An uncountable countable set
in [Math]
|
Prev: integral problem
Next: Prime numbers
From: Tony Orlow on 22 Oct 2006 05:15 Virgil wrote: > In article <453912c1(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > >> David Marcus wrote: >>> Tony Orlow wrote: >>>> David Marcus wrote: >>>>> Tony Orlow wrote: >>>>>> Virgil wrote: >>>>>>> In article <4533d315(a)news2.lightlink.com>, >>>>>>> Tony Orlow <tony(a)lightlink.com> wrote: >>>>>>>>> Then let us put all the balls in at once before the first is removed >>>>>>>>> and >>>>>>>>> then remove them according to the original time schedule. >>>>>>>> Great! You changed the problem and got a different conclusion. How >>>>>>>> very....like you. >>>>>>> Does TO claim that putting balls in earlier but taking them out as in >>>>>>> the original will result in fewer balls at the end? >>>>>> If the two are separate events, sure. >>>>> Not sure what you mean by "separate events". Suppose we put all the >>>>> balls in at one minute before noon and take them out according to the >>>>> original schedule. How many balls are in the vase at noon? >>>>> >>>> empty. >>> Suppose we put ball n in at 1/n before noon and remove it at 1/(n+1) >>> before noon. How many balls in the vase at noon? >>> >> At all times >=-1 there will be 1 ball in the vase. > > Which ball will that be at noon, TO? The last one inserted. You know, when the infinite set is finished, like you wanted. Duh.
From: Tony Orlow on 22 Oct 2006 05:28 MoeBlee wrote: > Tony Orlow wrote: >> I'm reading Non-Standard Analysis instead. > > What book? Uh, Robinson, Non-Standard Analysis, first edition. > > You really would be better served by having your basics in set theory > and mathematical logic in order first and then taking on non-standard > analysis. > Thanks for the advice. To be honest, I got rather bogged down in portions of the Tools from Logic section, and skipped to the next chapter. I'll probably have to go back later, but he appears to be taking the approach of developing the arithmetic on such numbers based on which wff's can be created, and assuming that they apply to *N as well as N. >> Robinson agrees there's no >> smallest infinity, > > Then that is not the same as the ordering of the ordinals we're talking > about. I very much doubt that Robinson claims that there is not a least > infinite ordinal. No, he doesn't discuss "ordinals", though I believe he used the term "countable" at points in the first section. > > Please tell me exactly what passages or theorems you are referring to > in Robinson's work so that I can see exactly what it is you are talking > about. In Chapter III, section 3.1.1, he states: "There is no smallest infinite number. For if a is infinite then a<>0, hence a=b+1 (the corresponding fact being true in N). But b cannot be finite, for then a would be finite. Hence, there exists an infinite numbers [sic] which is smaller than a." That sounds very logical to me. Subtract 1 from a number n. n-1<n. finite(n)<->finite(n-1). Infinite(n)<->infinite(n-1). Pretty simple. Look, Ma, no ordinals!! > > MoeBlee > ToeKnee
From: Ross A. Finlayson on 22 Oct 2006 05:33 Lester Zick wrote: > On 19 Oct 2006 17:37:49 -0700, "Ross A. Finlayson" > <raf(a)tiki-lounge.com> wrote: > > >Lester Zick wrote: > >... > >> > >> So what is it exactly that "set" theory allows us to do in mathematics > >> that we couldn't already do without it? Define infinity? Define > >> regularity? Define choice? Define ordinals? Define cardinals? You seem > >> to be of a psychological frame of reference prevalent among modern > >> mathematikers that arithmetic in the form of set theory represents > >> some kind of TOE. > >> > >> ~v~~ > > > >Lester, > > > >Only the null axiom theory could be the TOE. > > Sorry, Ross, but I don't rely on axioms and I have no idea what the > "null axiom theory" is or is supposed to explain. It seems that every > time I turn around someone is propounding some new axiom which is > supposed to explain something people can't explain without assuming > some new and completely trivial assumption. > > ~v~~ Hi Lester, I thought you already knew that the null axiom theory, A theory, refers to an axiomless system of natural deduction. I promote _less_ axioms. Also I thought you knew that the null axiom theory escapes Goedelian incompleteness and is the theory of everything. Snark is a boojum. Define definition. Ross
From: Tony Orlow on 22 Oct 2006 05:38 Lester Zick wrote: > On Fri, 20 Oct 2006 14:12:27 -0400, Tony Orlow <tony(a)lightlink.com> > wrote: > >> MoeBlee wrote: >>> Tony Orlow wrote: >>>> Also, upon which axioms is the definition of cardinality based? >>> The usual definition is: >>> >>> card(x) = the least ordinal equinumerous with x >>> >>> The definition ultimately reverts to the 1-place predicate symbol 'e' >>> (and the 1-place predicate symbol '=', if equality is taken as >>> primitive). For the definition to "work out" ('work out' is informal >>> here) in Z set theory, we usually suppose the axioims of Z set theory >>> plus the axiom of schema of replacement (thus we're in ZF) and the >>> axiom of choice (thus we're in ZFC). However, there is a way to avoid >>> the axiom of choice by using the axiom of regularity instead with a >>> somewhat different definition from just 'least ordinal equinumerous >>> with'. Also, we could adopt a "midpoint" between the axiom schema of >>> replacement and the axiom of choice by adopting the numeration theorem >>> (AxEy y is an ordinal equinumerous with x) instead, which would be a >>> method stronger than adopting the axiom of choice, but weaker than >>> adopting both the axiom of choice and the axiom schema of replacement. >>> As to the more basic axioms of Z, for the definition to "work out", I'm >>> pretty sure we need extensionality, schema of separation (or schema of >>> replacement if we go that way), union, and pairing (pairing is not >>> needed if we have the schema of replacement). I'm not 100% sure, but my >>> strong guess is that we don't need the power set axiom for this >>> purpose. And we don't need the axiom of infinity. >>> >>> Why don't you just a set theory textbook? >>> >>> MoeBlee >>> >> I'm reading Non-Standard Analysis instead. Robinson agrees there's no >> smallest infinity, > > Just out of curiosity, Tony, what's his rationale? I quoted it for MoeBlee, and you can see, but basically, he extends N to *N by applying the basic truths about finite numbers to the infinite. Since it is true for all n in N that, except for 0, every element has a predecessor n-1, then this must also be true for any infinite n. We assume some smallest infinite n. Since n-1<n, and since n-1 is infinite if n is infinite, we have n-1 being a smaller infinity than n, which is a contradiction. For any smallest n we can assume, there is a smaller n-1, so there is no smallest infinite, just like there is no greatest finite. > >> and that there are an uncountable number of countable >> neighborhoods with what he calls the '~' relation. I'm very encouraged >> to see essentially the same ideas as mine, put in technical terms. Maybe >> when I'm through with that, although I'm also reading Boole's treatise >> on logic where he eventually talks about probabilistic logic. > > ~v~~
From: Tony Orlow on 22 Oct 2006 06:14
cbrown(a)cbrownsystems.com wrote: > Tony Orlow wrote: >> cbrown(a)cbrownsystems.com wrote: >>> Tony Orlow wrote: >>>> cbrown(a)cbrownsystems.com wrote: >>>>> Tony Orlow wrote: >>>>>> cbrown(a)cbrownsystems.com wrote: >>>>>>> Tony Orlow wrote: >>>>>>>> cbrown(a)cbrownsystems.com wrote: >>>>>>>>> Tony Orlow wrote: > > <snip> > >>>> You have agreed with everything so far. At every point before noon balls >>>> remain. >>> To be precise, the assertions above all imply that at every time t = >>> -1/n, where n is a natural number, there are balls in the vase. >>> >>> But that *alone* does not even include every time t before noon; let >>> alone every time t. For example, notice that nowhere above do you or I >>> /explicitly/ assert: "at t=-2/3, the number of balls in the vase is a >>> positive finite number". >>> >>> We assert something specific about t = -2/4, and something specific >>> about t = -1/3, but nowhere do we directly state somthing about t = >>> -2/3. >>> >>> On the other hand, given the problem statement, I think we would both >>> /agree/ that there "should be" an obvious (perhaps even unique) >>> well-defined answer to the question : "what is the number of balls in >>> the vase at time t = -1/pi?" >>> >>> Assuming in the remaining statements that one agrees with the previous >>> statement, this leads us to the question: what are the unstated >>> assumptons that allow to agree that this must be the case? >>> >>> I attempted to describe those assumptions in my previoius post. Did you >>> read those assumptions? If so, do you agree with those assumptions? >> At this point I don't recall your previous post. I've been off a bit. > > Well, allow me to repeat them here (with two minor changes): > > In order to interpret the problem > > "At each time t = -1/n where n is a (strictly positive) natural number, > we place the balls labelled 10*(n-1)+1 through 10*n inclusive in the > vase, and remove the ball labelled n from the vase. What is the number > of balls in the vase at time t=0?" > > I make the following simple (and I would claim, fairly uncontroversial > and natural) assumptions: > > --- (object permanence) > > (1) When we speak of a time t, we mean some real number t. > > (2) If a ball is in the vase at any time t0, there is a time t <= t0 > for which we can say "that ball was placed in the vase at time t". > > (3) If a ball is placed in the vase at time t1 and it is not removed > from the vase at some time t where t1 <= t <= t2, then that ball is in > the vase at time t2. > > (4) If a ball is removed from the vase at time t1, and there is no time > t such that t1 < t <= t2 when that ball is placed in the vase, then > that ball is not in the vase at time t2. > > ---- (obedience to the problem constraints) > > (5) If a ball is placed in the vase at some time t, it must be in > accordance with the description given in the problem: it must be a ball > with a natural number n on it, and the time t at which it is placed in > the vase must be -1/floor(n/10). > > (6) If a ball is removed from the vase at some time t, it must be in > accordance with the description given in the problem: it must be a ball > with a natural number n on it, and the time t at which it is removed > from the vase must be -1/n. > > (7) If n is a natural number with n > 0, then the ball labelled n is > placed in the vase at some time t1; and it is removed from the vase at > some time t2. > > --- (very general definition of "the vase is empty at noon") > > (8) the number of balls in the vase at time t=0 is 0 if, and only if, > the statement "there is a ball in the vase at time t=0" is false. > > --- > > Perhaps you would add other assumptions (9), (10), etc.; but my > question is: > > Given the problem statement, do you agree that /each/ of these > assumptions, /on its own/, is reasonable and not just some arbitrary > statement plucked out of thin air? > > If not, which assumption(s) is(are) not reasonable or is(are) > unneccessarily arbitrary? > > <snip> Those all look reasonable to me as I read them. I don't see any statement regarding the fact that ten balls are added for every one removed, though that can be surmised from the insertion and removal schedule. That's the salient fact here. You never remove as many as you add, so you can't end up empty. > >>>> You claim nothing changes at noon. >>> Where, exactly, above do I claim that "nothing changes at noon"? >> Do you disagree with the other standard-bearers, and claim that >> something DOES occur at noon? > > That is not a response to /my/ question "where, exactly, above do I > claim that 'nothing changes at noon'?" > > /I/ don't claim that "something occurs at noon"; nor do /I/ claim that > "nothing occurs at noon". Uh, what would be your opinion on the matter. CAN something occur at noon in this experiment or not? Either way, you have a problem. > > /YOU/ are claiming that the truth of these statements follows logically > from our assumptions; but until you make some kind of mathematical > statement which corresponds to "something occurs at noon", I really > can't address it as a /mathematical/ question. 1/n=0. Happy? Que pasa aqui? Is n in N? > > <snip> > >>>>>>> Do you accept the above statements, or do you still claim that there is >>>>>>> /no/ valid proof that ball 15 is not in the vase at t=0? >> I said that any specific ball was obviously out of the vase at noon. >> > > That's good: we at least agree that it logically follows from (1) - (8) > that there are no labelled balls in the vase at t=0. No finite balls. > > What I honestly find baffling is your repeated claim that it doesn't > then logically follow from assumptions (2) and (5), that if a ball is > in the vase at /any/ time, it is a ball which is labelled with a > natural number; and so therefore the above statement is logically > equivalent to "there are no balls in the vase at t=0". At every time before noon, there are balls in the vase. If nothing occurs at noon, there are still balls in the vase. If balls are removed at noon, then balls are inserted such that 1/n=0, infinite balls. What really baffling is that a divergent sum could be considered to evaporate, and by so many apparently bright people. Are you all Christians or something? > > <snip> > >>>>> Therefore ball 15 is not in |