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Re: distinguishability - in context, according to definitions
Posted:
Feb 17, 2013 1:49 AM
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On Feb 16, 10:59 pm, fom <fomJ...@nyms.net> wrote:
> On 2/16/2013 5:55 AM, Dan wrote:
>
> > I can't concentrate enough to understand the whole post, however :
> > Machines can't decide whether 1.0... = 0.9999999... .
>
> No. But, people do. It is taken as a trivial "decision".
>
> The beginning of the post states that it was an exercise
> in logic. In that respect, there was no meaning attached
> to 1.000... = 0.999... .
>
> In the logical construction of the real numbers system
> using Dedekind cuts, one must fix a choice as to which
> kind of rational cut will be taken as canonical. The
> cuts corresponding to the rational numbers will either
> all be chosen relative to least upper bounds or they
> will all be chosen relative to greatest lower bounds.
>
> The reason for this is that the identity of real numbers
> relative to the construction is based upon the order
> relation inherited from the rational numbers used in the
> preceding logical construction.
>
> When one looks at this presentation in books on real analysis,
> the general tendency is to fix the rational cuts relative
> to least upper bounds. This comes from the tendency to view
> the construction relative to initial sequences. But, when
> one considers the canonical name we use because of the fact
> that some ratios form quotients that halt relative to the
> Euclidean long division, the canonical name for the rational
> cuts should be oriented from fixing the rational cuts
> relative to greatest lower bounds.
>
> > In general , machines can't decide the equality or inequality of real
> > numbers ,or infinite strings in general ,without the 0.(9) = 1
> > equivalence of real numbers.
> > This comes as a consequence that all computable functions are
> > continuous ,while equality is not.
>
> Nice remark. I have your link up in my browser and look forward
> to reading it.
>
> I just completed a discussion of identity
>
> news://news.giganews.com:119/88qdnU1ZNsB9j4LMnZ2dnUVZ_uudn...@giganews.com
>
> or look for "when indecomposability is decomposable" on
> sci.math/sci.logic
>
> Relative to the trichotomy of real numbers in relation to
> Dedekind cuts, the simplest view of what you are
> expressing is the quotient topology for a map from the
> real numbers into a three point set. The example from
> Munkres "Topology" goes something like:
>
> p(x)=a if x>0
> p(x)=b if x<0
> p(x)=c if x=0
>
> The quotient topology induced by this map is given by
>
> {{a},{b},{a,b},{a,b,c}}
>
> This example shows why the Dedekind cuts suffice to
> construct the real numbers, but they do not reflect
> the topology above because the logical construction
> requires that rationals correspond with a uniform
> choice with regard to the order. In fact, that choice
> must be uniform with respect to the ordinal sequencing
> of natural numbers as that order is what is held invariant
> in a full-blown construction. So, a function like the
> one above for a Dedekind cut rational has the form
>
> p(x)=a if x>=0
> p(x)=b if x<0
>
> Moreover, this is the choice is precisely the direction
> we make with respect to the trivial "decision" concerning
> canonical representation associated with
>
> 1.0... = 0.9999999...
>
> To address this problem, one must look at the metrization
> of pseudometrics. In "Topology" by Kelley, there is a
> somewhat strange circumstance in the proof of the metrization
> lemma in the chapter on uniform spaces. It seems that
> Dedekind cuts are logically prior to Cantorian fundamental
> sequences because metrization invokes the least upper bound
> property in its proof.
>
> Now, what is important about a pseudometric is that the
> relevant axiom is attaching a metric structure to an identity
> relation in the logical sense. That is,
>
> x=y -> d(x,y)=0
>
> For a metric that axiom is
>
> x=y <-> d(x,y)=0
>
> So, in the hierarchy of logical definition, one obtains the
> real numbers from Dedekind cuts relative to a logical identity
> relation. Then, a definition of least upper bound and greatest
> lower bound for that system may be defined. Then, provided that
> the nature of relations used in the metrization lemma are
> satisfiable, one uses the function constructed in that
> proof to put a metric on the system of Dedekind cuts.
>
> Given any metric structure, the system conforms to the
> definition of a metric space. Then, one can use the order
> relation of the "new" rational numbers (those that correspond
> with rational cuts) to define Cantorian fundamental sequences.
>
> It is at that point that the trichotomy expressed by
>
> p(x)=a if x>0
> p(x)=b if x<0
> p(x)=c if x=0
>
> is admitted.
>
> If you look in my other post, however, it is clear that
> one could use the "real numbers" of a near-field on 9
> symbols as the a, b, c in the example above.
>
> p(x)=1 if x>0
> p(x)=-1 if x<0
> p(x)=0 if x=0
>
> Or, looking at the canonical order for my labels in
> that post, and treating a finite subsequence circularly,
>
> p(x)=NAND if x>0
> p(x)=NOR if x<0
> p(x)=IMP if x=0
>
> That canonical order is given as
>
> <LEQ, OR, DENY, FLIP, NIF, NTRU, AND, NIMP, XOR, IMP, NAND, TRU, IF,
> FIX, LET, NOR>
>
> The subsequence in which I am interested is
>
> <IMP, NAND, TRU, IF, FIX, LET, NOR>
>
> And, since NOR is the last in the sequence, the order of the
> three elements I want relative to wrapping the subsequence
> is given by
>
> <NOR, IMP, NAND>
>
> In any case, the point is that your observation speaks directly
> to the fact that identity is a topologically complex matter
> and that the trichotomy of the real numbers cannot be taken
> for granted.
>
> >http://blog.sigfpe.com/2008/01/what-does-topology-have-to-do-with.html
>
> > Our language is countable , the real numbers are not . Thus we don't
> > work directly with "the plenum" , the real numbers as infinite strings
> > of digits . How could we? Who has time to read an infinite string?
>
> Well, (in all good humor) Alan Turing for one.
>
> Correct. While not an application one would have expected for
> automata, the fact is that a representation of the logical problem
> using automata led to reduced machines because of the Euclidean
> algorithm.
>
> > What we do work with are the 'finite definitions' of these 'infinite
> > numbers' , for all the real numbers we can think about .Whether any
> > other kind of number is "real" ,other than those we can think about,
> > depends on your "orientation" in mathematics , though I affirm they're
> > not 'real' .
> > Now , these 'finite definitions' "subdue" the infinite numbers, making
> > their contents accessible to our tiny, finite minds . Thus , equality
> > becomes decidable , and 1 = 0.(9) while 1 not = 0.9999998(9) .
>
> Thanks to my recent postings on sci.logic and sci.math, I
> define "subdue" to be the algebraic extension field over the
> rationals characterized by
>
> a+b*surd(5)
>
> > One question remains : Is anything lost when "replacing" these
> > "infinite objects" by their finite definitions?
>
> No, but something is lost when we treat
>
> 1.000... = 0.999...
>
> trivially if, in fact, one is interested in how one uses
> definition and the sign of equality to 'subdue'.
>
> That was an important point of the post. The decision turns
> a lossless representation into a lossy representation.
>
> What branch of mathematics other than finite-state automata
> could even represent that?
>
> > The beautiful fact is that the objects of mathematics are analytic ,
> > not synthetic , thus nothing is lost in terms of meaning by saying
> > 0.(9) instead of 0.99999.... and much is gained in terms of what we
> > could do with "the finite 0.(9)" , as opposed to "the infinite
> > 0.9999..." .
>
> The analytic/synthetic distinction? Are you sure you
> want to go there?
The problem of 1 = 0.(x) appears for any possible base of
numeration .
If you're bothered by the representation being 'lossy' , you can
always try continued fractions for the numbers in the interval [0,1] :
Each real number is represented by a (possibly infinite) sequence of
strictly positive integers :
You represent r by [a1,a2,a3 ..... an ...] meaning that
r = 0 + 1/ (a1 + 1 / (a2 + 1/ (a3 + .... )))
I'm pretty sure you can build up the whole of analysis this way ,
though nobody's bothered to do it, so it must be tedious.
That being said, I was never really bothered by the whole 0.(9) = 1
business , it's just a quirk in notation .
What seems far more troublesome is the representation of finite
fields , you always have to 'choose' one of many irreducible
polynomials if you want to work with them .
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