Journal of Mathematical Logic最新文献

Let $b\ge 2$ be an integer. We show that the set of real numbers that are Poisson generic in base $b$ is ${\mathrm{\text{Π}}}_3^0$-complete in the Borel hierarchy of subsets of the real line. Furthermore, the set of real numbers that are Borel normal in base $b$ and not Poisson generic in base $b$ is complete for the class given by the differences between ${\mathrm{\text{Π}}}_3^0$ sets. We also show that the effective versions of these results hold in the effective Borel hierarchy.

We address the question of consistency strength of certain filters and ultrafilters which fail to satisfy the Galvin property. We answer questions [Benhamou and Gitik, Ann. Pure Appl. Logic173 (2022) 103107; Questions 7.8, 7.9], [Benhamou et al., J. Lond. Math. Soc.108(1) (2023) 190–237; Question 5] and improve theorem [Benhamou et al., J. Lond. Math. Soc.108(1) (2023) 190–237; Theorem 2.3].

We present numerous natural algebraic examples without the so-called Canonical Base Property (CBP). We prove that every commutative unitary ring of finite Morley rank without finite-index proper ideals satisfies the CBP if and only if it is a field, a ring of positive characteristic or a finite direct product of these. In addition, we construct a CM-trivial commutative local ring with a finite residue field without the CBP. Furthermore, we also show that finite-dimensional non-associative algebras over an algebraically closed field of characteristic $0$ give rise to triangular rings without the CBP. This also applies to Baudisch’s $2$-step nilpotent Lie algebras, which yields the existence of a $2$-step nilpotent group of finite Morley rank whose theory, in the pure language of groups, is CM-trivial and does not satisfy the CBP.

According to a theorem due to Kenneth Kunen, under ZFC, there is no ordinal $\lambda$ and nontrivial elementary embedding $j:V_{\lambda +2}\to V_{\lambda +2}$. His proof relied on the Axiom of Choice (AC), and no proof from ZF alone is has been discovered.

$I_{0,\lambda }$ is the assertion, introduced by Hugh Woodin, that $\lambda$ is an ordinal and there is an elementary embedding $j:L(V_{\lambda +1})\to L(V_{\lambda +1})$ with critical point $<\lambda$. And $I_0$ asserts that $I_{0,\lambda }$ holds for some $\lambda$. The axiom $I_0$ is one of the strongest large cardinals not known to be inconsistent with AC. It is usually studied assuming ZFC in the full universe $V$ (in which case $\lambda$ must be a limit ordinal), but we assume only ZF.

We prove, assuming ZF $+$$I_{0,\lambda }$<

We identify a particular mouse, $M^{\mathrm{\text{ld}}}$, the minimal ladder mouse, that sits in the mouse order just past $M_n^{♯}$ for all $n$, and we show that $ℝ\cap M^{\mathrm{\text{ld}}}=Q_{\omega +1}$, the set of reals that are ${\mathrm{\Delta }}_{\omega +1}^1$ in a countable ordinal. Thus $Q_{\omega +1}$ is a mouse set.

This is analogous to the fact that $ℝ\cap M_1^{♯}=Q_3$ where $M_1^{♯}$ is the sharp for the minimal inner model with a Woodin cardinal, and $Q_3$ is the set of reals that are ${\mathrm{\Delta }}_3^1$ in a countable ordinal.

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