An uncountable countable set
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From: David Marcus on 2 Nov 2006 03:38 imaginatorium(a)despammed.com wrote: > cbrown(a)cbrownsystems.com wrote: > > Which is an absurd thing to say about a time; so either noon cannot > > properly be said to be an actual time at all ("noon doesn't > > exist/occur/happen"); or else something not specified in the problem > > actually does happen at noon (such as the removal/addition of an > > infinite number of infinitely labelled balls); or else the stopping of > > the frenzy and its cause both occur at a time which is strictly between > > all times before noon and noon itself (in which case, nothing happens > > at all /at/ noon; instead, something happens at a time which is > > indistinguishable from, but not the same as, noon). > > > > This is where/how Tony leaves the rails. > > > > The examples you give above of f(x) and g(x) are irrelevant; because > > there is no specified "physical action" (ball removal or insertion) in > > those examples; so the problem of "happenings" and "causes" is not an > > issue; f and g are simply distractions from the original problem. > > You might be missing my point slightly. I'm trying to point out that > the distinction between f() and g() is not one that could have a > physical interpretation. In a sense the mathematics overspecifies, > meaning that the use of functions over the reals is actually not quite > sufficiently abstracted. The Mathematics is more flexible than the English. However, we can choose a convention for how to interpret the English as Mathematics. For example, we could agree that if we say the ball is removed at a certain time, then we mean it is out at that time. -- David Marcus
From: Virgil on 2 Nov 2006 03:42 In article <1162453868.081267.272020(a)k70g2000cwa.googlegroups.com>, imaginatorium(a)despammed.com wrote: > Tony Orlow wrote: > > Mike Kelly wrote: > > > Tony Orlow wrote: > > >> Randy Poe wrote: > > <a lot of stuff> > > > >> I have seen and understood your argument. It "makes sense". It seems > > >> logical. All balls are inserted and removed before noon, the same set, > > >> it would seem. But the method of proof is not correct. > > > > > > Why? What are you basing this assertion on? That you don't agree with > > > the conclusion? > > > > Yes. I am exploring exactly why. This is just another "la(rge)st finite" > > argument. It doesn't "add up". > > Tony, many of us have heard this from you many times now, often > accompanied by your mangled version of some mantra or other (I really > mean a mantra, as in Om mani padme hum, not the popular meaning of > "mantra"). I am correct in characterising this by saying that you do > indeed accept and agree that there is no largest pofnat, no last > natural number (under the normal definition), yet though you accpt it > is true, you believe it is invalid to deduce anything from it. Is that > really the case? > > Elsewhere, you just referred to " the twilight zone between zero and > the smallest positive real". I suppose again you agree that there is no > smallest positive real (because if q is the smallest positive real, > then 0 < q/2 < q contradiction)? Yet you feel it makes sense to talk of > a "zone" between one thing (0) and another thing that doesn't exist? > > Can you perhaps understand the difficulty most of us have with > following your reasoning? > > Brian Chandler > http://imaginatorium.org Some of us, indeed, refuse to allow that it is reasoning at all in any mathematically comprehensible sense.
From: David Marcus on 2 Nov 2006 04:18 cbrown(a)cbrownsystems.com wrote: > David Marcus wrote: > > cbrown(a)cbrownsystems.com wrote: > > > David Marcus wrote: > I will say: at least a few posters "got it" in regards to the somewhat > related staircase problem (which I remaked on in a separate post), and > were kind enough to let me know that they had changed their "mental > images" accordingly. That's good to know. > > > Sure they do - at any /given/ time before noon, there's at least as > > > much water in the "start full" vase as the "start empty" vase, and then > > > some to boot; so that's more water. Not a hard mental image to > > > construct at all. > > > > See below... > > > > > > In one case, you've got > > > > a vase filling with water. In the other, you've got a full vase that is > > > > emptying. Clearly the vase that is emptying can't always have more water > > > > than the vase that is filling! > > > > > > But that is a /different/ mental image - the former was a mental image > > > /at/ a time; the latter is a mental image of /during some times/. Which > > > image we find "more compelling" is something that we learn to choose > > > (whether by some logical method, or some other method). > > > > Given a puzzle, the first step is to produce a mental image. > > ... i.e., a single, coherent mental image, consistent with the > "features" of the problem as you accept them... That's the best way to do it. But, I fear that many people simply accept whatever picture is initially conjured up in their brain. > > Then we > > look at the image to get the answer to the problem. > > .. as part of an iterative process of distinguishing "features" of the > problem which we deem relevant, yes. In this sense, what you are saying > is no different than the task of asking "here is an oil painting; what > is it a picture of?" The iterative process requires being able to distinguish the features of the problem implicit in the words and comparing them to the features in the picture. If the person can't do the former, then they are reduced to simply looking at the oil painting as it sprang full blown in their mind. All of Tony's answers seem to be consistent with a simple mental picture. I haven't seen any evidence that he has modified this picture as the discussion has gone along. > > Once we have the > > mental image, we don't go back to the problem. The mental image is all > > the info we have! > > "There are 10 differences between the picture on the left and on the > one on the right. Can you find them all?" > > As an aside, I note that most mathematicians with whom I have conversed > regarding this subject either class themselves as those who > predominantly operate by imagining some sort of pseudo-physical "space" > wherein the implications of A, B, and C can be logically /visualized/ > as "viscerally true"; versus those who consider the excercise to be > predominantly one of /language/: the puzzle is to find statements A, B, > and C such that semantically A follows from B follows from C; so since > C, therefore A. (N, B.: such self-identification is not at all > correlated to what I would call "mathematical ability"). > > I suspect you are firmly in the former camp. I'm not sure. I'm not sure what you mean by "viscerally true" or how "logically visualized" is different from "semantically following". > I find myself > predominantly there, but (and here I imply a generality from my > experience) at the same time I feel that the evolution of my concept of > 'viscerally true' has been directly affected by considerations of the > other camp's POV. I think I've lost you. I wasn't trying to explain how a mathematician approaches the problem. I was trying to explain how someone like Tony approaches the problem, and why, although his answers seem illogical, they can be explained by a simple model of how his mind works. > The unfortunately outdated phrase "dialectic" comes to mind. > > Certainly, when I add 147 to 63, I am not creating a "mental image" of > 210 "things". I simply apply the clear-cut, essentially > linguistic/mechanistic rules regarding what the combination of the > purely linguistic expressions "147", "added to", and "63" are supposed > to "mean" in that contraposition. > > On the other hand, when I consider the (purely linguistic/mechanistic) > statement "G is a divisble group", it clearly evokes certain > "visualisations" of what such a group would "look like". For example, > any sort of pseudo-physical space somehow filled with individual > ball-like "things" seems completely inappropriate; because G must be > much more "gooey" than that. > > So I agree that mental images are an important (nay, critical) part of > conceptualising a problem; but I disagree that this action of > "conceptualising" can be examined independently of some sort of > essentially linguistic/mechanical process of decoding "what do these > words mean in contraposition?". Sure. That's the right way to do it, if you want to be consistent/logical/mathematical. > > For the original problem, the mental image is an empty vase that fills > > by noon. For the modified problem where all the balls are put in first, > > then balls are removed, the mental image is a full vase emptying by > > noon. > > > > So, we have two mental images. Now, you ask whether one vase always has > > more water than the other. Obviously not! One starts full and becomes > > empty, while the other starts empty and becomes full. > > > > As for how Tony chooses a mental image, he reads the problem and accepts > > whatever mental image forms in his mind. Logic is not used. > > In this case, the statement "logic is not used" cannot itself, > logically, be refuted :). I think it could be refuted by experiment. It is a conjecture based on data. > > Now, if you could ask a problem which would generate a mental image of > > two vases at a particular time, maybe you could get a different answer. > > > > > > > But that is not a problem; because "those are different situations - > > > > > it's an obvious obfuscation of the original problem". > > > > > > > > > > Consistency - it's not everyone's cup of tea! > > > > > > > > If you aren't using logic, then you don't even understand what > > > > consistency is. > > On further reflection, I would propose that concepts such as > "consistency" arise naturally in human
From: Ross A. Finlayson on 2 Nov 2006 10:26 David Marcus wrote: > Ross A. Finlayson wrote: > > David Marcus wrote: > > > Mike Kelly wrote: > > > > Tony Orlow wrote: > > > > > Where are the iterations mentioned there? You're missing the crucial > > > > > part of the experiment. By your logic, you could put them in in any > > > > > order and remove them in any order, and when you say both processes are > > > > > done, nothing's left, but that's BS. It ignores the sequence specified. > > > > > This is just a distraction. > > > > > > > > Yes, if you insert and remove exactly the same balls then you get the > > > > same result when you're done, no matter what order you did it all in. > > > > Why is that BS? It seems blindingly obvious. > > > > > > > > But I forgot, you think that if you shift all the insertions 1 minute > > > > further back in time, you DO get an empty vase at noon, right? I really > > > > don't understand how your mind works. > > > > > > Try the mental picture with the water. We fill it up, then we start > > > letting it run out. No reason all the water shouldn't empty out of the > > > vase by noon. > > > > > > -- > > > David Marcus > > > > Hi, > > > > There are only sets in ZFC. When people talk about the real numbers in > > ZFC, it is as a construction of sets in ZFC has been found to be > > isomorphic in a relatively strong sense to the real numbers. > > Obviously, mathematicians want to find the best representation of > > everything the real numbers are or must be by their nature. > > > > Then, that gets into that some feel that mathematics can't assume what > > it sets out to prove. That's reasonable. By the same token, the real > > numbers have many, many roles to fill, and some of them have, for > > example, in the projectively extended real numbers, points at > > plus/minus infinity. > > > > It is relatively standard to define a real number as a Dedekind cut or > > Cauchy sequence, which are basically defined in terms of sequences of > > rationals, which resolve to generally the familiar decimal > > representation which is adequate in finitely expressing rationals, and > > with radicals, algebraics. > > > > If a Dedekind cut is as was recently stated some "initial segment" of > > the rationals, I wonder to what ordering that pertains. > > > > Consider how that is to describe an irrational number. Basically the > > sequence of elements is to converge towards the number. There's an > > irrational less than one and greater than .9, in decimal, less than one > > and greater than .99, less than one and greater than .999, etcetera. > > The general consensus here is that .999... = 1, yet for each > > .999...999, there is an irrational between it and one. So, does that > > not seem that there are irrationals unrepresentable via > > Dedekind/Cauchy? It would seem that certainly as the irrational is > > some finite distance from 1 that it would be between .999...998 and > > .999...999, and between that irrational and one are infinitely many > > more numbers forever, there always exist irrationals between .999...999 > > and 1, and, for Dedekind/Cauchy to represent them, they must have a > > unique representation. > > > > For no finite number of 9's or rep-units in binary can these > > irrationals in the diminishing remaining interval be represented, and > > for any infinite number of rep-units the result is said to be one. So, > > either between the finite and infinite those values are represented, or > > they're not, and due to the completeness of the reals, if they're not, > > then Dedekind/Cauchy, the standard set-theoretic method to construct > > real numbers, is insufficient to construct some real numbers. > > > > For any it's so, for all it's not, or vice versa. Don't worry I've > > heard of the transfer principle. > > > > Consider the representation of rational numbers, for example 9/10. > > That would be .9, .90, .900, ..., .9(0): 9/10's. There is no last > > element of that list, .9 could be an initial segment of a sequence for > > 9/10's or any irrational between .9 and 1.0, as above. The initial > > sequence .9, .90 could be an initial sequence for any irrational > > between .900 and .91. The initial segment .900 could be an initial > > sequence for any number between .9000 and .901. > > > > .90000 <= x <= .9001 > > .900000 <= x <= .90001 > > .9000000 <= x <= .900001 > > .90000000 <= x <= .9000001 > > .900000000 <= x <= .90000001 > > .9000000000 <= x <= .900000001 > > .90000000000 <= x <= .9000000001 > > .900000000000 <= x <= .90000000001 > > .9000000000000 <= x <= .900000000001 > > .90000000000000 <= x <= .9000000000001 > > .900000000000000 <= x <= .90000000000001 > > ... > > > > As the number of zeros diverges, the diminishing interval goes to zero, > > where the lower and upper bounds are a and b, lim n->oo b-a = 0. For > > any finite iteration there are obviously a continuum of elements that x > > could be, so for a value, x, to not obviously be among a continuum of > > possible values there must be infinitely many iterations. > > > > Keep in mind that there are printed counterexamples to standard real > > analysis with a least positive real. > > > > Obviously the ground around .999... vis-a-vis 1 is very well turned, > > that's the point, to some extent we're talking about significant > > ephemera. > > > > Look at the 1 on the right side above. Where does it go? > > > > .90001 <= x <= .9002 > > .900001 <= x <= .90002 > > .9000001 <= x <= .900002 > > .90000001 <= x <= .9000002 > > > > Standardly, equal. > > > > .90001 <= x <= .901 > > .900001 <= x <= .9001 > > .9000001 <= x <= .90001 > > .90000001 <= x <= .900001 > > > > .90001 <= x <= .91 > > .900001 <= x <= .901 > > .9000001 <= x <= .9001 > > .90000001 <= x <= .90001 > > > > .90001 <= x <= .91 > > .900001 <= x <= .901 > > .9000001 <= x <= .9001 > > .90000001 <= x <= .90001 > > > > .90001 <= x <= .91 > > .900001 <= x <= .91 > > .9000001 <= x <= .901 > > .90000001 <= x <= .9001 > > > > .90001 <= x <= .91 > > .900001 <= x <= .91 > > .9000001 <= x <= .91 > > .90000001 <= x <= .901 > > > > .90001 <= x <= .91 > >
From: Randy Poe on 2 Nov 2006 10:58
Lester Zick wrote: > On 31 Oct 2006 11:54:23 -0800, "Randy Poe" <poespam-trap(a)yahoo.com> > wrote: > > > > >Lester Zick wrote: > >> On 30 Oct 2006 19:47:06 -0800, "Randy Poe" <poespam-trap(a)yahoo.com> > >> wrote: > >> > >> > > >> >Lester Zick wrote: > >> >> On 30 Oct 2006 10:46:23 -0800, "Randy Poe" <poespam-trap(a)yahoo.com> > >> >> wrote: > >> >Why, do you think that there's a least integer? What, > >> >around -1000? > >> > >> If 1 is an integer then 1 would be least would it not? > > > >No, 0 is an integer with the property 0 < 1. > > Thanks for the heads up. What makes you think it's true? If you don't accept 0 < 1, we really aren't talking about the same mathematics at all. > >> >Proof: > >> >A least member x0 would have the property that > >> >x0 <= x for all other members x. > >> > > >> >Let x0 be any integer. x0-1 is also an integer, which is <x0. > >> >Thus x0 can't be a least member. > >> > > >> >Similar argument for x0 being any real. > >> > >> Not if the integers under discussion are positive: > > > >When we say "the least member of the set of integers" they > >are not all positive, since the set of integers is not all positive. > > If you say so. It's whoever says so. The phrase "the set of integers" includes everything that is an integer, whoever utters that phrase. > >When we say "the least member of the set of POSITIVE > >integers", they are all positive. > > No lie? Yes, the phrase "the set of positive integers" refers to everything which is not only an integer, but is also positive. > >> is 1 an integer? Is it positive or negative? > > > >It is a positive integer. > > > >But it isn't the smallest member of the set of integers. > > It's also not imaginary. Uh, right. Was that supposed to be relevant? > >> It certainly isn't negative unless so stated. > >> Ergo it is not negative nor are integers negative unless explicitly > >> qualified. You make one propositional logic error then try to sneak in > >> an implicit qualification to justify your original error. > > > >Eh? How is it an "implicit qualification" to mean "the set > >of integers" when the set specified is "the set of integers"? > > I don't recall as the original said anything about sets. > > >Wouldn't be adding the word "positive" when it is left out > >be considered adding a qualification that wasn't present? > > Beats me. OK, then I'll help you out. If somebody says "the set of integers" and you interpret this to mean "everything which is an integer and which is also positive" then you have added "which is also positive" to the original qualifications. > >How exactly does the "set of integers" have "implicit > >qualifications" that "the set of positive integers" doesn't? > >What additional restriction is added to Z+ to make it Z? > > Ask somebody who cares. Ah, my mistake. I thought your continued discussion of this topic meant that you cared, or at least cared about what you were writing. - Randy |