An uncountable countable set
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From: Tony Orlow on 17 Oct 2006 22:03 MoeBlee wrote: > Tony Orlow wrote: >> MoeBlee wrote: >>> Tony Orlow wrote: >>>> No, set theory confuses the issue with its concentration on omega. >>> Oh boy, here we go again with "Set theory confuses...". Please just way >>> which axioms of set theory you reject and which ones you use instead. >>> >>> MoeBlee >>> >> Uh, what axioms of set theory are specifically involved in your "proof"? >> I don't remember a deduction from those axioms. Perhaps you could >> refresh my memory. > > My proof of what? > > MoeBlee > Regarding the vase-ball problem. Also, upon which axioms is the definition of cardinality based? ToeKnee
From: Tony Orlow on 17 Oct 2006 22:07 MoeBlee wrote: > Tony Orlow wrote: >> MoeBlee wrote: >>> P.S. >>> >>> I lost the context, but somewhere you (Orlow) posted: >>> >>> "ZF and NBG don't handle sequences or their sums, but only unordered >>> sets" >>> >>> Z set theory defines and proves theorems about ordered tuples, finite >>> and infinite sequences, and infinite summations and infinite products >>> and many other things like that. >>> >>> MoeBlee >>> >> Huh! But I thought sets were unordered. > > Without the axiom of choice, we prove the existence of certain > orderings on certain sets. With the axiom of choice, we prove that > every set has a well ordering. > >> If the theory of infinite series >> is derived from set theory, > > I don't say that historically it is derived from set theory. But > infinite series, such as one finds in ordinary calculus and real > analysis are definable in set theory and the theorems about them can be > proven from the axioms of set theory. Theorems like, an infinite series with alternating positive and negative terms can be validly rearranged to have a sum as large or small as one likes? :) Sorry, I don't buy it. > >> how come they seem to contradict each other >> here? > > A set theoretic explication of an infinitary thought experiment (which > is not an explication that occurs IN set theory) differs from your own > explication of that thought experiment. That is not a contradiction > between set theory and any theorem regarding infinite series. Depends which assumptions you assume regarding such series. > >> I don't recall a derivation or proof of the empty vase from the >> axioms of set theory. > > Well, duh. Haven't people already made the point as clear as it can > possibly be made that there are no vases in set theory? But anyway, > even though there are no proofs about vases in set theory, you don't > recall ANY proof in set theory, since you've never studied a single > one. > > MoeBlee > If you want to ask which axioms I don't like that lead to my disagreeing with your conclusion, then state it as derived from the axioms, and I'll tell you. It's the Zeno machine that's the problem. You don't have an axiom for that, really, do you?
From: David R Tribble on 17 Oct 2006 22:14 Tony Orlow wrote: >> What is sum(n=1->oo: 9)? > Alan Morgan wrote: >> I think you actually mean, what is 10-1+10-1+10-1.... >> >> It was recognized long before Cantor that there isn't a simple answer to >> that question. > Tony Orlow wrote: > There is if you prohibit rearranging the terms to change the relative > frequencies of the two terms. Group all you like without rearranging. > This series is (+10-1)+(10-1)+(10-1)+... So you're saying that s = (10 - 1) + (10 - 1) + (10 - 1) + ... is just s = 9 + 9 + 9 + ... But surely then s = (10 - 1) + (10 - 1) + (10 - 1) + ... is exactly the same as t = 10 + (-1 + 10) + (-1 + 10) + ... (notice the "relative frequencies" of the 10's and 1's are the same), which is just t = 10 + 9 + 9 + ... So since the terms of s and t have the same relative frequencies, we conclude that s = t. Right, Tony?
From: David R Tribble on 17 Oct 2006 22:21 Han de Bruijn wrote: Virgil wrote: > 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. Huh. So what is the mass of a pair of braces? How many ergs are expended in adding a pair of braces around a set of objects?
From: Tony Orlow on 17 Oct 2006 22:32
imaginatorium(a)despammed.com wrote: > Tony Orlow wrote: >> imaginatorium(a)despammed.com wrote: >>> Tony Orlow wrote: >>>> imaginatorium(a)despammed.com wrote: >>>>> Tony Orlow wrote: >>>>>> imaginatorium(a)despammed.com wrote: >>>>>>> Tony Orlow wrote: >>>>>>>> David Marcus wrote: > > <snip> > >>> Suppose I define the following function, referring to sliver-1, which >>> is the area between y=-2/x and y=-1/x for x<0. ("sleight" stands for >>> 'sliver height', not 'sleight of hand'...) >>> sleight(x) = -2/x +1/x for x<0; 0 elsewhere >> Uh huh. For x<0 as opposed to x>=0. No declared point of discontinuity >> there.... > > OK. Let's see if it's possible to understand what, if anything, you > mean by "function". > > Do you agree that the graph of y=-1/x for x < 0 is one lobe of a > hyperbola? > > Do you agree that the graph of y=-2/x for x < 0 is one lobe of another > hyperbola? > > Do you agree that in the unbounded x-y plane, these two lobes define a > "sliver", a boomerang-shaped area, extending indefinitely 'left' and > also extending indefinitely 'upward' (using these directional terms in > the sense of looking at a conventional graph)? > > Do you agree that for any simply-connected area (think that's the right > term) within the x-y plane we could consider the function that maps x > to the vertical measure* of the area at the particular x value? > > ( * a term I've made up. If you don't understand ask; if anyone knows a > proper word, please tell me) > > By way of a different example, consider the circle radius 1, centre (0, > 0), and find its 'height()' function. For any value of x outside the > range (-1, +1), the vertical measure is zero, because, obviously, the > circle only extends horizontally from -1 to +1. Within that range, the > vertical measure is equal to the height of an ellipse centred on the > origin, of width 2 and height 4, so (if I calculate correctly) the full > function is given by: > > height(x) = 2 * sqrt(1-x^2) for -1 < x < 1 > height(x) = 0 otherwise > > Please tell me: is this a function? Is it a continuous function? If so, > does it have a "declared discontinuity"? > > You might like to do the same for the function height() of a rectangle > diagonal from (0,0) to (3, 57). > > If you somehow claim that there _is_ no function representing the > height of the sliver at a particular value of x, you really need to > give us your definition of "function". > > If you agree there is such a function, why not try to write it down? > > You may or may not agree that this function is discontinuous - in any > event, please explain whether my description above of the hyperbola > lobes and the "sliver" has already included a "declared discontinuity". > If not, does that mean there might be different ways of writing the > same function, possibly some including a "declared discontinuity", > others not. Your examples of the circle and rectangle are good. Neither has a height outside of its x range. The height of the circle is 0 at x=-1 and x=1, because the circle actually exists there. To ask about its height at x=9 is like asking how the air quality was on the 85th floor of the World Trade Center yesterday. Similarly, it makes little sense to ask what happens at noon. There is no vase at noon. Additionally, when it comes to 1/x and functions of that ilk, the discontinuity at x=0 is not declared, but surmised on the basis that oo<>-oo. When the number circle is considered, and oo as the inverse of 0 considered to be both positive and negative, your discontinuity disappears, and your sliver in the upper left and lower right can be said to be connected at a point above and below, and a point to the right and left, at oo. Just a conceptual picture to amuse you. > >>> <snip> > >>> On the contrary, the process *at noon* is completely well-defined. >> Then how come no one can say what happens "at noon", which doesn't >> happen "before noon"? You're stretching to the point of breaking. > > No, I'm slightly lost. Don't understand the relation between the > comma-separated clauses of the first sentence. What happens at noon doesn't happen before. Nothing happens at noon. Before noon there are balls. In between noon and "before noon" somethign changed. I don't understand it either. > >> A >>> ball is inserted in the vase or removed from the vase only at a time >>> that is -1/n for some pofnat n. There is no pofnat m such that -1/m = >>> 0. Therefore no ball is either inserted or removed at noon. (This >>> really is elementary, you know.) >> Well then, nothing can change at noon that was true at every time before >> noon, when there is a growing positive number of balls in the vase. What >> changed at noon? > > Every time *before* noon was a time at which a ball was still to be > removed. Give me a (real, genuine, numerical) value of a time before > noon, and I will give you the number of a ball that has yet to be > removed from the vase. *At* the time noon, there is no ball that has > yet to be removed. (In normal logic, this means the vase is empty.) Every specific ball n is removed at a specific time before noon. At every one of those times, balls n+1 through 10n remain. > >> Nothing, since nothing happened "at" noon. > > No ball movement, no. Anything besides ball movement? > >> So, the >> number of balls in the vase at noon is growing is growing, > > No, the number of balls at any time, however small, *before* noon is > growing. And the smaller the time before noon, the crazier it is > growing. > Right. That's what I was saying in the rest of that sentence. > >>>> ... since the distinction between >>>> iterations disappears. How do you know there are countably many >>>> iterations, and not some uncountably number? You don't. >>> Of course I do. >> Ah, Zeno told you.... > > Try being less obnoxious. In the end you might make yourself look less > silly. Says the pot to the kettle. <snippety snoppity snoop> > >> The problem explicitly says balls with natural numbers >>> (pofnats) on them. >> And this set ends where? Nowhere. Well, actually, at noon. Isn't that a >> tad artificial, and somewhat contradictory? > > Does the infinite series (gosh, it's amazing, but I believe there are > tiny fragments of mathematics you have actually managed to grasp) 1 + > 1/2 + 1/4 + 1/8 + 1/ |