An uncountable countable set
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From: Mike Kelly on 31 Oct 2006 13:57 Randy Poe wrote: > Mike Kelly wrote: > > Tony Orlow wrote: > > > Mike Kelly wrote: > > > Nothing is allowed to happen at noon in either experiment. > > > > Nothing "happens" at noon? I take this to mean that there is no > > insertion or removal of balls at noon, yes? Well, I agree with that. > > But what relevence does this have to the statement "noon does not > > exist"? What does that even *mean*? > > > > When you've been saying "noon doesn't exist", you actually mean to say > > "no insertion or removal of balls occurs at noon"? > > > > How about this experiment, does noon "exist" in this experiment : > > > > Insert a ball labelled "1" into the vase at one minute to noon. > > > > ? > > I think that when Tony and Han say "noon doesn't exist" they > really mean "there is no noon on the clock in that experiment", > as a way of saying "I have no idea how to answer questions about > noon in that experiment, so I'll say that there is no noon and that > way I don't have to answer any such questions." > > I've asked questions similar to yours. The answer is: "It's easy > for me to figure out there's a ball in the vase at noon. Therefore > I will allow noon to 'exist' in this problem." What really puzzles me though is that sometimes Tony is quite happy to answer what the state of things is at noon, even while he denies at the same time that noon "exists". Or perhaps sometimes he'll say that noon only "exists" if we "allow infinite iterations"; as if somehow the insertion of balls into the vase is what causes the clock to move forwards. -- mike.
From: MoeBlee on 31 Oct 2006 14:05 Tony Orlow wrote: > MoeBlee wrote: > > Tony Orlow wrote: > >> Intuitionistic logicians reject that a false premise > >> implies anything. > > > > Name such an intuitionistic logician and the work in which this > > appears. > > > > Intutitionistic logic DOES have the principle > > > > For all formulas P, > > > > f -> P > > > > ('f', the 'falsehood' symbol is often a primitive of intuitionistic > > (and certain formulations of classical) logic). > > > > Youv'e got it wrong, shooting of your big mouth on that which you know > > nothing, as usual. > > > > MoeBlee > > > > Yeah I misspoke. Sorry. Okay, thanks. MoeBlee
From: MoeBlee on 31 Oct 2006 14:08 Tony Orlow wrote: > When I say there are problems with the axiom of extensionality, I refer > to the application of the fact that two sets, when viewed statically, > contain the same elements. Yes, that means that, without regard to time > or order or anything else, the sets are, theoretically, the same. I > don't dispute that. But, I do dispute the application of that fact to > the exclusion of specifically stated time constraints and their > resulting definition of iterations. I am not saying that the axiom > itself is wrong, as a definition. It just doesn't capture the essence of > what's going on. Static sets are not sequences of arithmetical events. > Do you disagree? Okay, fair enough. But then you need to find axioms and definitions that capture your notion of dynamic sets. MoeBlee
From: Lester Zick on 31 Oct 2006 14:29 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. In my own opinion the proposition should at the very least read "there is a smallest integer but no non integer smallest real". ~v~~
From: Lester Zick on 31 Oct 2006 14:34
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" and "there is no positive least real" you're still stuck with the implication that there is a distinction between integers and reals. >> Hmmm. Curiouser and curiouser. > >Not at all, when you distinguish between propositions that >include that p-word and those that don't. ~v~~ |