fom
Posts:
1,037
Registered:
12/4/12
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Re: Distinguishability of paths of the Infinite Binary tree???
Posted:
Dec 26, 2012 2:44 PM
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On 12/26/2012 5:29 AM, Zuhair wrote:
> On Dec 26, 12:21 pm, Virgil <vir...@ligriv.com> wrote:
>>
>> For any finite set of such strings, finite initial segments suffice to
>> distinguish all of them from each oterhbut for at least some infinite
>> set, no finite set of finite initial segments suffices.
>>
> Yes but a countable set of them suffices! no?
>
No.
The diagonal argument generates a distinct logical
type given a particular assumption.
A completion of an incomplete dense topological space
populates the namespace of the new logical type.
The completed infinity of eventually constant infinite
strings is required to separate the completed infinity
of arbitrary strings.
Again, THE COMPLETED INFINITY OF EVENTUALLY CONSTANT
INFINITE STRINGS is required to separate the completed
infinity of arbitrary strings. These are not the
finite strings. The finite strings collect the
infinite strings into the open sets of the
topology.
This is what is meant by a dense set. Think of
the fact that there is a rational between every
two irrationals and an irrational between every
two rationals. The order relation of the rationals
grounds the order relation of the reals, and hence
gives an identity criterion for the reals in terms
of antisymmetry. But, this is not an existence
criterion.
A completed infinity is admitted as a base
assumption of the original argument.
A completed infinity is required to substantiate
any named particular of the new logical type.
There is no countable-to-uncountable magic until
one claims that there is a correspondence --
only type distinctions.
========================
These constructions trace back to the incommensurables
of Grecian geometry. The counter-intuitiveness
arises from geometric intuitions associated with
invariance under rigid motion and the measurement
of area.
The reason I mention the measurement of area is
because of an example from a statistics book I
once read. Area is a non-linear function of
linear measurements. As such, the random errors
of linear measurements can produce systematic
error in the subsequent calculation of area.
This does not happen with linear functions.
This example speaks to the general treatment
of differentials in calculus. Modern
differentials use the little-oh concept to
deal with small linear errors in the neighborhood
of a definite value.
dx=delta(x)
dy=f'(x)dx
delta(y)=f(x+delta(x))-f(x)
delta(y)=f'(x)h + o(h) where o(h)->0 as h->0
df=f'(x)h
Whereas it once seemed that the calculus
depended on the structure of the real
numbers, that dependency is less apparent
now. At the time Cantor and Dedekind did
their investigations, infinitesimals were
still actively being researched. Continuity
seemed to depend upon the geometric completeness
of the real line, but there had been a strong
desire on the part of late nineteenth century
mathematicians to get away from geometric
reasoning. As various algebraic systems
different from naturals, rationals, or reals
had been shown to have arithmetic relations,
the notion of "number" had been expanded.
Thus Cantor and Dedekind had little inhibition
against introducing logical types having the same
relations as the numbers with which everyone
had been using and then calling them "numbers."
But, if you need something intuitive
to ground what Cantor and Dedekind
were doing, look to the fundamental
theorem of algebra which asserts that
the field of complex numbers is
algebraically closed. The proof
depends upon the fact that every
positive real number has a real
positive square root and that every
polynomial of odd degree over the
real numbers has a root in the real
numbers. These assertions depend in
the first instance on the existence
of irrational numbers and in the
second instance on the intermediate
value theorem (meaning no "gaps" like
with the rational numbers).
So, in order for (x^2-2)=0 to have
a solution, there has to be something
that is not a rational number but
behaves like a number.
All known proofs of the fundamental
theorem of algebra depend on real
analysis (which is not the same
as calculus or approximation
theory).
Example on area referred to above:
http://books.google.com/books?id=BQDP5W4xAXUC&pg=PA69&lpg=PA69&dq=statistics+bias++%22area+of+a+circle%22&source=bl&ots=ZXi5Mdiu3Y&sig=Cxnp7akTywOLw2Vo0-ZoOsWJz4A&hl=en&sa=X&ei=ljnbUI-eGaLe2AWMnoHgBQ&ved=0CFkQ6AEwBg#v=onepage&q=statistics%20bias%20%20%22area%20of%20a%20circle%22&f=false
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