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.
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
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
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,
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
> 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?