Post by John GabrielPost by John Gabrielhttp://www.spacetimeandtheuniverse.com/math/4507-0-999-equal-one-772.html#post40616
The objects I use, that is, lim (n->oo) [L_n(m), U_n(m)] are *exactly* D. Cuts.
The poor idiot Dedekind could not describe them as precisely as I did.
Moron academics hate this proof because it exposes their stupidity.
Definition of D. Cut: It is a partition of the rational numbers into two non-empty
sets A and B, such that all elements of A are less than all elements of B, and A
contains no greatest element.
https://en.wikipedia.org/wiki/Dedekind_cut
A set partition of the rational numbers into two nonempty subsets S_1 and S_2
such that all members of S_1 are less than those of S_2 and such that S_1 has
no greatest member.
http://mathworld.wolfram.com/DedekindCut.html
The definition I use encapsulates these facts to prove that NO real number
can be represented this way.
You do not seem to have managed to correctly use the definitions
you quote. You are most likely blissfully unaware of your failure
(Dunning-Kruger effect) and you think that chanting some phrase
is all you need to do. ("By the Awesome Power of Dedekind Cuts,
I refute thee! I refute thee! I refute thee! Begone, thou foul
theorem!")
This could certainly be the reason that you think that "eliminating"
irrational numbers only to replace them with "incommensurable
magnitudes" which behave in every way like irrational numbers
has some sort of effect. On The Astral Plane, perhaps?
Talking to you about Dedekind cuts would be pointless.
Instead, I intend to talk to you about flying rainbow sparkle-ponies.
It's very likely to have just as little effect on you (Dunning-Kruger),
but there is the prospect of you chanting "By the Awesome Power
of Flying Rainbow Sparkle-Ponies ... "
Flying rainbow sparkle-ponies -- as I define them here -- have
all of the properties that are claimed for real numbers, and they
can be constructed from rational numbers using a bit of set theory.
Since you apparently are able to believe in the rational numbers,
and the only axiom that the reals have in addition to those of
the rationals is the least upper bound property, I will focus on
demonstrating the least upper bound property for the set of all
flying rainbow sparkle-ponies.
A _flying rainbow sparkle-pony_ is defined as a subset A of the
rational numbers Q which is not {} or Q, which is _closed downward_ ,
and which _does not contain a maximum_ .
The subset relation '=c' forms a _linear order_ on the set
of all flying rainbow sparkle-ponies.
For every bounded, non-empty set of flying rainbow sparkle-ponies /F/
there exists a least upper bound. That least upper bound is U /F/
---
Again, in more detail:
A _flying rainbow sparkle-pony_ is defined as a subset A of the rational
numbers Q which is not {} or Q, which is _closed downward_ , and which
_does not contain a maximum_ .
These are some examples of flying rainbow sparkle-ponies:
[-1] = {x| x e Q and x < -1 }
[355/113] = {x| x e Q and x < 355/113 }
[sqrt(2)] = {x | x e Q and ( x^2 < 2 or x < 0 ) }
"Closed downward": For all x e A, if y < x then y e A.
A is closed downward.
"Does not contain a maximum": For all x e A, there exists
some y e A such that y > x.
The subset relation '=c' forms a _linear order_ on the set
of all flying rainbow sparkle-ponies.
"Linear order":
Let A, B, and C be flying rainbow sparkle-ponies. Then
(i) ( A =c B & B =c C ) -> ( A =c C )
(ii) ( A =c B & B =c A ) -> ( A = B )
(iii) A =c B or B =c A
Proof of (iii):
Either (1) there is x such that x e A and x ~e B,
or (2) there isn't.
(1) Assume x e A and x ~e B.
For all y e B, y < x,
because, B is closed downward, so if there were
y e B and y > x, then x e B. But x ~e B.
For all y < x, y e A,
because A is closed downward.
Therefore, For all y e B, y e A.
B =c A.
(2) Assume there is no x such that x e A and x ~e B.
For all x e A, x e B.
A =c B.
-- B =c A or A =c B
Let /F/ be a _bounded_ , non-empty set of flying rainbow
sparkle-ponies.
"Bounded": There is some flying rainbow sparkle=pony B
such that for all A e /F/ , A =c B.
The union of /F/, U /F/ , is itself a flying rainbow
sparkle-pony.
-- U /F/ is closed downward.
Let x e U /F/ . Then x e A for some flying rainbow
sparkle-pony A e /F/ . For all y < x, y e A
and thus y e U /F/ .
-- U /F/ contains no maximum.
Let x e U /F/ . Then x e A for some flying rainbow
sparkle-pony A e /F/ . There exists some y > x, y e A
and thus y > x, y e U /F/ .
-- U /F/ is not empty
/F/ is not empty of flying rainbow sparkle-ponies,
and no flying rainbow sparkle-pony is empty.
-- U /F/ is not Q.
/F/ is bounded, so there is some flying rainbow
sparkle-pony B such that, for all A e /F/ , A =c B.
B is not Q, so there is some x ~e B.
Because for all A, A =c B, x ~e A and thus x ~e U /F/
And U /F/ is not Q
U /F/ is an upper bound for /F/
-- For all A e /F/ , A =c U /F/ .
Let B be any upper bound of /F/ . Then U /F/ =c B.
-- For all x e U /F/ , there is some A e /F/ such that
x e A. But A =c B, so x e B.
For all x e U /F/ , x e B.
Thus, U /F/ =c B.
For every bounded, non-empty set of flying rainbow sparkle-ponies /F/
there exists a least upper bound. That least upper bound is U /F/