Easton's theorem (Wikipedia) per ZFC allows a vast range of formulas for infinite powersets; a relevant "base case" (so to speak) is the allowance for forcing $2^{\aleph_0} = 2^{\aleph_1}$, a proposition also known as Luzin's hypothesis (Wikipedia).

Try as I might, I've not made as much progress as I'd hoped on understanding forcing theory (and I've been trying since early 2019). So I'm not currently in a position to answer the following question:

• Let $\mathfrak{X}$ be some non-aleph transfinite cardinal, e.g. perhaps one based on amorphous sethood (MathOF). Is there a theory, e.g. ZF, where we can force $2^{\mathfrak{X}} = 2^{\aleph_0}$?

Motivation: I'm trying to do at least one of either of the following:

1. Work out an introductory theory of "conjugated transfinite cardinals of different choice-theoretic flavors," e.g. maybe something like $\aleph_{\alpha} ⊕ \mathfrak{X}$ (I was asking Asaf Karagila about this on the MathSE and he indicated that some/most of these kinds of expressions would end up dominated by the alephic factor, though; or at least the Continuum's cardinality, if partly alephic, would be so dominated).

2. Physical, i.e. (for present purposes) evolving or at least oscillating, cardinality assignments to sets. So maybe we could justifiably write down formulas that look like $\mathfrak{C} ⇌ 2^{\aleph_0} ⇌ \aleph_{\alpha}$, where each horn of the formula represents a cardinality assignment that a continuous physical set "fluctuates" into/out of.

posted 7 months ago

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Ripheus24‭

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