Re: infinity

sci.math_031005_b:
Say two points start on a race from zero along the reals towards
infinity. One point has a velocity of 1 distance unit / time unit, the
other of 2 distance units / time unit. At any point in time point one
is at point x where point two is at point 2x.

sci.math_031006_b:
Then again I think the impulse function evaluates to "half infinity" at
zero, and consider the Gamma function on negative integers to have
values of various finite multiples of a scalar infinity.

sci.math_031006_b:
About the uniform probability distributions over intervals of reals,
that's not about making some new definition of what a probability
distribution is. It's about applying the characteristics of a
probability distribution to an infinite population. We were talking
about the probability of an infinite binary seqence having one element
being on, the rest off. That probability is expressed as n/2^n, as n
diverges to infinity. The probability of any possible sequence is
equal to 2^n/2^n, in the limit: one. So anyways out of those n
possible sequences with one on bit and the rest off bits, each is
equally probable. The probability of each among all possible infinite
binary sequences is being1/2^n, the probability of each among all
infinite binary sequences with one on bit is 1/n. So a theoretical
(read: thought experiment) method to generate an element of N is to
once again flip infinitely many coins. At this point it's a crazy, or
rather, unconventional thought experiment in that the first coin toss
says whether it is oo/2 or greater or less than oo/2. Assume it's a
long sequence of zeros, then it would be saying about whether the
result is greater than or equal to oo/4, oo/8, oo/16, etcetera.
Without a method to generate a sample from a uniform distribution over
the natural numbers, it's still that the probability of selecting any
is 1/|N|.

sci.math_031006_b:
I wrote a brief post that summarizes my understanding of the
infinitesimals, it talks about 1-infinitesimals, 2-infinitesimals,
etcetera, n-infinitesimals, with the oo-infinitesimal being zero.

sci.math_031009:
What's a proper subset of the rationals or irrationals that is dense in
the reals, where its complement is infinite?

sci.math_031009_b:
These are well-known and almost trivial, for example, the sum for i=1
to infinity of 1/25000^i is 1/24999, 1/9 in decimal is .1111...,
etcetera. I assume that for real x>1 that the sum of x^-i for i from 1
to oo is 1/(x-1).

sci.math_031009_b:
What do you think about the canonicalization of the infinite binary
sequences? That is to say, for the sequence .010101(01)... with half
zeroes and half ones, that you could exchange any two sequence elements
any number of times and not get .001001001(001).... Is there a rule to
exchange (permute) the elements of the sequence .0101(01)... to get
00010001(0001)...?

sci.math_031009_c:
About (a+b)/2 =/= a =/= b, my point is that any a and b that you have
selected ("unique" numbers) are separated by infinitely many other
numbers. Their average is as well unique and infinitely distant from
either between them in the "consecutive sequence of the continuous
reals."

sci.math_031009_c:
In a way it's about reconciling point-ness and line-ness. A point is
the intersection of two (and infinitely) many lines that intersect, a
line connects two (and infinitely many) points that are collinear. It
takes two points to describe a line, it takes two non-parallel
(coincident) lines to describe a point. That's obviously extravagant.
Then there are three non-collinear points for a plane, and three
coincident planes for a point, etcetera. The lines are not parallel.
They're correlated.

sci.math_031009_c:
The rationals and irrationals are each dense in the reals, they are
disjoint and their union is the reals. Then again, so are the
algebraics and transcendentals, or for example the rationals p/q where
p and q are relatively prime and q is even and that set's complement in
the reals. Also a constant function is everywhere discontinuous with
any of those sets as its domain. This causes me angst. I derive humor
from using the word angst. The model can't have density 1/2 for each
of the rationals and irrationals where the same reasoning applies to
the algebraics and transcendentals because the (set of) algebraics is a
proper superset of the rationals with infinitely many proper subsets
that are supersets of the rationals. That reminds me of about the
algebraics and transcendentals, with the rationals, algebraic
irrationals, and transcendentals.

sci.math_031009_d:
I consider .00010001(0001)... as an infinite sequence and how to
permute its elements to get .01010101(0101).... It seems almost
simple, move the 1 in the fourth place to the second place, and move
each one in the 4x'th place for x>1 to the 4(x-1)'th place. There's
certainly no dearth of ones and zeros, the amount of each is infinite.
Yet, the zero that was in the second place would have to go somewhere,
and necessarily displace another.

sci.math_031009_d:
Then there are the sequences of type b with infinitely many zeros and
ones and infinitely many more or less of of one than the other,
opposite symbol, eg sequences representing subsets of the naturals like
the primes, powers of two, etcetera. Consider the powers of the
primes.

sci.math_031009_d:
The sequences .(01)... and .(10)..., for half zeros and half ones, are
perhaps the simplest cases with infinite numbers of ones and zeros to
consider. Exchange each one and zero of the representative subsequence
and it becomes the other.

sci.math_031009_d:
Given instructions: form a sequence of infinitely many ones and
infinitely many zeros, there are many ways to describe a rule or method
to do that. Given instructions: given a sequence with infinitely many
ones and infinitely many zeros, exchange its elements to form another
sequence, there are less sequences.

sci.math_031009_e:
You don't have to flip infinitely many coins to tell if a real number
from [0,1) is less than 1/2, just one.

sci.math_031009_e:
Consider x and y, independent variables, x goes to infinity: y/x is
effectively zero or indeterminate, x/x is one, x is a dependent
variable of itself. If y is a dependent variable of x, then, y/x is
not necessarily zero or indeterminate (divergent).

sci.math_031009_e:
The probability of a binary sequence of length n having one "on"
element is n/2^n, and the sum of all the probabilities of the
possibilities is 2^n / 2^n=1. The sum from zero to infinity of zero is
zero. One half is definitely a real number. The probability of a
binary sequence of length n having one "on" element is the same as that
of it having one "off" element.

.

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