An uncountable countable set
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From: Virgil on 2 Oct 2006 18:35 In article <45216233(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > 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. But the "rules of the gedanken" specifically forbid any "last 10 balls", by specifying an ENDLESS sequence of 10 ball additions.
From: Virgil on 2 Oct 2006 18:49 In article <45216360(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > 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". There is a function, f, from the reals, R, representing time in suitable units to the set of von Neumann naturals,N = {0,1,2,3,...}, such that f(t) = the number of balls in the vase at time t, for all t. This function has an isolated jump discontinuity at each time at which any balli is inserted or removed from the vase and and at noon there is a condensation point of such discontinuities and the function is unbounded to the left in every neighborhood of noon, but is still right-continuous at noon. Is that what you're saying? There's "the > infinite case"? There is the function case, as described above. > > > > 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. Whoever said math had to be practical? Einstein, for one, expressed great surprise at how practical something so inherently impractical as mathematics kept turning out to be.
From: David R Tribble on 2 Oct 2006 19:28 Virgil wrote: >> 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. > Tony Orlow wrote: > I'm saying the if you iterate the negative integers starting at 0, in > that order, there is no infinite descending sequence. 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", and the axiom of choice aside, I don't see > there being any well ordering of the reals. The closest one can come is > the H-riffic numbers. :) Hardly. The H-riffics are a simple countable subset of the reals. Anyone mathematically inclined can come up with such a set.
From: David R Tribble on 2 Oct 2006 19:45 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. > David R Tribble wrote: >> "At some point". Is that at the last moment before noon, when the >> last 10 balls are added to the vase? >> > Tony Orlow wrote: > 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. The rules don't mention a last moment. The rules state that for each point in time, at 1/2^n seconds prior to noon, 10 balls are added to the vase, and then the ball that was previously inserted earlier than all the others is removed. (I.e., at time 1/2^n, balls 10n+1 thru 10n+10 are added, and ball n is removed.) Therefore the rules stipulate that balls are added and removed at each moment 1/2^n sec before noon, for each n = 1,2,3,... . By assuming that there is some last moment when the vase is emptied, or when the last ball is added, or whatever, you are assuming that there is a largest n. That's your assumption, because it's not mentioned in the rules. But it's an unwarranted assumption, by the simple fact that there is no largest n (as you yourself have proclaimed many times). [Personally, I think the problem is a bit simpler if only two balls are added and one removed at each step. But, whatever.]
From: Ross A. Finlayson on 2 Oct 2006 20:35
Tony Orlow wrote: > Ross A. Finlayson wrote: > > Hi Ross - > Nice to see you. I hope you don't mind my adopting the Finlayson Numbers > in my IFR sort of way. Cheers. > > > Tony Orlow wrote: > >> Virgil wrote: > >>> In article <45201554(a)news2.lightlink.com>, > >>> Tony Orlow <tony(a)lightlink.com> wrote: > >>> > >>>> Virgil wrote: > >>>>> A set is a container, and is not one of the objects that it contains. > >>>> It is nothing more or less than its contents. > >>> It is determined uniquely and entirely by its contents, as stated in the > >>> axiom of extentionality. > >> So we agree. There is nothing besides the members. > > > > Also, the members are only sets. > > Each member is a set? Well, yes, a set of property values which > distinguish the members. > > > > > I feel that I have been fair presenting construction of reason that > > enable a rational person to make their own decision that, for example, > > forceing uses a universal ordinal, > > I don't know much about "forcing", but it sounds kind of aggressive and > not very nice. ;) > > the generic extension of N contains > > no elements not in N yet bijects to R, > > Sequential form can be applied, I think, to any structure, if "forced". > (Is that what that means?) > > the Borel/Combinatorics impasse, > > and those as basically non-controversial. > > The impasse being over...what? Thanks in advance. :) > > > > > I see some truths in some of the posters with alternative opinions, > > also, I'm quite familiar with the standard viewpoint. I use my > > viewpoint, which I develop in part myself. > > How dare you even feign that right? The nerve! > > > > > These "infinities", as they are, are on the one hand various, as > > infinite more various than any finite set could ever be, on the other > > hand they each share certain properties and it seems at root they are > > each irregular. > > Not just regular old numbers? Well I suppose not... > > > > > Then, where some "least" infinite set has a given construction, there > > are what are to most implicit aspects of that construction, implicit in > > the sense of being true and not just unstated but known, to somebody, > > thus true thus implicit. > > Like the set of multiples of neN, as n->oo? That would be one definition > of the least, beyond the simplest. > > > > > There is no universe in ZF where there are only sets, and not all of > > them. There is no set of all sets, nor set of all ordinal (nor > > cardinal) numbers as sets, no collection of a variety of other things > > that are necessary for the establishment of certain formal arguments, > > in ZF. > > There are objects, properties, and relations. Is there else? > > > > > If you talk about and use those things, as most do with for universal > > quantification over sets, then thus necessarily ZF is not sufficient > > and is at once contradictory. > > With self, or reason? > > > > > Oh, I'm not a crank. > > I am. :) > Does that mean I'm wrong? > Define "crank". > I think it means someone who makes the entrenched "cranky". > > I think Goedel tells you the null axiom theory is > > the only possible theory, where if it's inconsistent or incomplete it's > > not A theory. > > > > Ross > > > > According to most, that would mean, in the "light" of Godel, that "there > is no theory". Is there a spoon? > > Have nice continuum, > > Tony Oh, I don't mind. You mention of Inverse, some years ago I was trying to figure out if a system like ZF had something along the lines of axiom of inverse. You might want to frame your inverse function rule in terms of a space of functions, for example the Hilbert space leading to the L^infinity with linear and non-linear differential operators and so on. In representing numbers as sets, or for that matter other mathematical constructs as sets, if in the set theory there are those objects there are all the rules about them mechanistally as sets. There are only sets in set theory, if it's not a set, it's not in the set theory. It's easy to say, "here's a set of numbers, here's how they're defined and operate", but in a set theory that entire self-contained description must be in the set theory, for example along the lines of a type theory, where various items fit into various categories. So, having the empty set be zero might be very useful transitively but the number zero is not the empty set. In some cases considering that it is leads to some useful results from vacuity, almost all of which are meaningless. (Quantify over sets: not a set in regular "set theories".) Set theory as applied to the finite is almost totally non-controversial. It is only in the infinite that for whatever reason various researchers and discutants have mutually controversial and various interpretations of those systems interpreted as formal mathematics. For example, half of the integers are even. Well-order the reals. Oh, forceing/forcing is just a word that basically means fiat, to basically axiomatize a completed infinite set. Borel-vs-Combinatorics refers to Borel's set being almost everywhere in the reals, and Combinatoricist's set being almost everywhere in the reals, yet they are disjoint, as I discuss in "Factorial/Exponential Identity, Infinity." Ross |