Uncountability of the Rationals?
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From: Tony Orlow on 11 Jun 2010 10:04 On Jun 11, 1:11 am, Brian Chandler <imaginator...(a)despammed.com> wrote: > David R Tribble wrote: > > Tony Orlow wrote: > > >> E 0 > > >> E 1 > > >> 0<1 > > >> x<y -> E z: x<z<y > > >> This leads to some level of infinites between counting numbers. > > <snip> > > > > For example, the binary fractions would surely fit his definition. > > > Surely they would, but Tony seems to be missing an axiom or > > two for defining anything beyond 0, 1, and <. Maybe I just missed > > it in the first reading. I'll check again. Maybe I can also find those > > counting numbers and levels of infinities he mentions. > > I think you are being *very* slightly unfair. Tony's writing style is > a bit strange, but surely the above can be seen as a definition of UI, > the Unit Interval. > > UI is a set of elements, with a total ordering < such that: > UI includes an element 0 > UI includes an element 1 > If x and y are elements of UI and x < y, then there exists an element > p of UI such that x < p < y. > > My _guess_ is that Tony thinks this gives him something whose > bigulosity he can "declare", despite the fact that there are any > number of sets of different cardinalities even which match this > definition. I believe he also "declares" that the bigulosity of UI is > in some simple relation to the bigulosity of the TNaturals, or > something similar. No, Brian. You are doing well in interpreting my logical statements above, and indeed, within the half-open unit interval I consider there to be Big'Un or a zillion points (not necessarily real numbers). I do not consider omega and c to be related by any finite formula, though I had compared Big'Un to some length of the real line such that the value Big'Un extsited on this line. Of course, I believed it extended beyind that still, so it wasn't really the total length of the line. I've long since abandoned that notion, and simply consider a zillion to be the number of points in a unit line segment. > > But this has become sadly repetitive. Reading the old threads is just > as entertaining, and less work. > > Brian Chandler The old threads are old. Tony
From: Jesse F. Hughes on 11 Jun 2010 10:17 Tony Orlow <tony(a)lightlink.com> writes: > On Jun 10, 10:25 pm, David R Tribble <da...(a)tribble.com> wrote: >> Transfer Principle (LWalker) wrote: >> >> In searching for a way to make the T-riffics more >> >> rigorous, we can start by letting z, the symbol for >> >> "zillion," be a primitive and go from there. >> >> Tony Orlow wrote: >> > Yes. >> > E 0 >> > E 1 >> > 0<1 >> > x<y -> E z: x<z<y >> >> > This leads to some level of infinites between counting numbers. >> >> That's hard to believe. The only thing that your definitions do >> is define two numbers, 0 and 1, and an order relation for them. >> Where are all of the rest of the reals? Or even the naturals? > > How do you read the last statement? Between any two reals lies another > real. Lather, rinse, repeat. Does this not lead to an infinite number > of reals, or even rationals, between any two naturals? (sigh) If I read the above "axioms" correctly, you don't have the naturals. You have 0 and 1. I assume that you intend < to be a linear order. You should say so, preferably by including the axioms for linear order. In that case, all you've axiomatized is a dense linear order with top and bottom elements. No arithmetic. No metric. No topology, aside from the various topologies that are standard for linear orders. Not much of anything that we expect from [0,1]. -- "Kim liked the math I did for her and gave me quite a few groceries... likely so many groceries that they would have cost Kim about what she pays for two whole packages of cigarettes. Few people have ever rewarded me for my work as much as Kim did." -- Usenet nut
From: Tony Orlow on 11 Jun 2010 11:30 On Jun 11, 10:17 am, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: > Tony Orlow <t...(a)lightlink.com> writes: > > On Jun 10, 10:25 pm, David R Tribble <da...(a)tribble.com> wrote: > >> Transfer Principle (LWalker) wrote: > >> >> In searching for a way to make the T-riffics more > >> >> rigorous, we can start by letting z, the symbol for > >> >> "zillion," be a primitive and go from there. > > >> Tony Orlow wrote: > >> > Yes. > >> > E 0 > >> > E 1 > >> > 0<1 > >> > x<y -> E z: x<z<y > > >> > This leads to some level of infinites between counting numbers. > > >> That's hard to believe. The only thing that your definitions do > >> is define two numbers, 0 and 1, and an order relation for them. > >> Where are all of the rest of the reals? Or even the naturals? > > > How do you read the last statement? Between any two reals lies another > > real. Lather, rinse, repeat. Does this not lead to an infinite number > > of reals, or even rationals, between any two naturals? (sigh) > > If I read the above "axioms" correctly, you don't have the naturals. > You have 0 and 1. So what? That wasn't supposed to be a demonstration of all of mathematics, but a demonstration that between two counting numbers exist some infinity of intermediate numbers. Do you complain at the Ford dealership that their cars don't have wings? > > I assume that you intend < to be a linear order. You should say so, > preferably by including the axioms for linear order. In that case, all > you've axiomatized is a dense linear order with top and bottom elements. Yeah, you get it. A dense linear order over a finite interval equals an infinite number of intermediate points. > > No arithmetic. No metric. No topology, aside from the various > topologies that are standard for linear orders. Right. So you don't object so far? That's refreshing. > > Not much of anything that we expect from [0,1]. I never said that was a statement of all the axioms required for mathematics. I was simply saying there is an infinity of points in the unit interval, and that I call that number a zillion, formerly known as Big'Un. Thus we can combine measure with the concept of some uncountable unit for a more unified theory of infinite set size. It really boils down to a segment-sequence topology, as opposed to point- set, as somewhat described in Infinite Induction and the Limits of Curves. Ack! I guess it was on my old Cornell webpage, which no longer exists. I'll have to get a new web page going anyway... Tony
From: Jesse F. Hughes on 11 Jun 2010 11:39 Tony Orlow <tony(a)lightlink.com> writes: > On Jun 11, 10:17 am, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: > >> Not much of anything that we expect from [0,1]. > > I never said that was a statement of all the axioms required for > mathematics. I was simply saying there is an infinity of points in the > unit interval, and that I call that number a zillion, formerly known > as Big'Un. Yes, any dense linear order with top and bottom element contains infinitely many points, though not necessarily an uncountable number. Surely everyone agrees with this simple observation. > Thus we can combine measure with the concept of some uncountable unit > for a more unified theory of infinite set size. It really boils down > to a segment-sequence topology, as opposed to point- set, as somewhat > described in Infinite Induction and the Limits of Curves. Ack! I guess > it was on my old Cornell webpage, which no longer exists. I'll have to > get a new web page going anyway... I suppose the proof is in the pudding. As it stands, I'm not sure what you're talking about here. -- "It's my belief that when religion and pseudoscience achieve an official status within a culture [...], then genocide, war, oppression, injustice, and economic stagnation are sure to follow." -- David Petry, on why |X| < |P(X)| is bad, bad, bad.
From: Jesse F. Hughes on 11 Jun 2010 11:45
Tony Orlow <tony(a)lightlink.com> writes: > On Jun 10, 5:09 pm, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: >> Tony Orlow <t...(a)lightlink.com> writes: >> > On Jun 10, 3:05 pm, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: >> >> Transfer Principle <lwal...(a)lausd.net> writes: >> >> > To me, the difference between ZFC and non-ZFC theories >> >> > is more analogous to the difference between a flight >> >> > on American Airlines from JFK to LAX, and a flight on >> >> > Virgin America from JFK to LAX. In this analogy, >> >> > MoeBlee is forcing everyone to fly only on American >> >> > Airlines and not Virgin or any other airline. >> >> >> No, he's not. Moe has *never* said that ZFC is the only acceptable >> >> theory. >> >> >> > In another thread, he claims that he doesn't consider ZFC to be the >> >> > best theory, but the fact that he compares ZFC to an airliner and >> >> > other theories to rubber balls speaks for itself. >> >> >> The fact is that Tony, AP, etc., have *not* offered any coherent >> >> mathematical theory at all *and you know it*. Thus, if Moe criticizes >> >> their blatherings, then it is extraordinarily disingenuous to claim this >> >> as evidence that he accepts only ZFC. >> >> >> -- >> >> Jesse F. Hughes >> >> >> "Really, I'm not out to destroy Microsoft. That will just be a >> >> completely unintentional side effect." -- Linus Torvalds >> >> > "Disingenuous" means "lying". I believe Transfer's comment falls into >> > the category of a best-guess interpretation of Moe's motives. You, >> > Moe, Virgie, The Tribble and others seem completely closed to the >> > concept of any improvement on the standard obfuscation. For, "none >> > shall drive us from the Garden which Cantor has created for us". If it >> > doesn't produce fruit, it's time to plant a new bed, or at least >> > fertilize. >> >> I have nothing against alternative definitions of set size, but you have >> offered no clear definition at all. >> >> Nor do I have anything against alternative theories. Why should I? >> Indeed, in a previous life, I did a bit of work in ZFA, which is an >> anti-well-founded variant of ZFC. >> >> The criticisms that you receive are based primarily on the fact that you >> have never offered a clear and complete definition of set size that >> really competes with cardinality at all. Instead, you have a definition >> that applies to *some* (not all) sets and --- according to your own >> description --- gives different answers depending on one's perspective. >> >> That's not a promising mathematical definition. > > Then I guess natural density is not very promising either. Huh! Not as a general notion of set size, no. But it is useful in certain contexts. Let E be the set of even numbers and S the set of squares. It's perfectly sensible to say that, while E, S and N have the same cardinality, the natural density of E in N is 1/2, while the natural density of S in N is 0. Who could criticize such a claim? Now, if I said instead that cardinality is foolish and we should use natural density as a measure of set size, then of course I am being silly. Natural density is no replacement for cardinality. It just doesn't do the same job. -- "Now, once [James's research] is accepted, number theory is the wild, wild, west of the intellectual field and the hottest field on the planet in terms of potential for new entries. [...] The future in number theory belongs to the kids." -- James S. Harris corrupts youth |