Narrow Results

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 $+$$I_{0,\lambda }$$+$ “$\lambda$ is an even ordinal”, that there is a proper class transitive inner model $M$ containing $V_{\lambda +1}$ and satisfying ZF $+$$I_{0,\lambda }$$+$ “there is an elementary embedding $k:V_{\lambda +2}\to V_{\lambda +2}$”; in fact we will have $k$ ⊆$j$, where $j$ witnesses $I_{0,\lambda }$ in $M$. This result was first proved by the author under the added assumption that $V_{\lambda +1}^{\#}$ exists; Gabe Goldberg noticed that this extra assumption was unnecessary. If also $\lambda$ is a limit ordinal and $\lambda$-DC holds in $V$, then the model $M$ will also satisfy $\lambda$-DC.

We show that ZFC $+$ “$\lambda$ is even” $+$$I_{0,\lambda }$ implies $A^{\#}$ exists for every $A\in V_{\lambda +1}$, but if consistent, this theory does not imply $V_{\lambda +1}^{\#}$ exists.

A transitive model $M$ of ZFC is called a ground if the universe $V$ is a set forcing extension of $M$. We show that the grounds of$V$ are downward set-directed. Consequently, we establish some fundamental theorems on the forcing method and the set-theoretic geology. For instance, (1) the mantle, the intersection of all grounds, must be a model of ZFC. (2) $V$ has only set many grounds if and only if the mantle is a ground. We also show that if the universe has some very large cardinal, then the mantle must be a ground.

We establish natural criteria under which normally iterable premice are iterable for stacks of normal trees. Let $\mathrm{\Omega }$ be a regular uncountable cardinal. Let $m<\omega$ and $M$ be an $m$-sound premouse and $\mathrm{\Sigma }$ be an $(m,\mathrm{\Omega }+1)$-iteration strategy for $M$ (roughly, a normal $(\mathrm{\Omega }+1)$-strategy). We define a natural condensation property for iteration strategies, inflation condensation. We show that if $\mathrm{\Sigma }$ has inflation condensation then $M$ is ${(m,\mathrm{\Omega },\mathrm{\Omega }+1)}^{\ast }$-iterable (roughly, $M$ is iterable for length $\le \mathrm{\Omega }$ stacks of normal trees each of length $<\mathrm{\Omega }$), and moreover, we define a specific such strategy ${\mathrm{\Sigma }}^{\mathrm{\text{st}}}$ and a reduction of stacks via ${\mathrm{\Sigma }}^{\mathrm{\text{st}}}$ to normal trees via $\mathrm{\Sigma }$. If $\mathrm{\Sigma }$ has the Dodd-Jensen property and $\mathrm{\text{card}}(M)<\mathrm{\Omega }$ then $\mathrm{\Sigma }$ has inflation condensation. We also apply some of the techniques developed to prove that if $\mathrm{\Sigma }$ has strong hull condensation (introduced independently by John Steel), and $G$ is $V$-generic for an $\mathrm{\Omega }$-cc forcing, then $\mathrm{\Sigma }$ extends to an $(m,\mathrm{\Omega }+1)$-strategy ${\mathrm{\Sigma }}^{+}$ for $M$ with strong hull condensation, in the sense of $V[G]$. Moreover, this extension is unique. We deduce that if $G$ is $V$-generic for a ccc forcing then $V$ and $V[G]$ have the same $\omega$-sound, $(\omega ,\mathrm{\Omega }+1)$-iterable premice which project to $\omega$.

If we replace first-order logic by second-order logic in the original definition of Gödel’s inner model $L$, we obtain the inner model of hereditarily ordinal definable (HOD) sets [33]. In this paper, we consider inner models that arise if we replace first-order logic by a logic that has some, but not all, of the strength of second-order logic. Typical examples are the extensions of first-order logic by generalized quantifiers, such as the Magidor–Malitz quantifier [24], the cofinality quantifier [35], or stationary logic [6]. Our first set of results show that both $L$ and HOD manifest some amount of formalism freeness in the sense that they are not very sensitive to the choice of the underlying logic. Our second set of results shows that the cofinality quantifier gives rise to a new robust inner model between $L$ and HOD. We show, among other things, that assuming a proper class of Woodin cardinals the regular cardinals $>{\aleph }_1$ of $V$ are weakly compact in the inner model arising from the cofinality quantifier and the theory of that model is (set) forcing absolute and independent of the cofinality in question. We do not know whether this model satisfies the Continuum Hypothesis, assuming large cardinals, but we can show, assuming three Woodin cardinals and a measurable above them, that if the construction is relativized to a real, then on a cone of reals, the Continuum Hypothesis is true in the relativized model.

We survey two of the many aspects of the standard braid order, namely its set theoretical roots, and the known connections with knot theory, including results by Netsvetaev, Malyutin, and Ito, and very recent work in progress by Fromentin and Gebhardt.

This is a survey paper on some recent developments in the study of higher degree theory, the theory of degree structure of generalized degrees at uncountable cardinals, in particular at those with countable cofinality.

The study of generalized degree notions, known as α-recursion theory, was initiated by Sacks in the early 70s. But the early works were mainly from recursion theoretical perspective, and the results are mostly limited to Gödel's constructive universe. Some recent works (see Refs. 1,2) in set theory indicate that there is a profound connection between the complexity of degree structures at singular cardinals of countable cofinality and the large cardinal properties at the vicinity. This paper shall exam degree structures cross through L-like models for various mild large cardinals, or under large cardinal assumptions, by which hopefully a new line of research will be depicted.