Large cardinals are certain infinite sets whose existence, There are many other large cardinals, with imaginative names like “almost-huge[9]”,

There are a couple things going on here. First let’s get a misconception out of the way: * Infinity is just Infinity! It’s always the same! This is simply not true. To understand why, you have to understand a very special definition of equality. G.

Recall that a cardinal is n-huge for a natural number n 1 if there is an elementary embedding j: V !Mwith crit(j) = and Mjn( ) M, and that 1-huge cardinals are simply called huge. Equivalently, is n-huge if there is a normal -complete ultra lter Uon some P( ) and cardinals = 0 < 1 < < n= such that for each i<n, fx2P( ).

A cardinal satisfying one of the rank into rank axioms is n-huge for all finite n. The existence of an almost huge cardinal implies that Vopenka’s principle is consistent; more precisely any almost huge cardinal is also a Vopenka cardinal.

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get the result, due to Scott, that if there is a measurable cardinal, then V 6=L. Concluding, we deﬁne normal measures, which become invaluable in the study of elementary embeddings. Chapter 4 goes a little bit backwards, since we study a weaker large cardinal hypothesis, the existence of 0 ♯. Some eﬀort is needed in order to deﬁne 0 but

An n + 1-huge cardinal is n-huge*. Proof Suppose that κ is n -huge*, and so for some α > κ , there is an elementary j : V α → V β with crit ( j ) = κ and j n ( κ ) < α.

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The term itself is uncommon in the literature I’m familiar with, but usually means nothing more than a cardinal n -huge for all n∈N, which any rank-into-rank cardinal satisfies (as noted on this very page). More specifically, this property is strictly weaker than WA0 – any cardinal satisfying WA0 is already ω -superhuge in this sense,

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And yet, almost every description of set theory plunges straight into the cardinal and ordinal numbers. Why? That’s a question that mystified me for quite a long time. Why do we take this beautiful.

Martin and Steel’s paper "A proof of projective determinacy" defines an $omega$-huge cardinal to be an I2 cardinal but more recently you see it being defined to be an I1 cardinal. Is there a stand.

For the converse, consistency of ZFC + (schema) {there is n-huge cardinal} n implies existence of a model M of ZFC + n-huge cardinal for a nonstandard number n. Starting with such M and an n-huge embedding j in M with κ = crit(j), let M’ = {x∈M: ∃m<ω x ∈ M j m (V κ M )}.

A cardinal satisfying one of the rank into rank axioms is n-huge for all finite n. The existence of an almost huge cardinal implies that Vopenka’s principle is consistent; more precisely any almost huge cardinal is also a Vopenka cardinal.

It turns out that remarkable cardinals is a very robust large cardinal notion that is equiconsistent with several other natural set theoretic assertions as well. Cheng and Schindler recently showed that third-order arithmetic together with Harrington’s principle is equiconsistent with a remarkable cardinal.

Martin and Steel’s paper "A proof of projective determinacy" defines an $omega$-huge cardinal to be an I2 cardinal but more recently you see it being defined to be an I1 cardinal. Is there a stand.

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Avirtually n-huge*cardinal is an n + 1-iterable limit of n + 1-iterablecardinals. If is n + 2-iterable, then V is a model of proper class many virtually n-huge* cardinals. Avirtually rank-into-rankcardinal is an!-iterable limit of !-iterablecardinals. An !+ 1-iterablecardinal implies the consistency of a virtually rank-into-rankcardinal.

$begingroup$ Joseph, I’m not sure what the best solution is. Your proposal might work, but my gut feeling (and I say this as a non-set-theorist) is that probably any one of these questions, say 1. to begin with, is going to be hard and would probably make a good MO question on its own. I’d actually be pleasantly surprised if someone at MO could answer such a question, as I would bet that.

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a cardinal that is super-n-huge for every n, to inconsistency. This hierarchy is parallel to the usual hie rarch y of large cardinal axiom s, and can be used in the

Although they are very large, there is a first-order definition which is equivalent to n-hugeness, so the $theta$-th n-huge cardinal is first-order definable whenever $theta$ is first-order definable. This definition can be seen as a (very strong) strengthening of the first-order definition of measurability. Elementary embedding definitions

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Rank-into-rank. Every I1 cardinal κ (sometimes called ω-huge cardinals) is an I2 cardinal and has a stationary set of I2 cardinals below it. Every I2 cardinal κ is an I3 cardinal and has a stationary set of I3 cardinals below it. Every I3 cardinal κ has another I3 cardinal above it and is an n – huge cardinal for every n <ω. Axiom I1.

And yet, almost every description of set theory plunges straight into the cardinal and ordinal numbers. Why? That’s a question that mystified me for quite a long time. Why do we take this beautiful.

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Indeed, the extensions of the theory ZFC+“j is a nontrivial elementary embedding ” form a hierarchy of axioms, ranging in strength from Con(ZFC) to the existence of a cardinal that is super-n-huge for every n, to inconsistency. This hierarchy is parallel to the usual hierarchy of large cardinal axioms, and can be used in the same way.

An n + 1-huge cardinal is n-huge*. Proof Suppose that κ is n -huge*, and so for some α > κ , there is an elementary j : V α → V β with crit ( j ) = κ and j n ( κ ) < α.

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