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I realized that there's a difference between proving "x is not a set of anything" and proving "x is a set of nothing," and I needed to prove that the ur-element is not a set of no elements of any set, including itself. I think I worked it out* but anyway, another result of the model was that while zero is an empty subset of other sets (a set with no elements of any other sets), so all other sets are empty subsets of zero (sets of no elements of zero). So this underscores why for all natural numbers n besides 0, the powerset is uniquely 2 to the power of n, whereas 0 to the power of 0, 1 to the power of 0, 2 the power of zero... 1 to the power of 1, 1 to the power of 2, 1 to the power of 3... = 0 ^^2 = 1 ^^ 2 = 1. <So zero is the only number before aleph-zero that has an infinite number of powerset expressions.>
*my idea is that "being a set of no elements of x" means being 0 (in relation to other numbers) and being other numbers (in relation to zero). But the relation is never defined on the ur-element. More, then, we can just define the ur-element such that it it is not a set of no elements of other sets, nor itself. In conjunction with the ur-element's not being a set of anything either, it follows that it is not a set of anything or nothing, which rules out the ur-element being a set.
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minotaur elements
minotaur ordinals
dragon ordinals
unicorn ordinals
Orpheus surreals
Osirus ordinals
Amaterasu cardinals [again]
Ragnarok cardinals
Michael cardinals
Raphael ordinals
Gabriel surreals
Asmodeus surreals
warlock ordinals
sorcerer ordinals
wizard cardinals
sortilege ordinals
demonic cardinals
leviathan transet
"kenotic transet" <said of the ur-element> and "pleromatic set"
happy cardinals
sad ordinals
priestly cardinals
hieratic ordinals
sword cardinals
sword ordinals
nuclear cardinals
nuclear ordinals
nuclear surreals
presbyter cardinals
deacon cardinals
"reformed cardinals" <so large only divine revelation can give knowledge of them>
Ezekiel cardinals
Oberon numbers
Aquinas cardinals
Nicean cardinals
irminsul ordinals
Odysseus ordinals
Aeneas surreals
Milwaukee cardinals ("a kind of joke cardinal")
cheddar ordinal
parmesan cardinal
cola ordinal
hemlock ordinal
Icarus transet
ambrosiac cardinal
cowardly cardinal
cowardly ordinal
helpless surreal
potato surreal
guitar cardinal
piano ordinal [haha]
Deseret cardinal
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