of Kelley [Kel]: "what every young analyst should know"