Uncountability of the Rationals?
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From: Tony Orlow on 16 Jun 2010 09:56 On Jun 16, 2:15 am, Transfer Principle <lwal...(a)lausd.net> wrote: > On Jun 15, 2:03 pm, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: > > > Tony Orlow <t...(a)lightlink.com> writes: > > > IF it has an algebraic bijection with N+ it can be compared therewith.. > > What's "algebraic bijection" mean? > > I assume TO means a bijection f: N+ -> S such that f is an > algebraic function (i.e., a polynomial, rational, radical). Actually, I meant a general formulaic relation, not necessarily algebraic, but with an inverse function that can be determined through algebra, not necessarily restricted to polynomials, but also including exponents and logs, etc. > > According to what TO calls his "Inverse Function Rule," if > such a function exists, then the set S has set size (or > Bigulosity) of f^-1(zillion). Not exactly. "Zillion" is the uncountable unit infinity. Omega, or 'tav' as Brian calls it, is the Bigulosity of N+. So, f^-1(tav)- f^-1(1) (where "f^-1" is the inverse of f). > > (This is the compositional inverse, not the reciprocal. I > actually prefer the notation f_o^-1, with a "o" subscript > to denote composition.) Right. > > The examples given by TO thus far include: > > S = {2,3,4,5,6,7,8,9,...} > f(x) = x+1, Big(S) = f^-1(zillion) = zillion-1 > > S = {2,4,6,8,10,12,14,16,18,...} > f(x) = 2x, Big(S) = f^-1(zillion) = zillion/2 > > S = {1,4,9,16,25,36,64,81,...} > f(x) = x^2, Big(S) = f^-1(zillion) = sqrt(zillion) > > S = (2,4,8,16,32,64,128,256,512,...} > f(x) = 2^x, Big(S) = f^-1(zillion) = lg(zillion) Yes, but replace "zillion" with omega or tav. :) TOny
From: Tony Orlow on 16 Jun 2010 10:07 On Jun 16, 9:53 am, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: > Tony Orlow <t...(a)lightlink.com> writes: > > The H-riffics are a bit of a sidebar. Besides, it occurs to me that > > within R there exists one element which cannot be a child node of an > > element in R, namely, 0. So, 0 can be taken to be the root case for > > the tree of values, the foundation. Additionally, if one wants to > > include all reals positive and negative then the H-riffics may be > > redefined as: > > > E(0) > > E(x) -> E(-2^x) ^ E(2^x) > > I suppose this means > > 0 exists > > if x exists, then so does -2^x and 2^x. > > But, it does not follow from this that[1], for instance, 3/4 exists. I > suppose that you want to include at least finite sums of things that > exist, so then 3/4 would indeed exist. > > But then you still wouldn't have 1/3, unless you include some infinite > sums. Of course, you don't want to include all infinite sums, but only > those that converge. I don't know whether it's easy to specify this > condition (convergence of an infinite sum) in the context you have in > mind. > > Also, you claimed that you'd get all of the reals positive and > *negative*, but you'll need another assertion to get the negatives. > > But obviously, you do *not* get all of the reals by only the two axioms > you wrote above. Again, you presume too much. > > > Just thoughts, really unrelated to Bigulosity, and apparently > > unqualified to be an explicit well order of the reals. > > Footnotes: > [1] I'm ignoring the fact that statements such as "0 exists" are not > really formalizable in classical FOL. If you prefer, for set H: 0eH xeH -> -(2^x)eH ^ (2^x)eH For all xeR except for x=0 exists parent element yeR:y=log2(abs(x)). The string denoting 3 is not a standard countable string but is uncountable, and then continues to branch through the value tree after 3 to produce more reals and a continuation of the already uncountable sequence of bits. Nonstandard, but somewhat interesting. > > -- > Conservative, n: > A statesman who is enamored of existing evils, as distinguished > from the Liberal who wishes to replace them with others. > -- Ambrose Bierce Tony
From: Jesse F. Hughes on 16 Jun 2010 11:08 Tony Orlow <tony(a)lightlink.com> writes: > On Jun 16, 2:15 am, Transfer Principle <lwal...(a)lausd.net> wrote: >> On Jun 15, 2:03 pm, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: >> >> > Tony Orlow <t...(a)lightlink.com> writes: >> > > IF it has an algebraic bijection with N+ it can be compared therewith. >> > What's "algebraic bijection" mean? >> >> I assume TO means a bijection f: N+ -> S such that f is an >> algebraic function (i.e., a polynomial, rational, radical). > > Actually, I meant a general formulaic relation, not necessarily > algebraic, but with an inverse function that can be determined through > algebra, not necessarily restricted to polynomials, but also including > exponents and logs, etc. That's not very explicit. -- Jesse F. Hughes "If the car stops and you're not getting out, then you have to start it again." -- Quincy P. Hughes on his father's skills with a manual transmission.
From: Jesse F. Hughes on 16 Jun 2010 11:23 "Jesse F. Hughes" <jesse(a)phiwumbda.org> writes: > So, 1/3 is in R because log_2(1/3) is in R, and that's in R because > log_2(log_2(1/3)) is in R and so on? Of course, that should be log_2(|log_2(1/3)|) and so on. -- Jesse F. Hughes "Basically there are two angry groups. I am a harsh force of one. Against me is a society of mathematicians. So far it's been a draw." -- JSH gives another display of keen insight.
From: MoeBlee on 16 Jun 2010 13:13
On Jun 15, 7:22 pm, Transfer Principle <lwal...(a)lausd.net> wrote: > On Jun 15, 3:15 pm, MoeBlee <jazzm...(a)hotmail.com> wrote: > > On Jun 15, 4:40 pm, Transfer Principle <lwal...(a)lausd.net> wrote: > Is it possible for one to accept > all of the axioms of a theory, without accepting all of > the consequents of those axioms? Only if one doesn't accept the particular consequence relation you have in mind when you say "the consequents". But Orlow says (to the effect) classical logic is okay. Orlow's confusion is over the role of definitions in mathematics and in formal mathematical theories. > And if one does reject > theorems proved from axioms that he finds unobjectionable, > does he deserve to be called by a five-letter insult? If a person says "axioms okay, logic okay" but "I reject certain theorems that have terminology in them but not in the axioms", then he reveals he does not understand how definitions work in mathematics, and if over and over, and over a period years, he fails to address explanations about how definitions work in mathematics, and yet he fails to inform himself about this through whatever other media such as books, then I think that is at least one point toward considering that person as a crank. For me, probably not enough in itself to justify saying the person is a crank, but with other things, it would be a strong prong showing the person is a crank. > So I believe that the best we can do with regards to > preventing as many ordinals as possible from existing > would be to switch from ZF to Z. But Orlow objects to LIMIT ordinals. And Z proves the existence of a limit ordinal, viz. w. (True, ZF proves that there exists limit ordinals other than w.) > If two sets have the same Bigulosity, then they have the > same cardinality as well. Your summer assignment is to construct rigorous definitions of all such terminology as "Bigulosity". Your report is due the start of the new semester on September 15. MoeBlee |