An uncountable countable set
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From: Virgil on 2 Oct 2006 14:59 In article <1159784963.257471.99490(a)i42g2000cwa.googlegroups.com>, Han.deBruijn(a)DTO.TUDelft.NL wrote: > Virgil wrote: > > > The problem as I recall it was this: > > > > Given an infinite set of balls numbered with the infinite set of > > naturals and an "infinitely large" initially empty vase, and a positive > > time interval in seconds, t, and a small positive time interval in > > seconds, epsilon ( much smaller than t/2). > > (1) At time t before noon balls 1 through 10 are put into the vase and > > at time t - epsilon before noon ball 1 is removed. > > (2) At time t/2 before noon balls 11 through 20 are put into the vase > > and at time (t - epsilon)/2 before noon ball 2 is removed. > > ... > > (n) At time t/2^(n-1) before noon balls 10*(n-1)+1 through 10*n are put > > in the vase and at time (t-epsilon)/2^(n-1) before noon, ball n is > > removed. > > ... > > > > The question is what will be the contents of the vase at or after noon. > > There is no noon in this problem. There is in mathematics. The problem does not exist in physics at any time. > Or: the problem is _undefined_ at noon. It is in mathematics. > > > Of course, there can be no physical analog of this thought experiment, > > Not a single thought experiment has a "physical analog" as you recall > it, but there are _many_ highly abstract thought experiments which can > nevertheless be attached to a physically meaningful arrangement. > In this problem, things remains meaningful through all times < noon. Not so. When the problem states that before the timing begins all those infinitely many balls must exist, there is no physical analog at any time. > > > but in mathematics the question has a clear answer. > > You think so. But it's just an illusion. All math is illusion in that same sense. Everything mathematical is purely mental, so can only match "reality" by some sort of analogy, not by any actuality.
From: Tony Orlow on 2 Oct 2006 15:00 MoeBlee wrote: > Han de Bruijn wrote: >> Constructively valid proof. Intuition precedes axioms. A mathematician >> is like an architect who builds his mathematics. Take a look at some >> material concerning intuitionism and constructivism in the first place. > > What particular books or articles do you endorse? > > And what passages do you cite as claiming that constructivism is > incompatible with axiomatic theory? > > MoeBlee > Good questions. :)
From: Tony Orlow on 2 Oct 2006 15:02 David R Tribble wrote: > Virgil wrote: >>> Except for the first 10 balls, each insertion follow a removal and with >>> no exceptions each removal follows an insertion. > > Tony Orlow wrote: >> Which is why you have to have -9 balls at some point, so you can add 10, >> remove 1, and have an empty vase. > > "At some point". Is that at the last moment before noon, when the > last 10 balls are added to the vase? > Yes, at the end of the previous iteration. If the vase is to become empty, it must be according to the rules of the gedanken.
From: Tony Orlow on 2 Oct 2006 15:07 Virgil wrote: > In article <1159784963.257471.99490(a)i42g2000cwa.googlegroups.com>, > Han.deBruijn(a)DTO.TUDelft.NL wrote: > >> Virgil wrote: >> >>> The problem as I recall it was this: >>> >>> Given an infinite set of balls numbered with the infinite set of >>> naturals and an "infinitely large" initially empty vase, and a positive >>> time interval in seconds, t, and a small positive time interval in >>> seconds, epsilon ( much smaller than t/2). >>> (1) At time t before noon balls 1 through 10 are put into the vase and >>> at time t - epsilon before noon ball 1 is removed. >>> (2) At time t/2 before noon balls 11 through 20 are put into the vase >>> and at time (t - epsilon)/2 before noon ball 2 is removed. >>> ... >>> (n) At time t/2^(n-1) before noon balls 10*(n-1)+1 through 10*n are put >>> in the vase and at time (t-epsilon)/2^(n-1) before noon, ball n is >>> removed. >>> ... >>> >>> The question is what will be the contents of the vase at or after noon. >> There is no noon in this problem. > > There is in mathematics. The problem does not exist in physics at any > time. So, the limit "exists". Is that what you're saying? There's "the infinite case"? > >> Or: the problem is _undefined_ at noon. > > It is in mathematics. It's undefined? I guess you meant "not". Your vernacular is purposely obtuse. >>> Of course, there can be no physical analog of this thought experiment, >> Not a single thought experiment has a "physical analog" as you recall >> it, but there are _many_ highly abstract thought experiments which can >> nevertheless be attached to a physically meaningful arrangement. >> In this problem, things remains meaningful through all times < noon. > > Not so. When the problem states that before the timing begins all those > infinitely many balls must exist, there is no physical analog at any > time. The question is not whether they "exist" somewhere, but whether they "exist" inside the vase. >>> but in mathematics the question has a clear answer. >> You think so. But it's just an illusion. > > All math is illusion in that same sense. Smoke and mirrors. As if! > > Everything mathematical is purely mental, so can only match "reality" by > some sort of analogy, not by any actuality. "Purity" is velocity of c, or zero volume for finite substance. It's unattainable, practically. Tony
From: Virgil on 2 Oct 2006 15:20
In article <45210227(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Virgil wrote: > > In article <45205fa9(a)news2.lightlink.com>, > > Tony Orlow <tony(a)lightlink.com> wrote: > > > >> Virgil wrote: > >>> In article <45203919(a)news2.lightlink.com>, > >>> Tony Orlow <tony(a)lightlink.com> wrote: > >>> > >>> > >>> > >>>>> Since ordinals are, by definition, well ordered, they cannot contain any > >>>>> endlessly decreasing sequences, which TO's models require. > >>>> Neither can the reals. > >>> How about the set of negative integers? > >>> How is that not an endlessly decreasing sequence of reals? > >> The origin is at a finite location. Order starts from the bottom, if > >> "decreasing" has any meaning. > > > > The set of negative integers has no "bottom". > > > > TO seems to be changing his tune when it is used against him. > > > > According to TO every set of numbers has a natural order, and it is > > within that natural order that we must view it, but now he wants to > > reject the natural order because it runs counter to another of his > > claims. > > > > TO blows hot and cold with the same breath. > > I'm saying the if you iterate the negative integers starting at 0, in > that order, there is no infinite descending sequence. But that is not their "natural" order, and TO elsewhere insists that we follow natural orderings. > On the other hand > I don't know why I said "neither can the reals". In any case, the only > way the ordinals manage to be "well ordered" is because they're defined > with predecessor discontinuities at the limit ordinals, including 0. > That doesn't seem "real" In what sense of "real". There are subsets of the reals which are order isomorphic to every countable ordinal, including those with limit ordinals, so until one posits uncountable ordinals there are no problems. > and the axiom of choice aside, I don't see > there being any well ordering of the reals. The point is that no one can see it even if, given the AC, it is there. > The closest one can come is > the H-riffic numbers. :) Which have long since been shown not to be anything like well ordered. |