Re:A Bad Trip to Infinity
by CrazyStat on Saturday March 25, 2023 @05:22PM
(#35307539)
This is a bad example. In fact those sentences have the same cardinality, much like the set of even natural numbers has the same cardinality as the set of all natural numbers even though only half the natural numbers are even.
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Re:A Bad Trip to Infinity
by bryanrasmussen on Saturday March 25, 2023 @06:15PM
(#35308024)
they may have the same cardinality, but one infinite sentence is obviously longer than the other infinite sentence.
on edit: noting depends on if we consider as elements the words, or the letters.
Things that are "obviously" true are often false.
The only way to compare the lengths of the two infinite sentences is by comparing their cardinalities. There is no other meaningful definition of sentence length for an infinite sentence. They have the same cardinality, so they're the same "length."
Longer/larger according to what ?
If its according to cardinality then no, they have the same cardinality, same number of elements, presumably letters. If its according to a measure, then sure, one can be longer than the other -- the subset of [0,1] of the real line has the same cardinality, same number of 'points' as [0, 100] but the latter is longer when you compare according to the Lebesgue measure, i.e. 'length'.