Galileo's Paradox
in [Math]
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From: Lester Zick on 6 Dec 2006 13:33 On Wed, 6 Dec 2006 00:38:59 -0500, David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: >Virgil wrote: >> In order to discuss anything, there must be agreement on the meaning of >> the terms to be used in that discussion. Thousands of mathematicians >> have agreed on the meanings of certain mathematical terms, so that those >> who wish to discuss things mathematical cast themselves into outer >> darkness by insisting on using those terms to mean things at variance >> with their standard meanings. >> >> If EB wants to express different meanings he must come up with unused >> word or phrases to carry those meanings. Mathematics will not allow EB >> to rewrite mathematical dictionaries to suit his whims. > >Or, clearly state what meaning he is giving to words. On the other hand those thousands of mathematicians might consider the truth of what they say. ~v~~
From: MoeBlee on 6 Dec 2006 13:58 Bob Kolker wrote: > David Marcus wrote: > > > > > > As Virgil pointed out, "no bijection to the naturals" is not a correct > > definition of "uncountable". > > Eh? > > See: http://en.wikipedia.org/wiki/Uncountable > > In mathematics, an uncountable or nondenumerable set is a set which is > not countable. Here, "countable" means countably infinite or finite, so > by definition, all uncountable sets are infinite. Explicitly, a set X is > uncountable if and only if there is an injection from the natural > numbers N to X, but no injection from X to N. > > From the wiki article on countability. > > Do you see in error in the wikipedia entry on uncountability Yes. Or at least I don't like the way they use 'nondenumerable'. A denumerable set is one that is equinumerous with N. So a nondenumerable set is not necessarily an uncountable set, since a set can fail to be denumerable not just by being uncountable but by being finite. Also, "a set X is uncountable if and only if there is an injection from the natural numbers N to X, but no injection from X to N." Don't we use the (countable) axiom of choice to show that N injects into any uncountable set? I prefer this set of definitions: x is finite <-> x is equinumerous with a natural number x is infinite <-> ~ x is finite x is Dedekind infinite <-> x is equinumerous with a proper subset of x x is Dedekind finite <-> ~ x is Dedekind infinite x is denumerable <-> x is equinumerous with N x is countable <-> x is finite or x is denumerable x is uncountable <-> ~ x is countable MoeBlee
From: Bob Kolker on 6 Dec 2006 13:58 Eckard Blumschein wrote: > > As long as one merely intends to perform sandpit-mathematics. Sandpit mathematics: Like manifold theory, topology, and category theory. Bob Kolker
From: Bob Kolker on 6 Dec 2006 14:01 Eckard Blumschein wrote: > > I did not say this. Please quote me carefully. > The set of existing Dedekind cuts is finite. The set of feasible cuts is > countable. The set of Dedikind cuts is not finite. There is a Dedikind cut for each rational number for starters. Why do you say such stupid things? Bob Kolker
From: David Marcus on 6 Dec 2006 14:02
Eckard Blumschein wrote: > On 12/5/2006 7:06 PM, Tony Orlow wrote: > > Eckard Blumschein wrote: > >> On 12/1/2006 9:59 PM, Virgil wrote: > >> > >>> Depends on one's standard of "size". > >>> > >>> Two solids of the same surface area can have differing volumes because > >>> different qualities of the sets of points that form them are being > >>> measured. > >> > >> Both surface and volume are considered like continua in physics as long > >> as the physical atoms do not matter. > >> Sets of points (i.e. mathematical atoms) are arbitrarily attributed. > >> There is no universal rule for how fine-grained the mesh has to be. > >> Therefore one cannot ascribe more or less points to these quantities. > >> > >> Look at the subject: Galileo's paradox: The relations smaller, equally > >> large, and larger are pointless in case of infinite quantities. > > > > Now, just a minute, Eckard. You're contradicting yourself, if these > > objects are infinite sets of points. They have different measure. > > I did not consider measures. Let's get concrete. Are there more naturals > than odd naturals? This question could easily be answered if there was a > measure of size. It can easily be answered once you say what you mean by "more". Why do you think that common English words have unambiguous mathematical meanings? -- David Marcus |