Many-times huge and superhuge cardinals

Abstract

In this paper we consider various generalizations of the notion of hugeness. We remind the reader that a cardinal κ is huge if there exist a cardinal λ > κ, an inner model M which is closed under λ-sequences, and an elementary embedding i: V → M with critical point κ such that i(κ) = λ. We shall call λ a target for κ and shall write κ → (λ) to express this fact. Equivalently, κ is huge with target λ if and only if there exists a normal ultrafilter on P=κ(λ) = {X ⊆ λ:X has order type κ}. For the proof and additional facts on hugeness, see [3].We assume that the reader is familiar with the notions of measurability and supercompactness. If κ is γ-supercompact for each γ < λ, we shall say that κ is < λ-supercompact. We note that if κ → (λ), it follows immediately that κ is < λ-supercompact.Throughout the paper, n shall be used to denote a positive integer, the letters α, β, and δ shall denote ordinals, while κ, λ, γ, and η shall be reserved for cardinals. All addition is ordinal addition. V denotes the universe of all sets.All results except for Theorems 6b and 6c and Lemma 6d can be formalized in ZFC.This paper was written while the first named author was at Rochester Institute of Technology, Rochester, New York. We wish to thank the department of mathematics at R.I.T. for secretarial time and facilities.

C (n)-cardinals

“…The following Proposition is the C (n) -cardinal version of similar results for m-huge and superhuge cardinals due to Barbanel-Di Prisco-Tan [2] (see also [5], 24.13).…”

“…Notice also that, by a similar argument, every cardinal in C (2) belongs to the 1 definable class Lim(C (1) ) of all limit points of C (1) and, moreover, it captures all 1 proper classes. However, the least ordinal λ in Lim(C (1) ) that captures all 1 proper classes is strictly less than the least ordinal μ in C (2) . The point is that, fixing an enumeration ϕ n (x) : n < ω of all 1 formulas that define proper classes, the sentence…”

“…It now follows by Lemma 3.1 that κ is < λ-supercompact. But this is impossible because V λ+2 , ∈, α, λ ∈ C. (2). Fixing an ordinal ξ , to show that there is a supercompact cardinal greater than ξ , we argue as above.…”

Ultrahuge cardinals

“…Superhuge cardinals (and their relatives) were introduced by Barbanel, Di Prisco, and Tan in 1984 (cf. ) and are placed near the highest layers of the large cardinal hierarchy, just below the so‐called rank‐into‐rank cardinals which are the strongest axioms of infinity not known to be inconsistent with set theory.…”

The ultrafilter characterization of huge cardinals

“…Recent results by Barbanel, DiPrisco and Tan extend this to include that k isy(K)-supercompact. Their method of proof requires the axiom of choice and is established in the following way: If k is huge andy: V -» M is a witness to this, theny(K) is measurable in V (see [1]). Using the fact that k is a-supercompact for k < a <j'(k), ultrafilters on PKa for k<ö<j(k)…”