An uncountable countable set
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From: Lester Zick on 1 Nov 2006 16:54 On 31 Oct 2006 11:56:20 -0800, "Randy Poe" <poespam-trap(a)yahoo.com> wrote: > >Lester Zick 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? Well, Randy, this is really a trivial problem. What was originally stated was something along the lines of "smallest positive integer" versus "no smallest positive real". Striking common predicates we arrive at "integer" versus "no real". Now if you can't figure out the problem here you might consider a new career. ~v~~
From: Lester Zick on 1 Nov 2006 16:58 On 31 Oct 2006 12:07:47 -0800, "Randy Poe" <poespam-trap(a)yahoo.com> wrote: > >Lester Zick wrote: >> On 30 Oct 2006 19:50:42 -0800, "Randy Poe" <poespam-trap(a)yahoo.com> >> 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? >> >> I don't ignore it. I considered it as understood > >You shouldn't consider it understood. That's plain wrong. When >we say "the real numbers" we certainly aren't restricting ourselves >to positive reals. Are you restricting yourself to non imaginary numbers? >> but even if you >> include it above such that you have "there is a positive least >> integer" and "there is no positive least real" you're still stuck with >> the implication that there is a distinction between integers and >> reals. > >That's certainly true. There is indeed a distinction between >integers and reals. The integers are a subset of the reals. Well thanks for the explanation that you consider integers a subset of the reals since nothing in the foregoing justifies that explanation. ~v~~
From: Lester Zick on 1 Nov 2006 17:00 On Wed, 1 Nov 2006 00:14:50 GMT, "Dik T. Winter" <Dik.Winter(a)cwi.nl> wrote: >In article <b29fk2drh5cg1120isnglpruj5mo6j6739(a)4ax.com> Lester Zick <dontbother(a)nowhere.net> writes: >... > > I don't ignore it. I considered it as understood but even if you > > include it above such that you have "there is a positive least > > integer" and "there is no positive least real" you're still stuck with > > the implication that there is a distinction between integers and > > reals. > >Yes, there is. Not all reals are created equal, some are superior and >also integer. That may well be and I don't doubt there is. It's just that based on the conjectures noted and striking common predicates we're left with "integers" versus "no reals" which strikes me as a trifle ambiguous on the subject. ~v~~
From: Lester Zick on 1 Nov 2006 17:03 On Tue, 31 Oct 2006 20:39:22 -0500, David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: >Lester Zick wrote: >> On 30 Oct 2006 19:50:42 -0800, "Randy Poe" <poespam-trap(a)yahoo.com> >> 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? >> >> I don't ignore it. I considered it as understood but even if you >> include it above such that you have "there is a positive least >> integer" > >Nope. There is a least positive integer. You can't rearrange the word >order without changing the meaning. Yeah well that really clears things up since using the original predicate sequence and striking common predicates we're still left with "integer" versus "no real". >> and "there is no positive least real" you're still stuck with >> the implication that there is a distinction between integers and >> reals. > >Well, duh! Of course, there is a distinction. Fess up: are you trolling? Only for you sweetheart. Of course there is a distinction between integers and reals. It's just that according to "integer" versus "no real" you don't know what it is. ~v~~
From: Lester Zick on 1 Nov 2006 17:08
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: >> >> >> >> > >> >> >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. Thanks for the heads up. What makes you think it's true? >-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. If you say so. >When we say "the least member of the set of POSITIVE >integers", they are all positive. No lie? >> 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. >> 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. >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. ~v~~ |