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Re: distinguishability  in context, according to definitions
Posted:
Feb 17, 2013 1:49 AM


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 fullblown 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 nearfield 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/whatdoestopologyhavetodowith.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 finitestate 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 .



