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Re: distinguishability - in context, according to definitions
Posted:
Feb 16, 2013 3:59 PM
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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_uudnZ2d@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?
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