An uncountable countable set
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From: David Marcus on 31 Oct 2006 20:39 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. > 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? -- David Marcus
From: Ross A. Finlayson on 31 Oct 2006 22:07 Randy Poe wrote: > > 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 In Spinoza's consideration of the naturals as a continuum it would seem that if there's one there are both. I still hope I can get MoeBlee, who I find personable and well-informed, to admit he reads. That's about a notion of in the standard reals the necessary existence via standard methods of a least positive real. Ross
From: Ross A. Finlayson on 31 Oct 2006 22:13 David Marcus wrote: > Mike Kelly wrote: > > Tony Orlow wrote: > > > Where are the iterations mentioned there? You're missing the crucial > > > part of the experiment. By your logic, you could put them in in any > > > order and remove them in any order, and when you say both processes are > > > done, nothing's left, but that's BS. It ignores the sequence specified. > > > This is just a distraction. > > > > Yes, if you insert and remove exactly the same balls then you get the > > same result when you're done, no matter what order you did it all in. > > Why is that BS? It seems blindingly obvious. > > > > But I forgot, you think that if you shift all the insertions 1 minute > > further back in time, you DO get an empty vase at noon, right? I really > > don't understand how your mind works. > > Try the mental picture with the water. We fill it up, then we start > letting it run out. No reason all the water shouldn't empty out of the > vase by noon. > > -- > David Marcus Hi, There are only sets in ZFC. When people talk about the real numbers in ZFC, it is as a construction of sets in ZFC has been found to be isomorphic in a relatively strong sense to the real numbers. Obviously, mathematicians want to find the best representation of everything the real numbers are or must be by their nature. Then, that gets into that some feel that mathematics can't assume what it sets out to prove. That's reasonable. By the same token, the real numbers have many, many roles to fill, and some of them have, for example, in the projectively extended real numbers, points at plus/minus infinity. It is relatively standard to define a real number as a Dedekind cut or Cauchy sequence, which are basically defined in terms of sequences of rationals, which resolve to generally the familiar decimal representation which is adequate in finitely expressing rationals, and with radicals, algebraics. If a Dedekind cut is as was recently stated some "initial segment" of the rationals, I wonder to what ordering that pertains. Consider how that is to describe an irrational number. Basically the sequence of elements is to converge towards the number. There's an irrational less than one and greater than .9, in decimal, less than one and greater than .99, less than one and greater than .999, etcetera. The general consensus here is that .999... = 1, yet for each ..999...999, there is an irrational between it and one. So, does that not seem that there are irrationals unrepresentable via Dedekind/Cauchy? It would seem that certainly as the irrational is some finite distance from 1 that it would be between .999...998 and ..999...999, and between that irrational and one are infinitely many more numbers forever, there always exist irrationals between .999...999 and 1, and, for Dedekind/Cauchy to represent them, they must have a unique representation. For no finite number of 9's or rep-units in binary can these irrationals in the diminishing remaining interval be represented, and for any infinite number of rep-units the result is said to be one. So, either between the finite and infinite those values are represented, or they're not, and due to the completeness of the reals, if they're not, then Dedekind/Cauchy, the standard set-theoretic method to construct real numbers, is insufficient to construct some real numbers. For any it's so, for all it's not, or vice versa. Don't worry I've heard of the transfer principle. Consider the representation of rational numbers, for example 9/10. That would be .9, .90, .900, ..., .9(0): 9/10's. There is no last element of that list, .9 could be an initial segment of a sequence for 9/10's or any irrational between .9 and 1.0, as above. The initial sequence .9, .90 could be an initial sequence for any irrational between .900 and .91. The initial segment .900 could be an initial sequence for any number between .9000 and .901. .90000 <= x <= .9001 .900000 <= x <= .90001 .9000000 <= x <= .900001 .90000000 <= x <= .9000001 .900000000 <= x <= .90000001 .9000000000 <= x <= .900000001 .90000000000 <= x <= .9000000001 .900000000000 <= x <= .90000000001 .9000000000000 <= x <= .900000000001 .90000000000000 <= x <= .9000000000001 ..900000000000000 <= x <= .90000000000001 ... As the number of zeros diverges, the diminishing interval goes to zero, where the lower and upper bounds are a and b, lim n->oo b-a = 0. For any finite iteration there are obviously a continuum of elements that x could be, so for a value, x, to not obviously be among a continuum of possible values there must be infinitely many iterations. Keep in mind that there are printed counterexamples to standard real analysis with a least positive real. Obviously the ground around .999... vis-a-vis 1 is very well turned, that's the point, to some extent we're talking about significant ephemera. Look at the 1 on the right side above. Where does it go? .90001 <= x <= .9002 .900001 <= x <= .90002 .9000001 <= x <= .900002 .90000001 <= x <= .9000002 Standardly, equal. .90001 <= x <= .901 .900001 <= x <= .9001 .9000001 <= x <= .90001 .90000001 <= x <= .900001 .90001 <= x <= .91 .900001 <= x <= .901 .9000001 <= x <= .9001 .90000001 <= x <= .90001 .90001 <= x <= .91 .900001 <= x <= .901 .9000001 <= x <= .9001 .90000001 <= x <= .90001 .90001 <= x <= .91 .900001 <= x <= .91 .9000001 <= x <= .901 .90000001 <= x <= .9001 .90001 <= x <= .91 .900001 <= x <= .91 .9000001 <= x <= .91 .90000001 <= x <= .901 .90001 <= x <= .91 .900001 <= x <= .91 .9000001 <= x <= .91 .90000001 <= x <= .91 On the left and right side each converges to 9/10, but as this continues the lhs is .90 and the rhs is .91. So, is the vase empty at noon? Ross
From: Ross A. Finlayson on 31 Oct 2006 22:23 Lester Zick wrote: > On 30 Oct 2006 19:31:56 -0800, "Ross A. Finlayson" > <raf(a)tiki-lounge.com> wrote: > > >Lester Zick wrote: > > > >> Lo mismo. > >> > >> ~v~~ > > > >Hi, > > > >Lester, you suggest to eschew axioms, or that axioms are unjustified. > > Hi Ross, what I say is that axioms are undemonstrated assumptions of > truth. Axioms can be justified only through appeals to plausibility > and dialectical sophistry. But there is no proof axioms are anything > more than assumptions. > > >I agree. I proffer the null axiom theory, it has no non-logical > >axioms, where logical axioms are the truth tables. > > Uh no, Ross, this isn't going to work. You can't just posit "truth" > tables where the concept of truth itself is ambiguous. This is nothing > more than axiomatic assumptions of truth by another name. > > >Then, with some introspection, all and only true statements are > >theorems. Goedel can't dam it, the tower of rain. > > > >I relate it to Yggdrasil it is very simple. It's, quite simple. > > > >There's quite a bit more exposition about it here and on sci.logic and > >physics. For, if it is, it's the T.o.E. There's only one theory with > >no axioms. > > Well it really doesn't matter where you post it. There is no "theory > of truth" which winds up with undemonstrated assumptions whether you > call them axioms or not. Just getting rid of the word doesn't help. > Either you can demonstrate "truth" in universal terms whether logical, > physical, or mathematical or you're barking up the wrong tree. > > ~v~~ Hi Lester, I think we can agree on something. That is that there is a truth, any truth. Then, all the other things eventually consistent with that statement, tautology, are truths. Truth is basically tautology, identity, self-sameness. Then, in the roots of perceived paradoxes of mathematical logic, there is basically a representation of the Liar. When it lies, all the binary ones go to zero and all the zeros go to one, and there's no difference. For every saying there's an equal and opposite saying. It's conservation, in a sense, and in a sense conservation of consistency and completeness. Consider this thing-in-itself, in consideration of N element of N. Every finite integer is the set of naturals, every finite integer is an element of the naturals, transitively the set of naturals is an element of the set of naturals. Then any infinite set is not well-founded. So, for similar reasons as to that there is no universe in ZF, as a representative well-founded theory and a good one, there is no set of cardinal nor ordinal numbers in ZF, no set of sets in ZF, then you might see why there are a variety of anti-foundational theories. Ross
From: Tony Orlow on 31 Oct 2006 22:52
Randy Poe wrote: > Tony Orlow wrote: >> Randy Poe wrote: >>> Tony Orlow wrote: >>>> Randy Poe wrote: >>>>> Tony Orlow wrote: >>>>>> Randy Poe wrote: >>>>>>> Tony Orlow wrote: >>>>>>>> Virgil wrote: >>>>>>>>> In article <4542201a(a)news2.lightlink.com>, >>>>>>>>> Tony Orlow <tony(a)lightlink.com> wrote: >>>>>>>>> >>>>>>>>>> cbrown(a)cbrownsystems.com wrote: >>>>>>>>>>> When you say "noon doesn't occur"; I think "he doesn't accept (1): by a >>>>>>>>>>> time t, we mean a real number t" >>>>>>>>>> That doesn't mean t has to be able to assume ALL real numbers. The times >>>>>>>>>> in [-1,0) are all real numbers. >>>>>>>>> By what mechanism does TO propose to stop time? >>>>>>>> By the mechanism of unfinishablility. >>>>>>> But that's why I asked you a question about variables labelling >>>>>>> times yesterday, when noon clearly occurred. >>>>>> The experiment occurred in [-1,0). Talk of time outside that range is >>>>>> irrelevant. Times before that are imaginary, and times after that are >>>>>> infinite. Only finite times change anything, so if something changes, >>>>>> it's at a finite, negative time. >>>>>> >>>>>>> I can define a list of times t_n = noon yesterday - 1/n seconds, >>>>>>> for all n=1, 2, 3, ... >>>>>> Are there balls in the vase for t<-1? No. >>>>> What balls? What vase? >>>>> >>>>> I'm naming times. They're just numbers. >>>>> >>>>>>> Clearly this list of times has no end. But didn't noon happen? >>>>>> Nothing happened at noon to empty the vase, \ >>>>> What vase? Why are you obsessed with vases? >>>>> >>>>> Do you deny me the ability to create a set of variables >>>>> t_n, n = 1, 2, ...? Why do vases have to come into it? >>>> I thought we were trying to formulate the problem. >>> No, we (some of us) are trying to formulate a completely >>> different problem, with balls and vases (possibly even >>> times) explicitly removed so that other aspects can be examined. >>> >>> Yet you keep trying to put balls and vases back in, after being told >>> that they are not present in the new problem. >> I am pointing out that your formulation doesn't match the original >> problem. > > It's a new problem. > > Are you capable of contemplating a second problem, > throwing away balls and vases and starting from scratch, > asking different questions about a different problem? > > - Randy > I am capable, if I have a reason. Does it shed light on the original problem, or is it simply a distraction from the logical question about the problem at hand? I'm not really interested in pointless distractions. |