Re: The reality of sets, on a scale of 1 to 10 [Was: The non-existence of "dark numbers"]

Am 24.03.2025 um 22:06 schrieb Alan Mackenzie:

Moebius <invalid@example.invalid> wrote:

Guess he's referring to this notion:
https://en.wikipedia.org/wiki/Natural_density

 OK, thanks.  That natural density is not a property of sets [in general]

Right. But of (certain) subsets of IN.
See: https://en.wikipedia.org/wiki/Natural_density#Properties_and_examples
Wikipedia: "... [a] method to measure how "large" a subset of the set of natural numbers is."
But let's not split hairs here (like Mückenheim).
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Reading something like this
"...it may seem intuitively that there are more positive integers than perfect squares, because every perfect square is already positive and yet many other positive integers exist besides. However, the set of positive integers is not in fact larger than the set of perfect squares: both sets are infinite and countable and can therefore be put in one-to-one correspondence. Nevertheless if one goes through the natural numbers, the squares become increasingly scarce. The notion of natural density makes this intuition precise for many, but not all, subsets of the naturals (see Schnirelmann density, which is similar to natural density but defined for all subsets of IN." (Wikipedia)
I'd question the claim "However, the set of positive integers is not in fact larger than the set of perfect squares" (lol) - here "larger" is interpreted in the sense (context) of /cardinality/. Yeah, IF we measure "the size" of a set using the notion of cardinality, clearly "the set of positive integers is not [...] larger than the set of perfect squares". :-)