An uncountable countable set
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From: Lester Zick on 25 Oct 2006 18:29 On 25 Oct 2006 14:00:15 -0700, "MoeBlee" <jazzmobe(a)hotmail.com> wrote: >Lester Zick wrote: >> You've already rejected the alternative I offer although I don't know >> why. > >What alternative? You referred me to a post you made. That post doesn't >include mathematics for real analysis. It doesn't? My mistake. I thought the basic consideration underlying real analysis was truth. But it turns out that modern mathematical set analysis just prefers assumptions of truth to truth itself. Oh well. >> Well see the problem here, Moe, is that you and standard set analysis >> have no demonstrable basis for truth. So "untrue" things cannot be >> said of a standard analytical method for sets which has no method of >> demonstrating "truth" except in reference to its own assumptions of >> truth the first among which would seem to be that what it assumes is >> true is true. > >I'm not talking about whether the axioms and theorems are true. I'm >talking about whether a theorem is indeed a theorem of a certain set of >axioms or what a particular formulation is or is not (not what it is >understood to say, but rather the question of the brute fact of just >what the formulation is as a sequence of symbols). Such questions, >again irrespective of the question of the truth of the axioms and >theorems, are ones that can be settled, at least in principle (and to a >great extent, in practice) by direct inspection of sequences of >symbols. Maybe what standard set analysts are really after is the determination of truth from the direct inspection of symbols.The problem is they use concepts like "true" as if they meant something self evidently true of the axioms and what they claim of the axioms without adducing any evidence of truth besides a lack of any inconsistency with the axioms. >> In other words the only standard of truth in standard >> set analysis would seem to be the lack of inconsistency with its own >> assumptions of truth. > >That is incorrect. The method of models defines 'true' in a way that is >not just reducible to consistency. See, Moe, here you doing the same thing again. You just go on to make an extraneous reference to some other "theory" of truth to justify what you can't define as true in absolute terms and then complain that others don't agree. Who cares what the "method of models" defines as true? You can't establish any reason anyone should care and yet you go on to claim that as justification for your own opinion on the subject. >> And when someone like me comes along and suggests the analytical >> techniques of standard set analysis are flawed you get all bent out of >> shape because you insist not that I accept individual assumptions but >> because I reject the techniques which standard set analysis insists >> are somehow "true" and if I don't I'm therefore saying things about >> standard set theory which are somehow "untrue". > >No, I call you on the mistatements you make about what the actual >formulations of set theory are and on your mischaracterizations of set >theory that accrue from your ignorance on the subject. What misstatements of set theory? What mischaracterizations of set theory? Technically there is no "set theory" because standard analysis of sets in modern mathematics can't be demonstrated true and only represents a series of problematic set analytic techniques. So how can I or anyone misrepresent something that isn't true to begin with? >I have little desire to convince anyone that the axioms and theorems of >set theory are true. And I welcome, as opposed to desiring to squelch, >alternative theories. I have a long range agenda of eventually learning >as much alternative mathematics as I can. But that does not entail that >I should abandon calling people such as you on misstating, >mischaracterizing, and utterly misunderstanding what the actual >formulations of set theory and mathematical logic are and calling >certain other people on the fact that their proposed theories are - for >the lack of a specified mathematical language, logic, primitives, and >axioms, and not even shown to be amenable to such specification - >lacking as rigorous or (in certain sad cases) even coherent >mathematics. So call me on it. Just don't try to demand I acknowledge the virtue of your views on mathematics and set analysis as revealed truth. >Hey, as to set theory and mathematical logic, IF you know ANYTHING >about it, then you should critique it to your heart's desire! But your >grumblings about set theory and mathematical logic are irrelevent since >they are not about set theory or mathematical logic, but rather about >what you, as a function of ignorance on the subject, merely IMAGINE set >theory and mathematical logic to be. So you want to argue standard set analytical techniques? Go right ahead. They just don't bear on the mathematical problem at hand which is the determination of truth in mathematically exhaustive terms. I have no interest in standard set analytical techniques unless they're false. ~v~~
From: stephen on 25 Oct 2006 18:31 David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: > stephen(a)nomail.com wrote: >> David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: >> > The vase problem violates Tony's mental picture of a vase filling with >> > water. If we are steadily adding more water than is draining out, how >> > can all the water go poof at noon? Mental pictures are very useful, but >> > sometimes you have to modify your mental picture to match the >> > mathematics. Of course, when doing physics, we modify our mathematics to >> > match the experiment, but the vase problem originates in mathematics >> > land, so you should modify your mental picture to match the mathematics. >> >> As someone else has pointed out, the "balls" and "vase" >> are just an attempt to make this sound like a physical problem, >> which it clearly is not, because you cannot physically move >> an infinite number of balls in a finite time. It is just >> a distraction. As you say, the problem originates in mathematics. >> Any attempt to impose physical constraints on inherently unphysical >> problem is just silly. >> >> The problem could have been worded as follows: >> >> Let IN = { n | -1/(2^floor(n/10) < 0 } >> Let OUT = { n | -1/(2^n) } > I think you meant > Let OUT = { n | -1/(2^n) < 0 } Yes, that is what I meant. Thanks. >> What is | IN - OUT | ? >> >> But that would not cause any fuss at all. > I wonder. Does anyone reading this think | IN - OUT | <> 0? > -- > David Marcus I am curious about that myself. Stephen
From: David Marcus on 25 Oct 2006 18:36 Tony Orlow wrote: > David Marcus wrote: > > Tony Orlow wrote: > >> David Marcus wrote: > >>> Tony Orlow wrote: > >>>> David Marcus wrote: > >>>>> Tony Orlow wrote: > >>>>>> stephen(a)nomail.com wrote: > >>>>>>> Also, supposing for the sake of argument that there are "infinitely > >>>>>>> number balls", if a ball is added at time -1/(2^floor(n/10)), and removed > >>>>>>> at time -1/(2^n)), then the balls added at time t=0, are those > >>>>>>> where -1/(2^floor(n/10)) = 0. But if -1/(2^floor(n/10)) = 0 > >>>>>>> then -1/(2^n) = 0 (making some reasonable assumptions about how arithmetic > >>>>>>> on these infinite numbers works), so those balls are also removed at noon and > >>>>>>> never spend any time in the vase. > >>>>>> Yes, the insertion/removal schedule instantly becomes infinitely fast in > >>>>>> a truly uncountable way. The only way to get a handle on it is to > >>>>>> explicitly state the level of infinity the iterations are allowed to > >>>>>> achieve at noon. When the iterations are restricted to finite values, > >>>>>> noon is never reached, but approached as a limit. > >>>>> Suppose we only do an insertion or removal at t = 1/n for n a natural > >>>>> number. What do you mean by "noon is never reached"? > >>>> 1/n>0 > >>> Sorry, I meant t = -1/n. So, I assume your answer is that -1/n < 0. > >>> > >>> But, I don't follow. Translating "-1/n < 0" back into words, I get "all > >>> insertions and removals are before noon". However, I asked you what > >>> "noon is never reached" means. Are you saying that "noon is never > >>> reached" means that "all insertions and removals are before noon"? > >> Yes, David. What else happens in this experiment besides insertions and > >> removals of naturals at finite times before noon? If the infinite > >> sequence of events is actually allowed to continue until t=0, then you > >> are talking about events not indexed with natural numbers, so you're not > >> talking about the same experiment. If noon is not allowed, and all times > >> in the experiment are finitely before noon, well, at none of those times > >> does the vase empty, as we all agree. This is why I am asking when this > >> occurs. It can't, given the constraints of the problem. > > > > Does your "Yes" at the beginning of your reply mean that you agree that > > "noon is never reached" means that "all insertions and removals are > > before noon". By "mean", I mean that that is what the words mean, not > > that the two statements are equivalent or deducible from each other. > > Yes, every event, every insertion or removal, happens at a specific time > before noon. At each of those times, the vase is non-empty. Nothing else > occurs, as far as insertions of removals. Is that clear enough? So, when > does the vase become empty, At noon. > and how? By removing all balls before noon, but leaving some balls in the vase at every time before noon and after one minute before noon. -- David Marcus
From: David Marcus on 25 Oct 2006 18:49 Tony Orlow wrote: > David Marcus wrote: > > For n = 1,2,..., suppose we have numbers A_n and R_N (the addition and > > removal times of ball n where time is measured in minutes before > > noon). For n = 1,2,..., define a function B_n by > > > > B_n(t) = 1 if A_n <= t < R_n, > > 0 if t < A_n or t >= R_n. > > Fine for each ball n. > > > Let V(t) = sum{n=1}^infty B_n(t). Let L = lim_{t -> 0-} V(t). Let S = > > V(0). Let T be the number of balls that you say are in the vase at > > noon. > > You are summing B_n(t) to oo? The sum is over all positive integers. There is no B_oo. I'm sticking to standard Calculus notation. Does that change your answers below? > > Problem 1. For n = 1,2,..., define > > > > A_n = -1/floor((n+9)/10), > > R_n = -1/n. > > > > Then L = infinity, S = 0, and T = undefined. > > I say that if noon exists, there are an infinite number of balls in the > vase. n=oo -> L=T. By "noon exists" do you mean there is a ball B_oo? There isn't. > > Problem 2. For n = 1,2,..., define > > > > A_n = -1/n, > > R_n = -1/(n+1). > > > > Then L = 1, S = 0, T = 1. > > Yeah, L=T again. > > > > > Problem 3. For n = 1,2,..., define > > > > A_n = -1, > > R_n = -1/n. > > > > Then L = infinity, S = 0, T = 0. > > L=lim(x->oo: oo-x) = 0 <> oo L is the limit of V(t) as t approaches zero from the left. So, L = oo. I don't know what you mean by "lim(x->oo: oo-x)". We don't normally define things like oo-x, for x an integer, in Calculus. Of course, the most natural definition would be for oo-x to equal oo, in which case your limit would also be oo. But, you say it is zero. > L=T > > > Tony, can you give us a general procedure to let us determine T given > > the A_n's and B_n's? > > You can keep track of the points between your time vortexes and what's > going on during those periods, for starters. I'm afraid I don't know what I'm supposed to keep track of. In truth, I thought that by calculating V, I was keeping track. But, neither of the two quantities I can get from V, i.e., L and S, seem to consistently match your value. -- David Marcus
From: Lester Zick on 25 Oct 2006 18:59
On 25 Oct 2006 14:22:46 -0700, "MoeBlee" <jazzmobe(a)hotmail.com> wrote: >Lester Zick wrote: > >> So all you're really doing is regressing your appraisal >> of standard set analysis and techniques to a group of terms which only >> appear to have meaning within standard set analysis. > >No, all the definitions revert to one undefined non-logical primitive >(two undefined non-logical primitives, if we take equality as >undefined.) That one non-logical primitive is 'is a member of'. The >logical primitives are 'for all', 'not-both' (or we could use 'neither >nor', or we could use more than one sentential connective as >primitive), and denumerably many variables. Yeah, Moe, look you do a lot of talking to justify one primitive. My primitive is "contradiction" or "differences" and I can justfy that as true in one sentence. You say all your contentions "revert" (whatever that may mean) to one "undefined non logical primitive" by which I assume you mean one outright non demonstrable assumption but I see no evidence to justify your opinion on the subject one way or the other. To me it looks just like one ambiguous claim after another. >> Which doesn't even address the problem of whether the primitives you >> argue from are actually true. They're just assumptions. > >No, primitives are individual symbols. So they're just symbolic assumptions of truth. Big deal. > They're not even statements nor >assumptions that can be appraised for truth. Oh that's just swell. So now we have a standard set of set analytical techniques which can't even be appraised for truth. That's just swell. >> >x is equinumerous with y <-> Ef(f is a bijection from x onto y). >> > >> >There is no assumption of having defined 'cardinality' prior to the >> >above definition. >> >> So if you don't actually say the word "cardinality" it isn't there? > >Yes, the term 'cardinality' is not in the above definition. So now if you don't say "cardinality" the concept isn't there? So if I start out to say "cardinality is cardinality" but instead I substitute "gleeb" for the second "cardinality" to produce "cardinality is gleeb" I've maintained my standard set definitional virtue intact? Kinda like "if I can't see you you aren't there" huh? >> So >> if you say the word "equinumerous" instead we can all go home and rest >> easy that there is no "primitive" implication > >I don't refer to a "primitive implication", and I have no idea what you >mean by it. So if equinumerosity is more primitive than cardinality because you use it to define cardinality and yet requires cardinality in order to be equally numerous you can't figure out what the implication of the circular definitional primitive is? What can I say besides 3.14159 . . >> than things which are >> equinumerous don't bear a cardinal relation to one another to begin >> with? > >To begin with in the "real world" (whatever you take that to be)? The >definitions are of symbols of the language, which are rendered for >convenience by English nicknames. The ordering of the definitions is >not ensured to match some consensus of concepts of real world >ontological or metaphysical relations and dependencies, whatever that >might even be. No clue what you're talking about. How did the "real world" suddenly enter the picture? Not to mention metaphysics and ontology. >Look, again, I have to say, it is just not productive for me to try to >explain this in the vacuum of your knowledge of anything on the >subject. After this post, I really need to execise some self-discipline >by not wasting my time composing explanations for you that do no good >since they presuppose at least some familiarity with the subject >matter, which you don't have. In other words if I can't intuit your faith based belief in standard set analytical techniques you need to get back to the practice of your religion and leave me to my own devices? Oh please don't abandon me to the forces of darkness, Moe. >Or, if you like, I can start at page 1 with you. But I know you're not >interested in that. I'm interested in truth whichever page it comes on and not in your primitive beliefs whether they come on page 1 or not. >So, if you're not going to start at page 1, then it is just not >suitable for me to try to explain to you what's going on at page 100. Nor would it appear suitable for you to explain the truth of much of anything you claim about standard set analytical techniques. ~v~~ |