Galileo's Paradox
in [Math]
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From: Bob Kolker on 29 Nov 2006 14:11 Lester Zick wrote: > Just out of curiosity, Bob, why is cardinality in set theory not a > measure? I mean if you ask "how much gas" and get the answer "two > gallons" you've certainly measured the gas. Or if you ask "how much > space" and get the answer "two inches" you've certainly measured the > space. It seems to me that you can obviously superimpose cardinality > on questions like "how much" without having to count or match things. Consider two measure sets, disjoint. The measure of the union is the sum of the measures of each. Now consider two sets of the same cardinality, disjoint. The cardinality of the union equals the cardinality of either. In short cardinality does not add like measure. Bob Kolker
From: Virgil on 29 Nov 2006 14:12 In article <456d9f90(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Mike Kelly wrote: > > If you think you have some wonderful notion of integrated count and > > measure that applies to all sets then even if you're correct (you're > > NOT) then cardinality is still a valid definition and still as useful > > as it is now. > > Yes, and the square wheel will always be as useful as it ever has. Any squareness is all in TO's wheels. > > > What about when there is more than one type of measure that can be > > applied to a set, or none at all? What happens then? > > Where count can be calculated from either of two measures, then one has > a choice in that matter. Hopefully, one gets the same result either way. > Do you have an example you'd like to explore? "Outer measure" of sets in R^n, defined as the LUB of the content of a covering by open intervals, for one.
From: Bob Kolker on 29 Nov 2006 14:13 Tony Orlow wrote: > > Uncountable simply means requiring infinite strings to index the > elements of the set. That doesn't mean the set is not linearly ordered, > or that there exist any such strings which do not have a successor. Uncountable means infinite but not of the same cardinality as the integers. For example the set of real numbers. It is an infinite set, but it cannot be put into one to one correspondence with the set of integers. Bob Kolker
From: Virgil on 29 Nov 2006 14:13 In article <456da004(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Bob Kolker wrote: > > Tony Orlow wrote: > > > >> It is a direct consequence of the notion that a proper subset is > >> always smaller, in some sense, than the base set. > > > > And in another sense a proper subset of an infinite set has the same > > count as the set. > > > > Bob Kolker > > > > It has the same cardinality perhaps, but where one set contains all the > elements of another, plus more, it can rightfully be considered a larger > set. It can be heavier without being taller, or vice versa.
From: MoeBlee on 29 Nov 2006 14:14
Bob Kolker wrote: > The cardinal number of a set is the equivalence class of sets > with the same cardinality as the the given set. In what theory is this? MoeBlee |