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"a subset of the x-y plane with a large cardinality need be neither measurable nor of large measure. For example, the entire plane with its 2^ℵ0 elements has an
by GolDDranks 5y ago
"a subset of the x-y plane with a large cardinality need be neither measurable nor of large measure. For example, the entire plane with its 2^ℵ0 elements has an infinite measure."
This should be ℵ0^2, since we are talking about plane defined by two axes, x * y, each with measure ℵ0, so the measure becomes ℵ0 * ℵ0 = ℵ0^2, which is the same as ℵ0, right?
Edit: maybe they don't mean elements as points in plane, but subsets of the plane, in which case the statement would apply?
- kevinventullo 5y agoNo, they really did mean 2^ℵ0. Note the earlier sentence “The set ℝ of real numbers (also called the real line) is as large as the power set of ℕ, and this cardinality is denoted 2^ℵ0, or “continuum.”” So the x-y plane has cardinality 2^ℵ0 * 2^ℵ0 = 2^ℵ0.
- GolDDranks 5y agoAh, indeed. Thanks for the correction. I read there, for some reason, an implication that a plane would have a larger cardinality than a line, which isn't true. But indeed, both are 2^ℵ0.