An uncountable countable set
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From: David R Tribble on 30 Oct 2006 13:30 David R Tribble wrote: >> No, read your own definition again. Each H-riffic is a finite node >> along a path in a binary tree. > Tony Orlow wrote: >> Where does it say anything about a node in my definition, or whether >> strings can be infinite? Your baseless declarations about my definitions >> don't fly. David R Tribble wrote: >> When you stated that >> 1 in H >> x in H -> 2^x in H >> x in H -> 2^-x in H >> >> The set H is a countable set. Each x in H corresponds to a node in the >> binary tree listing all the x's in H (where each left fork is 2^x and >> each right fork is 2^-x from the node of any x). >> >> In different terms, each x in H is a finite recursion >> x = 2^y or 2^-y for some other y in H >> where each recursion ends at >> y = 1 >> >> Your definition above does not allow for any infinite-length recursions >> or infinite-length paths in the binary tree. >> >> As I posted previously, if you want to extend your definition to >> include infinite-length paths in the tree (which I dubbed the >> H2-riffics), you need to define additional numbers using additional >> rules. Something akin to the way the irrationals are defined on >> top of the rationals (as infinite sequences of rationals) in order to >> define the complete set of reals. > Tony Orlow wrote: > Is that true also of the digital reals? Of course not, because they (specifically, the irrationals) are not defined as finite sequences. Your H-riffics are defined as finite sequences. Go back and look at your own definition again. > I disagree with the notion that > any sequence is countable. In order to prove that the H-riffics really > cover the reals I have to use a Cauchy- or Dedekind-like method to prove > that any element in the continuum can be specified, even if it requires > an infinite specification. But, there is nothing explicit or inherent in > my rules that limits such specifications to finite lengths. You are > carrying that over from the standard notion of sequences as always > countable. I don't adhere to that concept. But your definition is obviously countable and finite. You have no rule that allows them to be otherwise. It's not a question of "not limiting the sequence to being countable", but one of "not allowing the sequence to be uncountable" with an appropriate rule. Your sequence is not uncountable because you have no rule that makes it uncountable. It's truly a pity you can't see that. Tony Orlow wrote: >> What makes you think infinite-length strings are excluded? They're not, >> in either of my riffic number systems. > David R Tribble wrote: >> You're confused. Infinite-length fractions are not excluded, >> obviously. But we're not talking about fractions, we're talking about >> each H-riffic being a node in the binary tree that lists all of them. >> Each H-riffic is a node on a finite-length path in the tree. > Tony Orlow wrote: >> Who the hell said that? Is this your number system now, that you get to >> declare that my H-riffics are nodes in your tree? Get real. > David R Tribble wrote: >> It's what you did _not_ say that excludes them. There is no way to >> produce infinitely recursively-defined H-riffics from your existing >> definitions, so you must add another rule or two that allows such >> numbers to exist. Which gives you a different set, of course. > Tony Orlow wrote: > Which rules would you recommend that counteract rules that I didn't > state? You are applying a rule that says that any sequence is finite. I'm applying exactly the rules (and only the rules) you state. There is no way to derive an infinite sequence from your rules. > That's not true. Countably infinite sequences exist, as in 1/3 in > decimal. They exist here too. They exist by the way decimal fractions are defined, as sums of infinite decreasing terms (digits). Infinite sequences are included from their very definition. Your definitions don't have that.
From: David R Tribble on 30 Oct 2006 13:36 David R Tribble wrote: >> Every member of N has a finite successor. Can you prove that your >> "infinite naturals" are members of N? > Virgil wrote: >> The property of not being an infinite natural holds for the first >> natural, and holds for the successor of each non-infinite natural, so >> that it must hold for ALL naturals. > Tony Orlow wrote: > It holds for all finite naturals, but if there are an infinite number of > naturals generating using increment, then there are naturals which are > the result of infinite increments, which must have infinite value. Can you show us one of those infinite naturals? And while you're at it, show us the finite natural that is its predecessor. Of course, you first have to define what you mean by "infinite increments".
From: Randy Poe on 30 Oct 2006 13:44 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: > >> > >> > > >> >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'm providing explicit descriptions of why, as the original > >quote said, there is a least positive non-zero integer, > >but not a least positive non-zero real. > > Who cares? That has nothing to do with the question I asked. The question you asked was whether the fact that there is a smallest non-positive integer (i.e., 1) but no smallest non-positive real implies that integers are not real. The answer is no. > >> 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. > > > >Since the original text is above, we can actually see whether > >that is what was said. > >"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." > > > >Are you equating "least integer" with "smallest positive > >non-zero integer" and "least real" with "smallest > >positive non-zero real"? > > It certainly seemed to me that I was just asking a question regarding > implications of the point made. And I reinterpreted that question just > above for clarification. Well, it is not so implied. OK? There is indeed a smallest non-positive integer. It is 1. There is no smallest non-positive real. That does not imply the integers are non-real. - Randy
From: Randy Poe on 30 Oct 2006 13:46 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. > 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. - Randy
From: Lester Zick on 30 Oct 2006 13:47
On 29 Oct 2006 18:15:35 -0800, "Ross A. Finlayson" <raf(a)tiki-lounge.com> wrote: >Lester Zick wrote: >> Oops. My bad. Hit the wrong button on the duplicate reply. - LZ >> > >De nada. >>> ... > >> > >> >Various considerations of the natural integers have there being a point >> >at infinity. That can be useful in a form of nonstandard analysis, to >> >say that, for example, an infinite sum with a limit actually equals in >> >evaluation that limit. Consider for example something along the lines >> >of >> > >> > 1 >> >s = ----- >> > 2^n >> > >> >for n from zero to infinity. The limit exists and is two and then >> >consider replacing the 2 in the denominator with s. When you can >> >actually say that the sum equals that limit, you get the same >> >expression for s. >> > >> >Otherwise, s could never equal 2. >> >> Okay. I take the point, Ross. But what rule is there requiring 00 to >> be part of the same set as finites raised to a power of infinity? I >> think the power of infinity could be defined using the "number of >> infinitesimals" which is reciprocally defined with differentiation. >> >> >There are other reasons to consider the infinite naturals as containing >> >an infinite element, that N E N. >> >> N E N? >> >> >In these arguments here, if there is no infinite value for n, then the >> >process never completes. >> >> Well infinitesimal subdivision certainly never completes. >> > >Les, Lester, there is some consideration that it does. Is not the >differential intuitively the atomic subdivision of one? Well here, Ross, you're more strictly speaking epistemology than math. And no dr is just indicative of the ongoing process and not results of that process. That's why dr can be integrated and the naturals alone cannot except as constant multiples of dr. >In an interesting way, variously on what you consider interesting, the >notion of subatomic particles in physics is a similar one as the >consideration of sub-iota reals in mathematics. Not sure what subiota reals in math are, Ross. But I agree that a lot of what passes for physics and math is motivated by some desire for atomic resolution. >> > Re Zeno, the arrow goes half and then half >> >and then half again ad infinitum to reach the mark, that it does is >> >well-known. That's similar to the above equation with the domain over >> >(1, 2, ... (infinitely many times). The distance is one, the arrow >> >travels the distance, the distance can be decomposed in that manner to >> >partial distances, thus the sum is the distance. >> >> But the problem with Zeno's paradoxes is that what is established >> primarily is that the arrow or whatever goes the distance first. In >> other words the paradox establishes unity and attempts to subdivide >> the unity infinitesimally then claims the unity cannot exist unless >> the process of infinitesimal subdivision can complete and be finite. >> Infinitesimal subdivision is not a finite process. It's only used as a >> method of drawing tangents for the purpose of comparing otherwise >> infinites. >> > >Right, it's not a "finite" process in the sense that there are finitely >many integers, but I think you would agree that it is a "finite" >process to the extent that the above description is self-contained. Well hopefully any self contained description would be finite. The only problem comes when one tries to define the possible form of self containment. That's a little tougher and it doesn't get any easier by bending the rules and simply making axiomatic assumptions of truth. >> >Similarly for the real numbers there are considerations of points at >> >infinity, in the "projectively extended" real numbers for example. >> >Similarly as to how division by zero in the "complete" ordered field is >> >undefined, ie that every other number than zero has a multiplicative >> >inverse, there is some consideration and ready application of there >> >being points at infinity, and meaningfully that their multiplicative >> >inverses are defined in similar ways as zero's. >> >> In other words people are just trying to make zero a natural number. >> It isn't. It has very specialized restrictions when it comes to use in >> arithmetic contexts. >> > >When I say "naturals" it includes zero and its successors. Zero and >its successors is the naturals. Notice the verb is conjugated "is", >not "are." Do you see how that subtle play on words, representing >mathematical concepts, leads to N E N? You still haven't explained what "N E N" is supposed to represent. But no zero is not one of the naturals even though it shares some of their arithmetic properties. >> >There are applications in integral transforms and so on for points at >> >infinity, for which the "standard" real numbers are insufficient, and >> >they could only be real numbers. >> >> Well the so called "points at infinity" don't have to be defined as >> part of the same set as natural numbers. Natural numbers are finite. >> Infinites are not. They don't require combination in a common set. >> Everything supposedly true of infinitesimals is reciprocally true of >> infinitesimals and not the set of naturals. >> > >In number theory there is some consideration of a "prime at infinity" >or for that matter "composite at infinity." That's similar to this >notion of passing the bar whether "infinity" would have properties of >being prime and/or composite. Here I was talking about "points at >infinity" in the reals instead of the integers, but the discussion >surrounds both. I guess it does. But that doesn't vitiate what I say above. Simply calling it number theory doesn't make it so unless number theory is demonstrated true. And if it can't be it can't be a theory; it would just be an analytical number treatment. >> >When you have lim n-> oo, that "oo" is an "infinity", it's right there >> >in the expression. With lim x-> 0+ or lim x-> 0-, those can differ >> >with the sign, and do, and in signed numeric formats the sign takes a >> >bit, and in some there are dual representations of zero, >> >mechanistically different. >> >> I'm not quite sure yet what the mechanical implications of zero are. >> But basically if you're going to integrate something that something >> has been infinitesimally differentiated and if not you aren't doing >> definite integration. Either way |