The consistency of a club-guessing failure at the successor of a regular cardinal

I answer a question of Shelah by showing that if \k is a regular cardinal such that 2^{{<}\k}=\k, then there is a {<}\k--closed partial order preserving cofinalities and forcing that for every club--sequence \la C_\d\mid \d\in \k^+\cap\cf(\k)\ra with \ot(C_\d)=\k for all \d there is a club D\sub\k^+ such that \{\a<\k\mid \{C_\d(\a+1), C_\d(\a+2)\}\sub D\} is bounded for every \d. This forcing is built as an iteration with {<}\k--supports and with symmetric systems of submodels as side conditions

A forcing notion collapsing \aleph_3 and preserving all other cardinals

I construct, in ZFC, a forcing notion that collapses \aleph_3 and preserves all other cardinals. The existence of such a forcing answers a question of Uri Abraham from 1983

Few new reals

We introduce a new method for building models of CH, together with $\Pi_2$ statements over $H(\omega_2)$, by forcing over a model of CH. Unlike similar constructions in the literature, our construction adds new reals, but only $\aleph_1$-many of them. Using this approach, we prove that a very strong form of the negation of Club Guessing at $\omega_1$ known as Measuring is consistent together with CH, thereby answering a well-known question of Moore. The construction works over any model of ZFC + CH and can be described as a finite support forcing construction with finite systems of countable models with markers as side conditions and with strong symmetry constraints on both side conditions and working parts

Forcing lightface definable well-orders without the CGH

For any given uncountable cardinal $\kappa$ with $\kappa^{{<}\kappa}=\kappa$, we present a forcing that is $<\kappa$-directed closed, has the $\kappa^+$-c.c. and introduces a lightface definable well-order of $H(\kappa^+)$. We use this to define a global iteration that does this for all such $\kappa$ simultaneously and is capable of preserving the existence of many large cardinals in the universe

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