An uncountable countable set
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From: Han de Bruijn on 18 Sep 2006 03:32 Tony Orlow wrote: > Han de Bruijn wrote: >> Precisely! Mathematicians get confused by the idea of a "bijection", >> which is an Equivalence Relation, which in turn is a "generalization" >> of "common equality" (yes: the one in a = b). But the funny thing is >> that EQUALITY HAS NEVER BEEN DEFINED. So there is actually nothing to >> "generalize". Equivalence relations are a "generalization" of nothing. >> >> But, fortunately, reality is more simple than this. Every equality is >> an equivalence relation. And every equivalence relation is an equality. [ ... snip ... ] > > I agree with that last statement, but would disagree that equality is > not definable. It depends on difference, most basically, and where none > is detected, two things can be said to be equal. Panta rhei, ouden menei (= everything flows, nothing remains the same). There is no such thing as "difference, [...] where none is detected" in nature (and culture). All equality is "in some sense" and relative. But a picture says more than a thousand words: http://hdebruijn.soo.dto.tudelft.nl/fototjes/gezocht.htm Han de Bruijn
From: Han de Bruijn on 18 Sep 2006 03:34 MoeBlee wrote: > Tony Orlow wrote: > >>Unbounded but finite may >>be considered potentially, but not actually, infinite. > > That will be jiffy, once you give axioms and/or our definitions for > 'potentially infinite' and 'actual infinite'. Until then, it's pure > handwaving. Come on, Moeblee, don't be silly! Let Google be your friend! Han de Bruijn
From: Han de Bruijn on 18 Sep 2006 03:37 Virgil wrote: > In article <cf41b$450a799d$82a1e228$10666(a)news1.tudelft.nl>, > Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > >>But, fortunately, reality is more simple than this. Every equality is >>an equivalence relation. And every equivalence relation is an equality. >>So the bijection between tally marks or collected pebbles with counted >>objects means, indeed, that tally marks and moving pebbles ARE numbers. > > Not in mathematics. Not in _your_ mathematics. Han de Bruijn
From: Han de Bruijn on 18 Sep 2006 03:46 imaginatorium(a)despammed.com wrote: > _How_ would you draw a ball from a vase containing an infinite set of > balls. Yaaawn! This has been discussed, at length, as well: http://huizen.dto.tudelft.nl/deBruijn/grondig/natural.htm#bv Han de Bruijn
From: Han de Bruijn on 18 Sep 2006 05:30
Virgil wrote: > In article <450c3c65(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > >>What is the average value of the reals in [0,1]? > > By what definition of average? > > There are a whole bunch of such definitions. > > Note that many such definitions only apply to finite sets of numbers, > and thus won't work. > > Also, since the set [0,1] is invariant under x --> x^n, its average > should be also. Objection! One such definition is: Average(x) = integral(0,1) x dx / integral(0,1) dx = 1/2 And, do you suggest that: Average(x) = integral(0,1) x^n . x dx / integral(0,1) x^n dx = (n+1)/(n+2) = 1/2 for arbitrary n ? That's a weird suggestion. No? Han de Bruijn |