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Everything posted by Ripheus23
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Story idea: a universe of portal magic, where otherwise incontiguous sections of the universe are adjoined as 'harmonic landscapes,' with the attendant civilizations known as 'consortia.' one hlc where there is a factionpower known as the Apex, another the Nexus, and elsewhere the Matrix-abyss, has hundreds of adjoined regions.
There are also consortia that bent their portals to convey the 'door-ships' as these are called.
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So somewhere in my set-theory research, I read about the use of what seemed to be a color index for infinite cardinals. Or a description of a set of cardinalities that depended on coloration facts, or something. I wish I could find it again as easily as I stumbled upon it...
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Note to self: describe deontic cardinality re: Moore's error (in the additive relation of ought-facts).
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Set theory is whack

"Definition 5.4: NN is a mouse iff NN is a critical premouse, N′N′ is iterable and for each i∈Ordi∈Ord there is NiNi, a critical premouse, such that (Ni)′=N′i(Ni)′=Ni′ where 〈N′i,π′ij,κi〉〈Ni′,πij′,κi〉 is the iteration of N′N′, and n(Ni)=n(N)n(Ni)=n(N).
Definition 5.1: Premouse N=JUαN=JαU is critical iff P(κ)∩Σω(N)⊈NP(κ)∩Σω(N)⊈N and NN is acceptable.
Definition 3.1: For κ<ακ<α, N=JUαN=JαU is a premouse at κκ iff N⊨“U is a normal measure on κ”N⊨“U is a normal measure on κ”."
~~~Cantor's Attice, "Core Model"
In other words, wtf are mice and premice?!?!?! This sounds more whack than Kurepa trees or zero-sharp or what.
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So I was reading about the von Neumann universe, which the article (Wikipedia's) said was built up by transfinitely iterating the powerset operation. So unfortunately(ish), I had assumed in my version of set theory that the infinite ascension operation (equivalent to the powerset one at least in the simple minimum limit) would not lead to the "apex" of the hierarchy, at least not directly/necessarily/etc. I know there's no given apex to the vNu since the vNu is not supposed to be "the set of all sets" as such, but anyway, I also remember reading (can't find where right now...) that large cardinal axioms (I think) are conceived as "arbitrarily close to vNu" in the sense that a large cardinal is understood in the light of transfinality (my term, but their procedure) first, from which some descension is possible.
Now, this might not make much of a difference one way or another, although it seems to indicate (to me) that simplifying the axiom system so that large cardinal axioms are reduced to theorems of the axioms of transfinality and transcardinality (as such), is a possible "way to go" (as I have been going, to be sure). So in other words, either I'm going to find out that these topics have already been deeply addressed
or that I'm on to something interesting.
