Uncountability of the Rationals?
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From: David R Tribble on 25 Jun 2010 19:22 Tony Orlow wrote: >> Secondly, any uncountable set, when ordered linearly (or with a finite >> number of child nodes each), must contain an element which is >> infinitely distant from the foundation of the set. A countably >> infinite set never does, which is why it's not "actually" infinite. > David R Tribble wrote: >> Since R+ (the set of positive reals) is an uncountable set, which >> element of it is "infinitely distant" from its foundation? And what >> is the "foundation" of set R+? > Tony Orlow wrote: > How far is H-riffic 3 from 2 (in base 2 H-riffic) ? That fails to answer my questions. I'll ask again. R+ is the uncountable set of positive reals. We'll order it "linearly" with the usual ordering ('<' over the reals). So it meets the two requirements you state above. So then, according to you, R+ "must contain an element which is infinitely distant from the foundation of the set". So: 1. Which element of R+ is that? 2. What is the foundation of set R+?
From: Jesse F. Hughes on 25 Jun 2010 19:37 Tony Orlow <tony(a)lightlink.com> writes: > On Jun 25, 12:41 pm, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: >> Tony Orlow <t...(a)lightlink.com> writes: >> > On Jun 25, 10:48 am, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: >> >> Tony Orlow <t...(a)lightlink.com> writes: >> >> > For the purposes of IFR, N+, the positive naturals, is the standard >> >> > countably infinite set. >> >> >> Can you tell us exactly how you define N+? This question may seem >> >> silly, but it really is important. >> >> > 1 e N+ >> > x e N+ -> n+1 e N+ >> ^n >> >> That "definition" does not specify a single set. There are many sets >> that satisfy that definition. The set R of real numbers, the set Q of >> rationals, the set Z of integers *all* satisfy the definition you've given. > > Uh, what? > I don't know where the "^n" came from, but it wasn't mine. Please > don't insert deliberate obfuscation. 1 is an element of N+ and if x is > an element of N+ then so is x+1. It wasn't an obfuscation. I was trying to point out a typo in your formula. You wrote "x e N+ -> n+1 e N+". It is obvious that you meant to use the same variable in both places. And, of course, that typo is irrelevant to my comment. You didn't pick out a defining characteristic of N+, since the sets Z, Q, R, w+w, and so on, all satisfy the condition you wrote. >> >> For comparison, I would define N+ to be the least inductive_1 set, where >> >> >> ind_1(x) <-> 1 in x & (Ay)(y in x -> y+1 in x). >> >> > I like mine better, but same difference. >> >> You're confused. I haven't yet defined N+, so it does not compare to >> your "definition". >> > > If that wasn't the full definition then you have little business > drawing conclusions prematurely. Er, that was the full definition of ind_1. The definition of N+ is just below. >> >> Thus, N+ has the property >> >> >> ind_1(N+) & (Ax)(ind_1(x) -> N+ c x). >> >> > How do you draw that conclusion? >> >> That's what I mean when I say that N+ is the *least* inductive_1 set[1]. (I >> haven't proved there *is* a least inductive set here, but I will if you >> wish. It's perfectly standard.) > > Mmmmmm....standard. Nummies! > > So, your proof is that N+ is a member of the set of inductively > defined and well-founded sets, and supposedly the "least" among them. "Inductive", not "inductively defined". And while it happens to be well-founded, that's not something that I've mentioned. It is a consequence of the definition, not a requirement. > It's a member, but far from the least. COnsider the size of the set of > natural tetrations. Is that set an inductive set? That is, does it satisfy the property 1 in x & (Ay)(y in x -> y+1 in x)? >> >> >> Is this also your definition? >> >> > Apparently not. >> >> I guess I wasted my time with this question. You don't really >> understand the need for specifying N (or N+) as a *least* set satisfying >> some condition, because you mistakenly think that your two conditions >> already define N+. I hope that my examples showed you why your >> so-called definition does not suffice, but I honestly don't expect so. > > What doesn't suffice for your purposes? What can't you prove? That there is a unique set satisfying the so-called definition. There are just oodles of sets containing 1 and closed under successor. Lots of them. Z contains 1 and is closed under successor. So your definition does not suffice and you still don't understand why. >> >> Footnotes: >> [1] That is, "least" modifies "inductive_1 set", not "inductive_1". N+ >> is the smallest set which is inductive_1. This is the definition of N+. > > Consider any subset of N+, with a somewhat naive perspective, if only > for a moment. Any proper subset S of N+ either does not contain 1, or contains an n such that n+1 is not in S. Thus, no proper subset of N+ is inductive_1. -- Jesse F. Hughes "You're terrified of your daughters dreaming about me." -- James S. Harris, on why mathematicians fear him
From: Virgil on 25 Jun 2010 20:25 In article <6361b7a1-0c04-4457-92a8-29a4da6b64c6(a)5g2000yqz.googlegroups.com>, Tony Orlow <tony(a)lightlink.com> wrote: > On Jun 25, 12:41�pm, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: > > Tony Orlow <t...(a)lightlink.com> writes: > > > On Jun 25, 10:48�am, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: > > >> Tony Orlow <t...(a)lightlink.com> writes: > > >> > For the purposes of IFR, N+, the positive naturals, is the standard > > >> > countably infinite set. > > > > >> Can you tell us exactly how you define N+? �This question may seem > > >> silly, but it really is important. > > > > > 1 e N+ > > > x e N+ -> n+1 e N+ > > � ^n > > > > That "definition" does not specify a single set. �There are many sets > > that satisfy that definition. �The set R of real numbers, the set Q of > > rationals, the set Z of integers *all* satisfy the definition you've given. > > Uh, what? > I don't know where the "^n" came from, but it wasn't mine. Please > don't insert deliberate obfuscation. 1 is an element of N+ and if x is > an element of N+ then so is x+1. That does not specify N+ uniquely. For example, both the set of rationals and the set of reals satisfy both conditions. What you need in addition is that N+ is a subset of every set, S, for which both 1 e S and (x e S -> x+1 e S)
From: Tony Orlow on 25 Jun 2010 23:17 On Jun 25, 5:51 pm, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: > Tony Orlow <t...(a)lightlink.com> writes: > > Where you have a minimum positive difference between successive > > elements (like 1), then any infinite number of them means an infinite > > difference between the first and last. > > Is it your opinion that N+ has a last element? No, and that's beside the point. If there are an infinite number of unit increments between any two values then there is an infinite difference between them. If not, then it's a countable sequence. N+ is countable. Some subsets of R *might* be measurable even though not continuous (perhaps the Cantor set?), but most are countable, and therefore without real measure. Between 0 and 3 in H-riffic base 2, there are an uncountable number of values, along a path that seems rather difficult to determine. That is an infinite difference between two distinct elements. > > -- > "And God Himself won't help you if this goes bad as despite your > beliefs I can assure you that angry people will against all law if > necessary tear their rage out of your hides if it goes badly." > -- James S. Harris, on the dangers of criticizing his mathematics I can feel my hair standing on end Tony
From: Tony Orlow on 25 Jun 2010 23:39
On Jun 25, 7:22 pm, David R Tribble <da...(a)tribble.com> wrote: > Tony Orlow wrote: > >> Secondly, any uncountable set, when ordered linearly (or with a finite > >> number of child nodes each), must contain an element which is > >> infinitely distant from the foundation of the set. A countably > >> infinite set never does, which is why it's not "actually" infinite. > > David R Tribble wrote: > >> Since R+ (the set of positive reals) is an uncountable set, which > >> element of it is "infinitely distant" from its foundation? And what > >> is the "foundation" of set R+? > > Tony Orlow wrote: > > How far is H-riffic 3 from 2 (in base 2 H-riffic) ? > > That fails to answer my questions. I'll ask again. I have already answered this question to a large extent with the H- riffics. There is some proof left to do, but the fact is that taking log(|x|) repeatedly tends around 0, and so the probability of the root of any number becoming 0 through this recursive process eventually approaches 1. > > R+ is the uncountable set of positive reals. We'll order it > "linearly" with the usual ordering ('<' over the reals). So it > meets the two requirements you state above. Okay. > > So then, according to you, R+ "must contain an element > which is infinitely distant from the foundation of the set". > So: > 1. Which element of R+ is that? > 2. What is the foundation of set R+? In the H-riffics the foundation is 0, and in base 2, 3 is infinitely distant from the foundation (and vice versa). You may require proof of this fact, and I'll work on that, because it's getting pretty interesting again, but for now, I'm just pretty convinced, personally. If you're not, I understand. Tony |