Re: infinity

sci.math_20040121.rtf:In partitioning the rationals into a set B
greater and a set A less than an irrational p, for any a and b there is
an infinite set of irrationals Q(a,b){p}, where Q(a,p) is an infinite
set of rationals between a and p and Q(p,b) an infinite set of
rationals between p and b.
sci.math_20040121.rtf:Virgil writes: "Thus, in the set of rationals, an
increasing sequence converging to sqrt(2) and a decreasing sequence
converging to the same value (that is to say equivalent as Cauchy
sequences), do not have any rational number between them.", and I
disagree. The sequence is in a state of convergence. Here we are
talking about monotone sequences, a(x)<a(x+1) and b(x)>b(x+1). For any
integer x, there exists a(x+1)<c and b(x+1)>c. When Virgil claims "no
rational number is between them", he means that for any rational number
q between them that there is some probably large value y thus that
a(x+y)>q or b(x+y) < q. By the same token, as Virgil's is an
incomplete and thus untrue statement, there exist infinitely many
rationals between q and c for any q he may select, and indeed in the
neighbohood of c.
sci.math_20040121.rtf:Now I am thinking about this and I get to a
difference between "for each and every" and "for every and all". For
each rational a and b there exists c. For all elements of the sets of
rationals {a_1, a_2, ...} and {b_1, b_2, ...} then I wonder that for
all of them there exists rational c. This is where although for no
finite x is a(x) not different than c, there are infinitely many x, the
integers, the limit of their difference is zero. Then again, each
integer is itself finite.
sci.math_20040121.rtf:Let us reiterate Cantor's proof: assume a
sequence x that is the range of a function X from N onto R. X is not
monotone, and in fact must infinitely change directions. Construct two
sequences a and b as so: a_1 (for N={1, 2, ...) is set equal to x_1.
b_1 is the value of x_i for the first value of i such that x_i <> x_1.
a_(n+1) is the first value of x x_i such that the index, i, is greater
than then index of the x sequence term for a(n), and a(n)<a(n+1)<b(n).
b_(n+1) is then the first value of x x_i such that i_b(n+1) is greater
than i_a(n+1) and a(n+1)<b(n+1)<b(n).
sci.math_20040121.rtf:So the proof begins by assuming some sequence x
has all of R as its range and also that x is not monotone and the sign
of x_n - x_(n+1) changes infinitely many times for a and b to be
infinite sequences, etcetera.
sci.math_20040121.rtf:The two monotone sequences a and b "move toward
each other", that is to say, for index n, b(n)-a(n) > b(n+1)-a(n+1).
Then the statement of the proof states that because of the gaplessness
of R, that [a(n), b(n)] always contains an infinite subset of R. The
same holds true for the density of Q and Q[a(n), b(n)] for sequences a
and b constructed from a sequence x of N onto Q.
sci.math_20040121.rtf:What if the function X from N to R was monotone?
Then the sequence a would have exactly one element and the sequence b
one or zero. Thus a necessary statement of that proof may be X is not
monotone and the sign of the difference of consecutive elements changes
infinitely many times.
sci.math_20040121.rtf:About the sequence x, there are further
assumptions on it than it is not monotone. The assumption is that its
limit is c, or rather, the sequence constructed from c-a1, b1-c, c-a2,
b2-c, ..., converges to c, and that that sequence is infinite and does
not equal c for any value.
sci.math_20040121.rtf:The existence of a monotone function from N to R
would not be disqualified by said proof, thus the proof must say "there
are no functions from N to R that both change signs in the differences
of infinitely many consecutive values and otherwise enable the
construction of these sequences a and b by absolutely converging in a
way to c."
sci.math_20040121.rtf:So there are several things to consider. One is
the "for each" vs. "for all". Another is the specific non-monotone and
infinitely changing consecutive difference sign change of the function
generating X. Another is the form of A and B thus that their
convergent values differ by a non-zero value.
sci.math_20040121.rtf:Assuming a suitable function from N to Q, able to
generate infinite sequences a and b that each converge monotonically to
a point c, c may be irrational. The limit of the difference b(n)-a(n)
is zero.
sci.math_20040122.rtf:In brief the proof would not generate infinite
sequences a and b if the function X N->R is monotone, or piecewise
monotone.
sci.math_20040122.rtf:Please explain why any function from N to R would
necessarily be non-monotone in the requisite way to generate two
infinite sequences a and b.
sci.math_20040123.rtf:If the function's range does not form the
sequence x as was described as specifically non-monotone, changing
signs of consecutive differences infinitely many times, generating
infinite sequences a and b, and with a form of convergence of each of a
and b to some value, then it does not fulfill the conditions (for any
function N<->R) and thus imply the conclusions presented therein.
sci.math_20040123.rtf:In this case, the function is assumed to generate
two infinite sequences a and b, yet there are classes of functions that
would not.
sci.math_20040123.rtf:So actually a question is whether any function X
would necessarily provide the infinite, monotone, convergent sequences
a and b, and why or why not.
sci.math_20040123_b.rtf:Here you'll notice I assume a mapping from the
naturals to the reals, leading into concepts of the indefinitely and
variously ordered sequence of real numbers as points on the real number
line and the vague fugue of infinitesimality. This is where we have
discussed similar concepts at length before.
sci.math_20040123_b.rtf:Let's examine that result. Is it the sum of
the area of many narrow rectangles between the curve and the axis? In
a way, it is. Yet, we can not get an exact result for any finite
number of those rectangles. Instead it is said that the width of the
rectangle goes to zero as the number of rectangles goes to infinity.
Is it a rational or real zero, or infinity? The points within the unit
interval are dense because they are uniformly distributed on any finite
scale, and infinumerous.
sci.math_20040124.rtf:A Google(tm) search for "iota infinitesimal"
provides some references to the consideration of iota as an
infinitesimal quantity, particularly with its quality of being the
least real quantity.
sci.math_20040124.rtf:http://groups.google.com/groups?as_q=iota+infinitesimal&as_ugroup=sci.*
sci.math_20040124.rtf:In the light of this context, about sequencing
the reals, I use iota to help describe the reals as a sequence of
points. Then, I posit a sequencing with zero, then the positive iota,
then the negative iota, then the next positive value, and the next
negative, ad infinitum.
sci.math_20040125_b.rtf:The reals are not necessarily the powerset of
the integers. Is there a way to construct, say, the unit interval of
reals from the powerset of the naturals, with assigning explictly each
member of the powerset of integers to be a unique real number? I guess
there is an obvious example of making the infinite binary sequences
that represent each subset of N, then there are issues about dual
representation. That's not compatible with the ordinal representation
of singletons or forms of composites, pairs with zero, as ordinals and
integers, various other forms of ordinal constructions. The idea with
that is to figure out constructions of the empty set to uniquely
identify not only each number but each mathematical concept, to
mechanistally, digitally, operate upon them.
sci.math_20040126.rtf:Iota's an infinitesimal. If you try to divide it
by two you can't treat it as anything except zero. The result of that
operation is undefined. Iota is atomic.
sci.math_20040126.rtf:The value of iota is that of the reciprocal of an
infinite quantity, the infinitesimal quantity that if you add together
that infinitely many values of iota the result is equal to one, it is
non-zero.

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