Galileo's Paradox
in [Math]
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From: Lester Zick on 29 Nov 2006 19:51 On Wed, 29 Nov 2006 14:09:19 -0500, Bob Kolker <nowhere(a)nowhere.com> wrote: >Lester Zick wrote:> >> Actually not because "infinity" a well defined mathematical concept >> whereas "cardinality" is only an ambiguously defined concept >> mathematically restricted to undemonstrable set analytical techniques. > >Nonsense. Nonsense that infinity is a well defined mathematical concept or that the definition of cardinality is an ambigously defined concept restricted to set analytical techniques? I mean unless you have some as yet undemonstrated idea that parochial set analytical techniques represent some kind of paradigm for mathematics as a whole I see no necessity to restrict the concept of cardinality in one for the other. > Two sets have the same cardinality if and only if there exists >a one to one onto mapping from one to the other. That is a plain >definition. Same cardinality produces and equivalence relation defined >on sets. The cardinal number of a set is the equivalence class of sets >with the same cardinality as the the given set. Yeah, yeah, yeah, Bob. You seem to confuse set analtyical techniques and definitions with mathematics. >Don't give up your day job Lester. But this is my day job, Bob. I mean why else would anyone pander to a bunch of congential idiots trying to make sense out of set "theory"? ~v~~
From: Virgil on 29 Nov 2006 21:21 In article <1164835003.596078.281470(a)n67g2000cwd.googlegroups.com>, "MoeBlee" <jazzmobe(a)hotmail.com> wrote: > Virgil wrote: > > In article <1164828291.381109.186960(a)n67g2000cwd.googlegroups.com>, > > "MoeBlee" <jazzmobe(a)hotmail.com> wrote: > > > > > Bob Kolker wrote: > > > > MoeBlee wrote: > > > > > > > > > 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? > > > > > > > > Standard set theory. > > > > > > Wrong. > > > > > > The usual definition is that the cardinality of a set (or the cardinal > > > number of a set) is the least ordinal equinumerous with the set. That > > > is not the class of sets with the same cardinality as the given set. In > > > Z set theories, the class of sets having the same cardinality as the > > > given set is not even itself a set, let alone a cardinal number. > > > > > > MoeBlee > > > > There are set theories in which equivalence classes need not be proper > > classes or cause other problems, and the equivalence class definition is > > commonly used in them. > > Other than type theories such as PM or a stratification theory such as > NF, what is an example of a set theory in which the class of all sets > equinumerous with a given nonempty set is a set? (I don't deny that > such a thing is possible; I'm just curious as to an example.) I was thinking of NF or variations on that theme. > > > In ZF or NBG, the least ordinal definition works nicely. > > Right, and they don't use the class of all sets equinumerous with a > given nonempty set to serve as the cardinality of the given set. > > MoeBlee
From: Virgil on 29 Nov 2006 21:22 In article <4t6dreF12hmccU2(a)mid.individual.net>, Bob Kolker <nowhere(a)nowhere.com> wrote: > Virgil wrote: > > > > > > This only holds for sets whose cardinality is either 0 or not finite. > > Precisely. However you can take the measure of the union of two disjoint > infinite (in cardinality) sets each with finite measure the the measure > of the union is the sum of the measures of the constituent set. In > short, measure is additive in the algebraic sense and cardiality (for > infinite sets) is not. > > That is one of the reasons that how much and how many are different > questions. > > Bob Kolker Right!
From: Tony Orlow on 29 Nov 2006 21:47 Virgil wrote: > 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. Where standard measure is the same, there still may be an infinitesimal difference, such as between (0,1) and [0,1], if that's what you mean.
From: Tony Orlow on 29 Nov 2006 21:49
Bob Kolker wrote: > 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 Yes, Bob, I know that. It boils down to the same thing. Any uncountable set with element indexes expressed as digital numbers will require infinitely long indexes for most elements, such as is the case for the reals in any nonzero interval. Standard integers each require only a finite number of digits. Tony |