Galileo's Paradox
in [Math]
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From: Bob Kolker on 29 Nov 2006 14:15 Eckard Blumschein wrote: > > No. Infinite quantities include e.g. an infinite amount of points. > Infinite means: The process of quantification has not been finished or > cannot be finished at all. A non-empty set is infinite if and only if it can be put in one to one correspondence with a proper subset of itself. That is the standard definition of infinite for sets. Bob Kolker > >
From: Virgil on 29 Nov 2006 14:18 In article <456da0d5(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Banach-Tarski is a proof by contradiction that set theory is out of > whack. What axiom(s) does the theorem contradict? A paradox is not a self-contradiction, it is merely something that runs counter to one's intuition. > Point set topology loses measure, since points have no measure. It does not "lose" measure, it merely ignores it. > The axiom of choice is abused. On the contrary, it is used to its fullest. > > Additive measure of infinite sets is possible with infinitesimals. Or would be if any infinitesimals could exist in standard analysis. But as they cannot, the issue is moot.
From: Bob Kolker on 29 Nov 2006 14:19 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. Bob Kolker
From: Virgil on 29 Nov 2006 14:21 In article <456DCC3F.5020700(a)et.uni-magdeburg.de>, Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > On 11/29/2006 3:56 PM, Tony Orlow wrote: > > > Cardinality is generalized from the simple count of finite sets to the > > infinite case. In the finite case, the cardinality of a set is exactly a > > natural number, a quantity. In the infinite case, cardinality becomes > > something more ephemeral, > > Epheremal means shortlived. We have a saying: Lies live short. > > but it still has its roots in the count of a set. > > Let's rather say in Cantor's illusion of allegedly being able to count > the uncountable. > > > >> 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? > > Then perhaps a red light will indicate logical error. Does volume completely determine mass? Different measures measure different things and need have any correlation.
From: MoeBlee on 29 Nov 2006 14:24
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 |