Ripheus23
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OK, so one way the argument goes is: given how simple the function 2^aleph-zero is, it's not likely that its output is particularly exotic. That is, the options seem constricted to aleph-1, aleph-aleph-zero, and maybe aleph-2 or aleph-aleph-1. Beyond that, it would be peculiar to go from 2 to the power of aleph-zero, to wherever we had gone.
Now, I also seem to have a disproof of the Generalized Continuum Hypothesis available. Namely, and stipulating that under the circumstances (the countable infinity of the list of aleph-numbers from aleph-zero to aleph-n), it is possible to replace aleph-zero with {n + 1}, so that we read the GCH as
2^[aleph-n] = aleph-[n + 1] = aleph-aleph-zero.
This doesn't imply that any n gives us aleph-aleph-zero, but only n relative to {n + 1}.
Anyway, aleph-aleph-zero is the successor of all the aleph-numbers on the first list. So it should be given from 2 to the power of aleph-[n + 1] if the GCH is true. However, this amounts to saying that 2 to the power of aleph-aleph-zero is aleph-aleph-zero. But now the GCH says that 2 to the power of aleph-aleph-zero should be 2 to the power of aleph-aleph-1. So the GCH leads, here, to a contradiction, it seems, unless we assume that aleph-aleph-zero is supposed to be equivalent in cardinality to aleph-aleph-1, counter to the principle of Cantor's paradise in general.
Now, there could be a map from 2^aleph-zero, to aleph-1, without the GCH being true, in which case 2^aleph-1 might be something like aleph-aleph-zero, for example. But it seems dubious that the question of the CH in particular should be resolved just by means of "which" axiom scheme one uses. Indeed, the principle of ascent is in equilibrium if 2^aleph-zero = aleph-aleph-zero, &c., I think, so...
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https://en.wikipedia.org/wiki/Kurepa_tree
Has the craziest sentences I've heard in a while:
QuoteMore precisely, the existence of Kurepa trees follows from the diamond plus principle, which holds in the constructible universe. On the other hand, Silver (1971) showed that if a strongly inaccessible cardinal is Lévy collapsed to ω2 then, in the resulting model, there are no Kurepa trees. The existence of an inaccessible cardinal is in fact equiconsistent with the failure of the Kurepa hypothesis, because if the Kurepa hypothesis is false then the cardinal ω2 is inaccessible in the constructible universe.
A Kurepa tree with fewer than 2ℵ1 branches is known as a Jech–Kunen tree.
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OMG zippers are real [set-theoretically, that is]! From "Non-wellfounded Set Theory," on the SEP:
QuoteThere is a natural operation of “zipping” two streams. Also called “merging”, it is defined by
(3) zip(s, t) = 〈 head(s), zip(t, tail(s)) 〉 So to zip two streams s and t one starts with the head of s, and then begins the same process of zipping all over again, but this time with t first and the tail of s second. For example, if x†, y†, and z†are the solutions to the system in equation (2) above, then we might wish to consider, for example,zip(x†, y†). In unraveled form, this is
(0,1,1,2,2,0,0,1,1,2,2,0,…).
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Hmm this is interesting:
QuoteThis theory is a proposal of ours, which elaborates on a suggestion of Rudy Rucker. We (and many others) have observed that of all the orders of infinity in Cantor’s paradise, only two actually occur in classical mathematical practice outside set theory: these are ℵ0 and c, the infinity of the natural numbers and the infinity of the continuum. Pocket set theory is a theory motivated by the idea that these are the only infinities (Vopenka’s alternative set theory also has this property, by the way).
~from "Alternative Axiomatic Set Theories," SEP