"But if points have zero extension then even an infinity of them cannot sum to anything greater than zero."
Infinities are weird; anybody who wants to learn math has to accept that. 0.99999… does equal 1, there are as many even numbers as integers, etc. these things are 'true' not because they make sense initially, but because they make the most sense of all the other things we have thought of so far. Similarly, a set of Aleph-0 points can completely cover a line.
"a set of Aleph-0 points can completely cover a line"
Aleph_0 is the cardinality of the integers. I don't think that'll cover a line. For that, you need the cardinality of the reals, C, which may or may not be Aleph_1.
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"But if points have zero extension then even an infinity of them cannot sum to anything greater than zero."
Infinities are weird; anybody who wants to learn math has to accept that. 0.99999… does equal 1, there are as many even numbers as integers, etc. these things are 'true' not because they make sense initially, but because they make the most sense of all the other things we have thought of so far. Similarly, a set of Aleph-0 points can completely cover a line.
"a set of Aleph-0 points can completely cover a line"
Aleph_0 is the cardinality of the integers. I don't think that'll cover a line. For that, you need the cardinality of the reals, C, which may or may not be Aleph_1.
OOPS. Thanks
Infinities are weird.