Galileo's Paradox
in [Math]
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From: Lester Zick on 4 Dec 2006 12:09 On 3 Dec 2006 11:22:56 -0800, cbrown(a)cbrownsystems.com wrote: >Tony Orlow wrote: >> cbrown(a)cbrownsystems.com wrote: >> > Tony Orlow wrote: >> > >> > <snip> >> > >> >> Anyway, ala Leibniz, each object IS the set of properties which it >> >> possesses, so any two objects with the exact same set of properties are >> >> the same object. >> > >> > But this begs the question: what do we mean, exactly, by a SET of >> > properties? What exactly are we trying to say when we say "This set of >> > properties is the same as this other set of properties"? >> > >> > Cheers - Chas >> > >> >> Well, what we really mean is that there is a set of universal >> properties, each of which is a set of values, and that each object is >> defined by a set of values, one from each set of property values, such >> that any two distinct objects differ in at least one property value. Was >> that specific enough? > >Well, you used the term "set" four times in your above definition of >what we mean by a "set". That's why I said "this begs the question, >what do we mean, exactly, by a set of properties?". > >There's something that we intuitively seem to think of as a "set"; but >unless such a thing is carefully defined, we end up with the >contradictions of naive set theory: I think the more basic question is whether non naive sophisticated set "theory" represents all of mathematics and whether it has any pre emptive prerogative to the definition of terms such as "cardinality" etc. in mathematics generally? ~v~~
From: Lester Zick on 4 Dec 2006 12:11 On Mon, 04 Dec 2006 05:32:20 -0500, Bob Kolker <nowhere(a)nowhere.com> wrote: >Eckard Blumschein wrote: >> >> Is it really justified to blame an allegedly insufficient definition of >> the term set for obvious antinomies of set theory? > >Set theory has been modified so the antinomies cannot arise. No new >contradictions have been found since the repairs to set theory have been >done. Such as the integration of points into lines, Bob? ~v~~
From: Eckard Blumschein on 4 Dec 2006 12:29 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. > Sets can have the same cardinality but different 'subsettedness' because > different qualities are being measured. I do not know a German equivalent to subsettedness. Wiki did not know subsettedness or at least subsetted either. So I have to gues what you possibly meant. Proper subset means included but not all-including. While I consider cardinality a rather illusory notion, I distinguish between counted (alias finite), countable (alias potentially infinite) and uncountable (alias fictitious). Well, the counteds are a subset of the countables. So far, the mislead bulk of mathematicians regard the countables (i.e. genuine numbers) a subset of the uncountables, too. TO>> You do? Do you mean that the addition of elements not already in a set >> doesn't add to the size of the set in any sense? > > In the subsettedness sense yes, in the cardinality sense, not > necessarily. In the sense of well-ordered subsettedness, not necessarily. > > TO seems to want all measures to give the same results, regardless of > what is being measured. Standard mathematics may lack solid fundamentals. At least it is understandable to me. However, I admit being not in position to likewise easily understand what you mean with well-ordered subsettedness.
From: Eckard Blumschein on 4 Dec 2006 12:31 On 12/1/2006 9:52 PM, Virgil wrote: > In article <45701F81.8050901(a)et.uni-magdeburg.de>, > Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > > Mathematics will survive >> set theory. > > Set Theory will survive Eckard. I survived the nazis and the communists. Mathematics will survive me.
From: imaginatorium on 4 Dec 2006 12:31
Lester Zick wrote: > On 4 Dec 2006 04:31:30 -0800, imaginatorium(a)despammed.com wrote: > > > > >Lester Zick wrote: > >> On Sat, 02 Dec 2006 13:26:13 -0500, Tony Orlow <tony(a)lightlink.com> > >> wrote: > >> >Lester Zick wrote: > >> >> On Fri, 01 Dec 2006 11:39:57 -0500, Tony Orlow <tony(a)lightlink.com> > >> >> wrote: > >> >>> Lester Zick wrote: > >> >>>> On Thu, 30 Nov 2006 12:08:03 +0100, Eckard Blumschein > >> >>>> <blumschein(a)et.uni-magdeburg.de> wrote: > >> >>>>> On 11/29/2006 8:13 PM, Bob Kolker wrote: > >> >>>>>> Tony Orlow wrote: > > > ><snip all that> > > > >> Well, Tony, I don't really think any of this matters. These just seem > >> to be word strings having nothing to do with the matters at hand. > > > >Well, indeed. Yet curiously it manages to be entertaining from time to > >time. > > And at least in my case more than a little true. > > But come now, Brian, a little question in zen math to brighten the day > as Bob seems unwilling to fess up: in modern math does set "theory" > represent all of mathematics as Bob seems to think... What is the difference between "set theory" and "set 'theory'"? Most of present-day mathematics is founded on set theory; but this is only a matter of convenience. and does it have a > pre emptive prerogative with respect to the definition of terms like > "cardinality" and so on in mathematics? Uhm "pre-emptive prerogative"? No idea. Don't think so - you, or rather, anyone with rather more inclination to mathematical ideas than you possess is perfectly free to build a different foundation. Has to be good though, and 99.999...% of crank stuff just doesn't cut the mustard. [spot joke] > .... I'm inclined to consider that > set "theory" is really only a parochial group of analytical techniques > which are inexhaustive at best and problematic in any event and not > really a theory at all. I wouldn't worry about it. Your "inclination to consider" reminds me of an old joke about Big Ben and the Leaning Tower of Pisa, which is probably out of place here - nothing you say will have much effect, so feel pretty free. > ~v~~ Right. Brian Chandler http://imaginatorium.org |