An uncountable countable set
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From: Han de Bruijn on 13 Oct 2006 02:57 Randy Poe wrote about the Balls in a Vase problem: > Specifically, that for each particular ball (whatever you > want to label it), there is a time when it comes out. Yes. And, at the same time, 10 others come in. So what? Han de Bruijn
From: Han de Bruijn on 13 Oct 2006 03:01 Mike Kelly wrote about the Balls in a Vase problem: > Ah, but noon is not a part of the sequence of iterations. No more than > 0 is an element of the sequence 1, 1/2, 1/4, 1/8, .... Thus the question is whether the sequence (number of balls) converges. > The question asks how many balls are in the vase at noon. Not at some > iteration. Well, it does not converge. So this question of yours is meaningless. Han de Bruijn
From: Han de Bruijn on 13 Oct 2006 03:04 Ross A. Finlayson wrote about the Balls in a Vase problem: > I describe some conditions on the ball and vase problem that can help > make it more realistic. > > The golem with the marker in the vase, where you can't reach into the > vase, if you want one ball out for putting ten in, there would need to > be infinitely many golems if each can only hold one ball. > > Recently in this discussion about infinite sets and so on one of the > talking points about Cantor that has emerged is that he counts > backwards from infinity. > > The empty-vasers construct the argument that for any ball labelled n, > where each ball has some factory serial, they can denote some time > 1/2^n where that number has been retrieved from the vase. By the same > token, at time 1/2^n, ten balls were just placed in the vase. For each > of those, the various times they are retrieved from the vase are > exactly specified, and, at each of those ten more new ones are added to > the vase. At each constructed time, for n many iterations, the count > of balls in the vase is 9n. > > The count of balls in the vase is the difference of two divergent > series. That wouldn't be bad if the difference would be a convergent sequence. But the difference is divergent as well. Han de Bruijn
From: Han de Bruijn on 13 Oct 2006 03:13 Michael Stemper wrote: > In article <1160613324.049353.270480(a)k70g2000cwa.googlegroups.com>, Ross A. Finlayson writes: > >>David Marcus wrote: >> >>>Ross A. Finlayson wrote: >>> >>>>David Marcus wrote: > >>>>>So, you aren't saying that ZF is inconsistent. You are saying that you >>>>>prefer to use a different system of axioms. Is that correct? > >>>>No, David, Dave, I say ZF is inconsistent. That doesn't mean all its >>>>results are false. It just lets me say whatever I want about a less >>>>inconsistent set of "axioms". > >>>So, you are saying that ZF is inconsistent. By "inconsistent", do you >>>mean that ZF proves both P and not P, for some statement P? If so, >>>please be explicit: what is the statement and what is the proof in ZF of >>>it? > >>Build a set: {x: true}, it's a set, > > No, "{x: true}" is gibberish. Just putting down a couple of braces and > throwing random text inside them does not generate a set. At least, no > according to the axioms of ZF that you're trying to refute. No more glibberish than defining the ordinals by putting ever more curly braces around the empty set: 0 = {} , 1 = {{}} , 2 = {{},{{}}} , 3 = {{},{{}},{{},{{}}}} ... http://www.jboden.demon.co.uk/SetTheory/ordinals.html Han de Bruijn
From: Han de Bruijn on 13 Oct 2006 03:27
Virgil wrote: > http://en.wikipedia.org/wiki/ZFC > Axiom of infinity: There exists a set x such that the empty set is a > member of x and whenever y is in x, so is S(y). Which is actually the construction of the ordinals. Right? Suppose we write 0 = { } then 1 = { { } } = { 0 } and 2 = { { } , { { } } } = { 0, 1 } and 3 = { { }, { { } }, { { }, { { } } } } = { 0, 1, 2 } http://www.jboden.demon.co.uk/SetTheory/ordinals.html So the axiom of infinity says that you can get everything from nothing. This is contradictory to all laws of physics, where it is said that you pay a price for everything. E.g. mass and energy are conserved. Han de Bruijn |