Galileo's Paradox
in [Math]
|
From: Tony Orlow on 30 Nov 2006 14:39 Virgil wrote: > In article <456e4621(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > >> 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. > > The outer measure of those two sets is exactly the same. Right, and yet, the second is missing two elements, and is therefore infinitesimally smaller in measure. That doesn't show up on the standard ruler. 0.999...=1.
From: Tony Orlow on 30 Nov 2006 14:39 Virgil wrote: > In article <456e46b7(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > >> 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. > > There is nothing in being a set, including being a set of reals, that > requires its members to be indexed at all. To establish an explicit bijection between infinite sets does require an ordering on the sets, at least in general.
From: Tony Orlow on 30 Nov 2006 14:45 Virgil wrote: > In article <456e475e(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > > >> Given a >> set density, value range determines count. > > Compare the "set densities" of the set of naturals, the set of > rationals, the set of algebraics, the set of transcendentals, the set of > constructibles, and the set of reals. Rather difficultt o formulate relations between those in standard theory. In the name of IST, I'll avoid any criteria including the notion of "standard" and state the following. The size of the set of hypernaturals is the square root of the size of the set of hyperreals. The set of hyperrationals corresponds to the square of the set of hypernaturals, minus all those pairs that are redundant, such as 2/4 or 6/18. That number of the hyperreals are the hyperirrationals. I am not sure how to relatively quantify transcendentals, constrictibles, or algebraics. Those are probably considered all "countable" by you, which doesn't say much about their relative sizes.
From: Tonico on 30 Nov 2006 14:46 Lester Zick wrote: > On Thu, 30 Nov 2006 07:32:58 -0500, Bob Kolker <nowhere(a)nowhere.com> > wrote: > > >Eckard Blumschein wrote: > > > >> > >> > >> I consider Dedekind wrong, and he admitted to have no evidence in order > >> to justify his basic idea. > > > >What sort of evidence? Surely not empirical evidence. Mathematics done > >abstractly has no empirical content whatsoever. > > Except apparently for axioms and definitions. ****************************************************************** What axioms of what part of maths have "empirical" evidence in the sense Eckard is tryuing to convey?!? For him, and for other trolls, Cantor "not having evidence" for his idea (what stupid this sounds!) means that he (cantor) never foiund an aleph_null under his bed, or that so far no one can buy aleph_beith apples out there. What "empirical evidence" are there in group theory's axioms? Or in Topology? Tonio
From: Tonico on 30 Nov 2006 14:46
Lester Zick wrote: > On Thu, 30 Nov 2006 07:32:58 -0500, Bob Kolker <nowhere(a)nowhere.com> > wrote: > > >Eckard Blumschein wrote: > > > >> > >> > >> I consider Dedekind wrong, and he admitted to have no evidence in order > >> to justify his basic idea. > > > >What sort of evidence? Surely not empirical evidence. Mathematics done > >abstractly has no empirical content whatsoever. > > Except apparently for axioms and definitions. ****************************************************************** What axioms of what part of maths have "empirical" evidence in the sense Eckard is trying to convey?!? For him, and for other trolls, Cantor "not having evidence" for his idea (what stupid this sounds!) means that he (cantor) never foiund an aleph_null under his bed, or that so far no one can buy aleph_beith apples out there. What "empirical evidence" are there in group theory's axioms? Or in Topology? Tonio |