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Cardinality

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Description

Summary

Sizes of sets, Sullivan and Mackey's §7.6: card of an infinite set is an aleph or a
power of 2 of one, sums and products of sizes are the larger, and sizes compare.

Remarks

PP, ZZ+, ZZ*, ZZ and QQ are aleph(0) in size.
RR, CC and every interval with two distinct ends are 2^aleph(0), the size
of the power set of ZZ+. That is an aleph too, but which one is the continuum
hypothesis, which ZFC does not decide, so it is not written as aleph(1). A power set
is 2^card(S); a union with an infinite side is its larger side; and an infinite
A less a smaller B keeps the size of A, so the irrationals are the size
of RR.
 A size read here is a finite count, or a tower 2^(...^(2^aleph(k))) of height
t, t = 0 being aleph(k) itself. Only what ZFC proves is used:
aleph(j) < aleph(k) for j < k; s < 2^s for every size, Cantor's
theorem; 2^s >= aleph(k + 1) for s >= aleph(k), so a tower of height
t over aleph(k) is at least aleph(k + t); and 2^s <= 2^u for
s <= u. What these leave open stays written: 2^aleph(0) = aleph(1) is
consistent with ZFC and so is its negation. A subset is no larger than its superset, which
settles one side of a comparison between two sizes that are not read.
 https://github.com/asc-community/AngouriMath/issues/1409

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