Uncountability of the Rationals?
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From: Tony Orlow on 25 Jun 2010 12:26 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+ > > 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. > > Thus, N+ has the property > > ind_1(N+) & (Ax)(ind_1(x) -> N+ c x). How do you draw that conclusion? > > Is this also your definition? Apparently not. > -- > Jesse F. Hughes > > "I have written many words to sci.math, some of them are not even > meaningless." --Ross Finlayson Some of Ross' words are not at all, but with some it's hard to tell. I like Ross. Tony
From: Jesse F. Hughes on 25 Jun 2010 12:41 Tony Orlow <tony(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. >> 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". >> >> 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.) >> 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. 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+. -- Jesse F. Hughes "That's what's brutal about mathematics! When you're wrong, you can have spent years, and lots of effort, and come out at the end with nothing." -- James S. Harris on the path of self-discovery (?)
From: David R Tribble on 25 Jun 2010 12:54 Tony Orlow wrote: > Okay, I should clear something up, here, as a misconception seems to > be spreading. Bigulosity does not just measure subsets of N+. It > measures any set formulaically bijected with N+, which may include > elements outside of N+. So by your definition, bigulosity is a relative measure of countable sets only. Uncountable sets (such as R) can't have bigulosities. > 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. 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+?
From: David R Tribble on 25 Jun 2010 13:17 David R Tribble wrote: >> You left out the part where Tony decides that there are >> "finite but unbounded" sets, like N. That is, they contain >> only finite elements (numbers), but since they have no >> "identifiable" largest element, they are "unbounded". > Tony Orlow wrote: > Foul: Misrepresentation > I never said any of that. Here are a few quotes from days gone by: http://groups.google.com/group/sci.math/msg/9ab9be0a33ae6119 [Jun 29 2005] |>| By having an unbounded range, meaning that there is no maximum distance |>| bertween members but there is also no infinite distance between members. |>| |> Then you do not have an infinite set, but an "unbounded" set. | |>| What is TO's definition of boundedness of a set in itself that allows it |>| to have members larger that its upper bounds? |>| |> Okay, I think i misspoke in those three words. |> The standard set N has maximum member N, [...] http://groups.google.com/group/sci.math/msg/2781e91c5911cd29 [Jan 03 2006] |> Of course there is no identifiable end to the finite naturals, |> but that doesn't mean there are an infinite NUMBER of them. |> Any infinite number of naturals will include infinite values, |> since any two naturals, a and b, with an infinite number of |> naturals between, them will differ by an infinite value, meaning |> that |a-b| is infinite, so that either a or b (or both) is |> infinite. If all values in the set are finite, then the |> difference between any pair of them is finite, and there are |> only an infinite number of intermediate values between any two. |> You have a finite set, there, pal. Tony Orlow wrote: > There are uncountable sets of only finite numbers such as {xeR: 0<x<1}. > There are countably infinite sets which are bounded such as > {xeQ,neN: q=1/n}, with a largest element. > 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. Last what? David R Tribble wrote: >> The situation is, and has always been since Tony started >> posting, that he wants everything to act like "numbers" >> including set "sizes", and specifically that they can be mapped >> to points on the real number line and obey the usual rules >> of (finite) arithmetic. Thus his constant mention of properties >> "without regard to finiteness", "infinite induction", "formulaic" >> infinities, infinite set members, "unit infinities", "bogus" >> bijections, and so forth. > Tony Orlow wrote: > Excuse me for trying to make math use "numbers". My bad. Your bad is not using consistent logic, nor seeing the inconsistencies and illogic. David R Tribble wrote: >> Look, Tony has demonstrated that he's a Crank many times >> over, with all of the usual symptoms: unwillingness to change his >> ideas even in the face of blatant illogic, abusive replies to >> disbelievers in his faith, proofs by declaration, assumed authority >> on terminology and topics he has no clue about, expectation of >> the impact of his ideas on the world, etc. > Tony Orlow wrote: > This isn't an "abusive reply" or obnoxious personal attack, is it? No, in point of fact. These are all verifiable facts. A simple search through the group archives provides pages of evidence to support everything I've said. > The sliver you perceive in my eye is but a reflection of the log in yours. Please present evidence that I've ever been guilty of any of the traits I listed above. David R Tribble wrote: >> I admit to finding the last few years occasionally entertaining, >> but lately it's become, well, boring. Nothing new is being offered, >> and more importantly, I'm not learning much new from the >> posters who actually know something about set theory. > Tony Orlow wrote: > I left for a few years. Maybe that's why you were bored. [...] Yes, you are the sole source of amusement to me.
From: Brian Chandler on 25 Jun 2010 13:37
Tony Orlow wrote: > On Jun 24, 3:55 am, Transfer Principle <lwal...(a)lausd.net> wrote: > > On Jun 21, 9:03 pm, Brian Chandler <imaginator...(a)despammed.com> > > wrote: > > > Transfer Principle wrote: > > > > On Jun 20, 9:29 am, Brian Chandler <imaginator...(a)despammed.com> > > > > wrote: Right, well, dipping down into this lengthy wossname... I wrote: > > > Here's a start for you: consider the sets > > > A = { "0", "10", "11", "100", ... } of (two-ended!) strings over > > > alphabet {0,1} starting with 1 > > > B = N ... the set of naturals (including 0), which we might represent > > > in binary > > > C = { 0, 10, 11, 100, ... } of integers whose decimal representation > > > only includes digits 0 and 1 (no sign) > > > I hope you can immediately see canonical bijections A <-> B and A <-> > > > C. Tony claims that B and C have different bigulosities, so your job > > > is to say which of the bijections (or both!) is not "strong". > > > > As I mentioned before, any alleged bijection between a set and one > > of its proper subsets indicates that the set contains extra elements > > which aren't mapped to unique elements. > > No it doesn't. There are no elements unmapped. There is a difference > in frequency of elements as the range increases. That is all IFR > detects, but it does it accurately. Ah, yes, the range. I remember that... > > In particular, we know that B contains extra elements. There is no > > set N of pofnats in this theory, but instead the set tav of tnats: This seems to be Transfer's attempt to address my question... > No, tav is the Bigulosity of N+, not a set. .... but your response seems to indicate he has missed the point. So let's skip again... > If the strings of B are bijected string-wise with those of C, they may > be considered equibigulous, OK, well, here's another copy of the definitions of B and C. Please explain what it means to "biject string-wise" the set of naturals and the set of integers whose decimal representation only includes digits 0 and 1. > > > B = N ... the set of naturals (including 0), which we might represent > > > in binary > > > C = { 0, 10, 11, 100, ... } of integers whose decimal representation > > > only includes digits 0 and 1 (no sign) Next: > ... but when the strings of C are specified to > be binary naturals, then one needs to decide how one is comparing the > two sets, and decimal notation comes into play with N=S^L. Well, the elements of C (sets normally have *elements* Tony, not "strings") are integers. A particular proper subset of the non- negative integers. What would it mean to "specify" that they are something else. Brian Chandler |