An uncountable countable set
in [Math]
|
Prev: integral problem
Next: Prime numbers
From: Lester Zick on 31 Oct 2006 14:42 On Tue, 31 Oct 2006 00:51:19 -0500, David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: >Randy Poe wrote: >> Lester Zick wrote: > >> > It doesn't? My mistake. So there's "no least real" >> >> There's no least real. >> >> > but a "least integer real"? >> >> There's no least integer. >> >> There's a least POSITIVE integer. Is there some reason you >> keep ignoring the critical word POSITIVE? > >Conjecture: People not trained in mathematics don't realize that each >and every word is important. Conjecture: People not trained in logic don't realize the proposition "least positive integers" and "no least positive reals" have mutually exclusive implications unless properly qualified since each and every word is critical and I daresay there are quite a few possibly infinite unstated qualifications to every proposition. And if based on the foregoing predicates alone you can assume integers are real I'm certainly allowed to assume that the integers you're talking about are positive since you're discussing them as "least" in relation to zero. ~v~~
From: Lester Zick on 31 Oct 2006 14:48 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: >> >> > >> >Lester Zick wrote: >> >> On 30 Oct 2006 08:34:21 -0800, "Randy Poe" <poespam-trap(a)yahoo.com> >> >> wrote: >> >> >> >> > >> >> >Lester Zick wrote: >> >> >> On 28 Oct 2006 12:54:51 -0700, "Randy Poe" <poespam-trap(a)yahoo.com> >> >> >> wrote: >> >> >"There is a least integer" and "there is a least real" >> >> >are both false. >> >> >> >> They are? >> > >> >Yes. If you disagree, perhaps you can name me the minimum >> >element of the sets Z and R. >> >> Or perhaps you can show us how it is "there is a least integer" and >> "there is a least real" are both false since it's your contention not >> mine. >> >> >> Perhaps you should take that up with theologians then. >> > >> >We are discussing mathematics. In the mathematical objects >> >called "the set of integers" and "the set of reals" there is no >> >least member. >> >> Well perhaps you can just prove that since it's your contention not >> mine? > >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? >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: is 1 an integer? Is it positive or negative? 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. There are all kinds of qualifications possible. You don't state them and you can't use them to argue truth. ~v~~
From: imaginatorium on 31 Oct 2006 14:48 Tony Orlow wrote: > imaginatorium(a)despammed.com wrote: > > Virgil wrote: > >> In article <1162268163.368326.64650(a)m73g2000cwd.googlegroups.com>, > >> imaginatorium(a)despammed.com wrote: > >> > >>> Virgil wrote: > >>>> In article <45462ba0(a)news2.lightlink.com>, > >>>> Tony Orlow <tony(a)lightlink.com> wrote: > >>>> > >>>>> stephen(a)nomail.com wrote: > >>>>>> Tony Orlow <tony(a)lightlink.com> wrote: > >>>>>>> stephen(a)nomail.com wrote: > >>>>>>>> Tony Orlow <tony(a)lightlink.com> wrote: > >>>>>>>>> stephen(a)nomail.com wrote: > >>> <snipola> > >>> > >>>>>>> OH. So, sets don't have sizes which are numbers, at least at particular > >>>>>>> moments. I see.... > >>>>>> If that is what you meant, then you should have said that. > >>>>>> And technically speaking, sets do not have sizes which are numbers, > >>>>>> unless by "size" you mean cardinality, and by "number" you include > >>>>>> transfinite cardinals. > >>>>> So, cardinality is the only definition of set size which you will > >>>>> consider.....your loss. > >>>> It is the only definition of set size that is known to produce a valid > >>>> partial ordering on sets. > >>> Huh? I thought cardinality produced a valid *total* ordering on sets. > >> The cardinalities are totally ordered, but the sets are not. > >> A total order on sets would require that when neither of two sets was > >> "larger than" the other that they must be the same set, not merely the > >> same size. > > > > Oh, right. But - and I'm not quite sure how to say this, but the > > cardinalities _are_ totally ordered; for any two sets A and B, c(A) < > > c(B), or c(A) = c(B), or c(A) > c(B). If you "reduce" the sets by the > > cardinality equivalence relation, they are totally ordered. The subset > > relation doesn't lead to an equivalence relation, only a partial > > ordering: so there is no s(A) = s(B) unless A=B; but for most pairs of > > sets, the subset relation simply says nothing at all. (Until His > > Master's Voice is heard, telling us something totally arbitrary.) > > > > Anyway, your claim was clearly wrong, since the subset relation > > provides a valid partial ordering on sets. > > If that's what a partial ordering vs. a total ordering is, Bigulosity is > a partial ordering on sets, not total ordering. Different sets can have > the same Bigulosity. Virgil was muddling everything up, as usual, but the difference between a partial and total ordering is basically whether there are pairs of elements for which the order is undetermined. The subset relation is a very obvious example, where (for example) the set of reals in [0, 1] and the set of prime integers cannot be compared, because neither is a subset of the other. "Bigulosity" has never been sufficiently clearly defined to tell, but since you get very steamed up about subsets, and since the only known coherent claim is that A proper subset of B -> b(A) < b(B), it's extremely unlikely Bigulosity could be extended to become a total ordering. Please compare the Bigulosities of the set of polygons with vertices on integral x-y coordinates and the set of topologically distinct polyhedra. Show your working. (Of course you don't need to come up with an "answer" like "ratio of 5 pi^2", but you need to show how such a task would be approached. One of the things you still don't seem to have realised is that before anything can be "maths" it has to be teachable to other people. I don't think anyone but you has the faintest idea what Bigulosity is really supposed to be, except in a ragbag of specific cases.) Brian Chandler http://imaginatorium.org
From: Randy Poe on 31 Oct 2006 14:54 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: > >> > >> > > >> >Lester Zick wrote: > >> >> On 30 Oct 2006 08:34:21 -0800, "Randy Poe" <poespam-trap(a)yahoo.com> > >> >> wrote: > >> >> > >> >> > > >> >> >Lester Zick wrote: > >> >> >> On 28 Oct 2006 12:54:51 -0700, "Randy Poe" <poespam-trap(a)yahoo.com> > >> >> >> wrote: > >> >> >"There is a least integer" and "there is a least real" > >> >> >are both false. > >> >> > >> >> They are? > >> > > >> >Yes. If you disagree, perhaps you can name me the minimum > >> >element of the sets Z and R. > >> > >> Or perhaps you can show us how it is "there is a least integer" and > >> "there is a least real" are both false since it's your contention not > >> mine. > >> > >> >> Perhaps you should take that up with theologians then. > >> > > >> >We are discussing mathematics. In the mathematical objects > >> >called "the set of integers" and "the set of reals" there is no > >> >least member. > >> > >> Well perhaps you can just prove that since it's your contention not > >> mine? > > > >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. -1000 is an integer with the property that -1000 < 1 -100000 is an integer with the property that -100000 < 1. There are many integers less than 1. > > >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. When we say "the least member of the set of POSITIVE integers", they are all 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 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"? Wouldn't be adding the word "positive" when it is left out be considered adding a qualification that wasn't present? 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? - Randy
From: Randy Poe on 31 Oct 2006 14:56
Lester Zick wrote: > On Mon, 30 Oct 2006 17:32:40 -0500, David Marcus > <DavidMarcus(a)alumdotmit.edu> wrote: > > >Lester Zick wrote: > >> On Sun, 29 Oct 2006 15:10:18 -0500, David Marcus > >> <DavidMarcus(a)alumdotmit.edu> wrote: > >> > >> >Lester Zick wrote: > >> >> On 28 Oct 2006 12:54:51 -0700, "Randy Poe" <poespam-trap(a)yahoo.com> > >> >> wrote: > >> >> > >> >> > > >> >> >Lester Zick wrote: > >> >> >> On Fri, 27 Oct 2006 14:23:58 -0400, David Marcus > >> >> >> <DavidMarcus(a)alumdotmit.edu> wrote: > >> >> >> > >> >> >> >Lester Zick wrote: > >> >> >> >> On Fri, 27 Oct 2006 16:30:04 +0000 (UTC), stephen(a)nomail.com wrote: > >> >> >> >> >A very simple example is that there exists a smallest positive > >> >> >> >> >non-zero integer, but there does not exist a smallest positive > >> >> >> >> >non-zero real. > >> >> >> >> > >> >> >> >> So non zero integers are not real? > >> >> >> > > >> >> >> >That's a pretty impressive leap of illogic. > >> >> >> > >> >> >> "Smallest integer" versus "no smallest real"? Seems pretty clear cut. > >> >> > > >> >> >You must be joking. I can't believe even you can be this dense. > >> >> > >> >> Oh I dunno. I can be pretty dense. Just not as dense as you, Randy, > >> >> but that's nothing new. > >> >> > >> >> >Is 1 the smallest positive non-zero integer? Yes. > >> >> > > >> >> >Is it the smallest positive non-zero real? No. 1/10 is smaller. > >> >> >Ah well, then is 1/10 the smallest positive non-zero real? No, > >> >> >1/100 is smaller. Is that the smallest? No, 1/1000 is smaller. > >> >> > > >> >> >Does that second sequence have an end? Can I eventually > >> >> >find a smallest positive non-zero real? > >> >> > > >> >> >How about the first? Is there something smaller than 1 which > >> >> >is a positive non-zero integer? > >> >> > >> >> See the problem here, Randy, is that you're explaining an issue I > >> >> didn't raise then pretending you're addressing the issue I raised. I > >> >> don't doubt there is no smallest real except in the case of integers. > >> >> But that is not what was said originally. What was said is that there > >> >> is a least integer but no least real. Now these strike me as mutually > >> >> exclusive predicates. But then who am I to analyze mathematical > >> >> predicates in logical terms especially when there are self righteous > >> >> neomathematikers around who prefer to specialize in name calling > >> >> rather than keep their arguments straight in reply to simple queries. > >> > > >> >I'll probably regret asking, but what the heck. Are you saying that the > >> >following two statements are contradictory? > >> > > >> >1. There is a smallest positive integer. > >> >2. There is no smallest positive real. > >> > >> No. In response to these two propositions I'm simply asking whether > >> you consider integers real? > > > >Yes, integers are real. > > As I have no doubt. > > >> Or you might try reading what I originally > >> asked which you pronounced illogical without apparently bothering to > >> read what I wrote. > > > >You wrote, "So non zero integers are not real?". I've no idea why you > >would think that. It doesn't seem to follow from anything that was said. > > The difficulty is that the proposition "there is a smallest integer > but no smallest real" would seem to indicate otherwise. Nobody has stated that proposition but you. Actual proposition: "There is a smallest positive integer". What you read: "There is a smallest integer." Do you really see no difference between those two propositions? You can't find a word present in the first that is absent in the second? - Randy |