相关论文: On the Consistency Strength of MM($\omega_1$)
The axioms of ZFC provide a foundation for mathematics, however, there are statements independent of ZFC, such as the Continuum Hypothesis (CH). We discuss Martin's axiom, which is an alternative to CH that roughly states that if there is a…
It is shown that the consistency strength of ZF + DC + "the closed unbounded ultrafilter on omega_1 is an ultrafilter" is exactly ZFC + one measurable cardinal.
It is shown that if BMM (= Bounded Martin's Maximum) holds then each set is contained in an inner model with a strong cardinal. This answers a question that has been asked by various people. It follows that BMM has a much larger consistency…
We investigate fragments of generic absoluteness principles known as Maximality Principles. We determine the consistency strength of $\Sigma_n$-$\mathsf{MP}(\mathbb R)$ and $\Pi_n$-$\mathsf{MP}(\mathbb R)$, the boldface Maximality Principle…
I show that it is consistent relative to the consistency of a Mahlo cardinal that Martin's axiom holds at $\omega_2$, but the weak Kurepa Hypothesis fails. This answers a question posed by Honzik, Lambie-Hanson and Stejskalov\'a. The…
In \cite{MV} we defined and proved the consistency of the principle ${\rm GM}^+(\omega_3,\omega_1)$ which implies that many consequences of strong forcing axioms hold simultaneously at $\omega_2$ and $\omega_3$. In this paper we formulate a…
We prove the consistency of the statement $\mathfrak{u}_{\aleph_\omega}<2^{\aleph_\omega}$. We show that the consistency strength of this statement is exactly a measurable cardinal $\mu$ so that $o(\mu)=\mu^{++}$.
We prove the consistency result from the title. By forcing we construct a model of g=aleph_1, b=cf(Sym(omega))=aleph_2.
In the absence of the Axiom of Choice, the "small" cardinal $\omega_1$ can exhibit properties more usually associated with large cardinals, such as strong compactness and supercompactness. For a local version of strong compactness, we say…
We study a strengthening of Bounded Martin's Maximum which asserts that if a \Sigma_1 fact holds of \omega_2^V in a stationary set preserving extension then it holds in V for a stationary set of ordinals less than \omega_2. We show that…
Combining stationary reflection (a compactness property) with the failure of SCH (an instance of non-compactness) has been a long-standing theme. We obtain this at $\aleph_{\omega_1}$, answering a question of Ben-Neria, Hayut, and Unger: We…
We prove from the existence of a Mahlo cardinal the consistency of the statement that $2^\omega = \omega_3$ holds and every stationary subset of $\omega_2 \cap \mathrm{cof}(\omega)$ reflects to an ordinal less than $\omega_2$ with…
The Maximality Principle MP is a scheme which states that if a sentence of the language of ZFC is true in some forcing extension V^P, and remains true in any further forcing extension of V^P, then it is true in all forcing extensions of V.…
We present several results relating the general theory of the stationary tower forcing developed by Woodin with forcing axioms. The main results is that the forcing axiom MM^{++} (also known as MM^{+\omega_1}) decides the \Pi_2-theory of…
We show that the consistency strength of the system NFUB, a variant of Quine's "New Foundations" recently introduced by Randall Holmes, is precisely that of [ZFC - Power Set] + "There is a weakly compact cardinal''. This is a preliminary…
We isolate here a wide class of well founded orders called tame orders and show that each such order of cardinality at most $\kappa$ can be realized as the Mitchell order on a measurable cardinal $\kappa$, from a consistency assumption…
In this paper we prove the equiconsistency of ``Every omega_1 tree which is first order definable over H_{omega_1} has a cofinal branch'' with the existence of a Pi^1_1 reflecting cardinal. The proof uses a definable version of Ramsey…
We introduce combinatorial principles that characterize strong compactness and supercompactness for inaccessible cardinals but also make sense for successor cardinals. Their consistency is established from what is supposedly optimal.…
We define a countable antichain condition (ccc) property for partial orderings, weaker than precalibre-$\aleph_1$, and show that Martin's axiom restricted to the class of partial orderings that have the property does not imply Martin's…
We study various classes of maximality principles, $\rm{MP}(\kappa,\Gamma)$, introduced by J.D. Hamkins, where $\Gamma$ defines a class of forcing posets and $\kappa$ is a cardinal. We explore the consistency strength and the relationship…