We consider classical representations of integers: Church's function iterators, cardinal equivalence classes of sets, ordinal equivalence classes of totally ordered sets. Since programs do not work on abstract entities and require formal representations of objects, we effectivize these abstract notions in order to allow them to be computed by programs. To any such effectivized representation is then associated a notion of Kolmogorov complexity. We prove that these Kolmogorov complexities form a strict hierarchy which coincides with that obtained by relativization to jump oracles and/or allowance of infinite computations.
We show that the existence of a Pi^{1}_{N}-indescribable cardinal over the Zermelo-Fraenkel's set theory ZF is proof-theoretically reducible to iterations of Mostowski collapsings and lower Mahlo operations. Furthermore we describe a proof-theoretic bound on definable countable ordinals whose existence is provable from the existence of second order indescribable cardinals over ZF.
This paper explores several topics related to Woodin's HOD conjecture. We improve the large cardinal hypothesis of Woodin's HOD dichotomy theorem from an extendible cardinal to a strongly compact cardinal. We show that assuming there is a strongly compact cardinal and the HOD hypothesis holds, there is no elementary embedding from HOD to HOD, settling a question of Woodin. We show that the HOD hypothesis is equivalent to a uniqueness property of elementary embeddings of levels of the cumulative hierarchy. We prove that the HOD hypothesis holds if and only if every regular cardinal above the fi…
The generic Vopěnka principle, we prove, is relatively consistent with the ordinals being non-Mahlo. Similarly, the generic Vopěnka scheme is relatively consistent with the ordinals being definably non-Mahlo. Indeed, the generic Vopěnka scheme is relatively consistent with the existence of a $Δ_2$-definable class containing no regular cardinals. In such a model, there can be no $Σ_2$-reflecting cardinals and hence also no remarkable cardinals. This latter fact answers negatively a question of Bagaria, Gitman and Schindler.
We investigate the existence of strong colorings on successors of singular cardinals. This work continues Section 2 of [Sh:413] (math.LO/9809199), but now our emphasis is on finding colorings of pairs of ordinals, rather than colorings of finite sets of ordinals.
We will consider a number of new large-cardinal properties, the $α$-tremendous cardinals for each limit ordinal $α>0$, the hyper-tremendous cardinals, the $α$-enormous cardinals for each limit ordinal $α>0$, and the hyper-enormous cardinals. For limit ordinals $α>0$, the $α$-tremendous cardinals and hyper-tremendous cardinals have consistency strength between I3 and I2. An $ω$-enormous cardinal has consistency strength greater than I0, and also all the large-cardinal axioms discussed in the second part of Hugh Woodin's paper on suitable extender models, not known to be inconsistent with ZFC an…
This paper contributes to the theory of large cardinals beyond the Kunen inconsistency, or choiceless large cardinal axioms, in the context where the Axiom of Choice is not assumed. The first part of the paper investigates a periodicity phenomenon: assuming choiceless large cardinal axioms, the properties of the cumulative hierarchy turn out to alternate between even and odd ranks. The second part of the paper explores the structure of ultrafilters under choiceless large cardinal axioms, exploiting the fact that these axioms imply a weak form of the author's Ultrapower Axiom. The third and fin…
We obtain results on the condensation principle called local club condensation. We prove that in extender models an equivalence between the failure of local club condensation and subcompact cardinals holds. This gives a characterization of $\square_κ$ in terms of local club condensation in extender models. Assuming $\gch$, given an interval of ordinals $I$ we verify that iterating the forcing defined by Holy-Welch-Wu, we can preserve $\gch$, cardinals and cofinalities and obtain a model where local club condensation holds for every ordinal in $I$ modulo those ordinals which cardinality is a si…
It is shown that the existence of a measurable cardinal is equiconsistent to a model of ZFC in which there is no ordinal-definable, stationary, costationary subset of $ω_1$
Under large cardinal hypotheses beyond the Kunen inconsistency -- hypotheses so strong as to contradict the Axiom of Choice -- we solve several variants of the generalized continuum problem and identify structural features of the levels $V_α$ of the cumulative hierarchy of sets that are eventually periodic, alternating according to the parity of the ordinal $α$. For example, if there is an elementary embedding from the universe of sets to itself, then for sufficiently large ordinals $α$, the supremum of the lengths of all wellfounded relations on $V_α$ is a strong limit cardinal if and only if…
We present two ways in which the model $L({\mathbb R})$ is canonical assuming the existence of large cardinals. We show that the theory of this model, with {\em ordinal} parameters, cannot be changed by small forcing; we show further that a set of ordinals in $V$ cannot be added to $L({\mathbb R})$ by small forcing. The large cardinal needed corresponds to the consistency strength of $AD^{L({\mathbb R})}$; roughly $ω$ Woodin cardinals.
We study several cardinal, and ordinal--valued functions that are relatives of Hanf numbers. Let kappa be an infinite cardinal, and let T subseteq L_{kappa^+, omega} be a theory of cardinality <= kappa, and let gamma be an ordinal >= kappa^+. For example we look at (1) mu_{T}^*(gamma, kappa):= min {mu^* for all phi in L_{infinity, omega}, with rk(phi)< gamma, if T has the (phi, mu^*)-order property then there exists a formula phi'(x;y) in L_{kappa^+, omega}, such that for every chi >= kappa, T has the (phi', chi)-order property}; and (2) mu^*(gamma, kappa):= sup{mu_T^*(gamma, kappa)| T in L_{k…
For any regular cardinal $κ$ and ordinal $η<κ^{++}$ it is consistent that $2^κ$ is as large as you wish, and every function $f:η\to [κ,2^κ]\cap Card$ with $f(α)=κ$ for $cf(α)<κ$ is the cardinal sequence of some locally compact scattered space.
For each natural number $n$, let $C^{(n)}$ be the closed and unbounded proper class of ordinals $α$ such that $V_α$ is a $Σ_n$ elementary substructure of $V$. We say that $κ$ is a \emph{$C^{(n)}$-cardinal} if it is the critical point of an elementary embedding $j:V\to M$, $M$ transitive, with $j(κ)$ in $C^{(n)}$. By analyzing the notion of $C^{(n)}$-cardinal at various levels of the usual hierarchy of large cardinal principles we show that, starting at the level of superstrong cardinals and up to the level of rank-into-rank embeddings, $C^{(n)}$-cardinals form a much finer hierarchy. The natur…
We define a generic Vopěnka cardinal to be an inaccessible cardinal $κ$ such that for every first-order language $\mathcal{L}$ of cardinality less than $κ$ and every set $\mathscr{B}$ of $\mathcal{L}$-structures, if $|\mathscr{B}| = κ$ and every structure in $\mathscr{B}$ has cardinality less than $κ$, then an elementary embedding between two structures in $\mathscr{B}$ exists in some generic extension of $V$. We investigate connections between generic Vopěnka cardinals in models of ZFC and the number and complexity of $\aleph_1$-Suslin sets of reals in models of ZF. In particular, we show tha…
We investigate the extent to which ultrapowers by normal measures on $κ$ can be correct about powersets $\mathcal{P}(λ)$ for $λ>κ$. We consider two versions of this questions, the capturing property $\mathrm{CP}(κ,λ)$ and the local capturing property $\mathrm{LCP}(κ,λ)$. $\mathrm{CP}(κ,λ)$ holds if there is an ultrapower by a normal measure on $κ$ which correctly computes $\mathcal{P}(λ)$. $\mathrm{LCP}(κ,λ)$ is a weakening of $\mathrm{CP}(κ,λ)$ which holds if every subset of $λ$ is contained in some ultrapower by a normal measure on $κ$. After examining the basic properties of these two notio…
Let $F_{ω_1}$ be the countable admissible ordinal equivalence relation defined on ${}^ω2$ by $x \ F_{ω_1} \ y$ if and only if $ω_1^x = ω_1^y$. It will be shown that $F_{ω_1}$ is classifiable by countable structures and must be classified by structures of high Scott rank. If $E$ and $F$ are equivalence relations, then $E$ is almost Borel reducible to $F$ if and only if there is a Borel reduction of $E$ to $F$, except possibly on countably many $E$-classes. Let $E_{ω_1}$ denote the equivalence of order types of reals coding well-orderings. It will be shown that in the constructible universe $L$ …
We show that assuming modest large cardinals, there is a definable class of ordinals, closed and unbounded beneath every uncountable cardinal, so that for any closed and unbounded subclasses $P, Q$, $\langle L[P],\in ,P \rangle$ and $\langle L[Q],\in ,Q \rangle$ possess the same reals, satisfy the Generalised Continuum Hypothesis, and moreover are elementarily equivalent. The theory of such models is thus invariant under set forcing. They also all have a rich structure satisfying many of the usual combinatorial principles and a definable wellorder of the reals. One outcome is that we can chara…
In this paper the Erdos-Rado theorem is generalized to the class of well founded trees. We define an equivalence relation on the class rs(infty)^{< aleph_0} (finite sequences of decreasing sequences of ordinals) with aleph_0 equivalence classes, and for n< omega a notion of n-end-uniformity for a colouring of rs(infty)^{< aleph_0} with mu colours. We then show that for every ordinal alpha, n< omega and cardinal mu there is an ordinal lambda so that for any colouring c of T=rs(lambda)^{< aleph_0} with mu colours, T contains S isomorphic to rs(alpha) so that c rest S^{< aleph_0} is n-end uniform…
In this paper we prove: Theorem 1. Let $\mathcal{K}$ be an abstract elementary class which satisfies the joint embedding and amalgamation properties. Suppose $λ>μ\geq LS(\mathcal{K})$ and $θ$ is a limit ordinal $<λ^+$. If $\mathcal{K}$ is $μ$ superstable and $μ^+$-superstable and satisfies $μ^+$-symmetry, then for any increasing sequence $\langle M_i\mid i<θ\rangle$ of $μ^+$-saturated models of cardinality $λ$, the model $\bigcup_{i<θ}M_i$ is $μ^+$-saturated.
