Re: JSH: A simple error
- From: sg552@xxxxxxxxxxxxx
- Date: 27 May 2007 10:47:16 -0700
James Harris wrote:
Posters in attacking my proofs showing inconsistency with the ring of
algebraic integers routinely move outside the ring. Here is a post
meant to show you how they do it.
In the ring of integers, consider x^2 + 3x + 2 = 0, which of course
factors as
x^2 + 3x + 2 = (x+2)(x+1)
and now solve for it using the quadratic formula, but kind of weird by
NOT resolving the square root then
x = (-3 +/- sqrt(1))/2
and now make the substitution, x=2y, so you get
4y^2 + 6y + 2 = 0, so you can divide by 2 to get
2y^2 + 3y + 1 = 0,
and your solution now becomes
y = (-3+/- sqrt(1))/4
which is two solutions where one is not an integer.
So you moved outside the ring of integers.
But isn't this completely analogous to what you do in your paper? You
define 7*g(x) = 5*a_2(x), and then show that a_2(x) and therefore
5*a_2(x) can be chosen to be an algebraic integer. You then claim that
the fact that g(x) is not an algebraic integer means you are "pushed
outside the ring". How is this different to what you have done above,
where you show that x is an integer and then define x = 2y?
[...]
But as I've noted repeatedly, the ring of algebraic integers is
inconsistent, and you cannot prove that is is from within the ring!
It is too weak as a ring, to allow you to prove that certain results
are not within it, so posters are forced to go outside the ring to
try and make their objections.
Not necessarily. Look at Dale's post in another of your threads, and
my subsequent simplification:
http://groups.google.co.uk/group/sci.math/msg/7727c857f87a67c6?hl=en&;
http://groups.google.co.uk/group/sci.math/msg/7a57f7c30fab856a?hl=en&;
http://groups.google.co.uk/group/sci.math/msg/3a804aa6d17f9dcb?hl=en&;
[...]
This result is one of the biggest in mathematical history
demonstrating an actual inconsistency with a well-known mathematical
object, which mathematicians have unknowingly used for over a hundred
years without understanding how it can lead to false arguments that
appear to be proofs when they are not.
You are claiming to have a proof that the algebraic integers lead to
inconsistency. There are then three possibilities:
1) Your "proof" is wrong.
2) Your proof is correct, but its conclusion is wrong, because you
have relied on some previously accepted theorem which is false (for
example perhaps the theorem that the AIs form a ring is false). If so
then it is your job to tell us what theorem of which you have made use
you believe to be false, or at the very least confirm that this is how
the apparent contradiction you speak of arises.
3) The conclusion of your proof is correct. This means that the
algebraic integers lead to a contradiction. However, since the term
"algebraic integer" is merely shorthand for "root of a monic
polynomial with integer coefficients", that would mean that the mere
existence of roots of monic polynomials with integer coefficients
leads to a contradiction. Since the axioms of set theory allow you to
define "roots of monic polynomials with integers coefficients" and
show that they exist, it therefore follows that the axioms of set
theory (on which all the mathematics you have ever seen can be based)
are inconsistent.
Those are the only possibilities. I know which one my money is on, but
I'm guessing you would disagree with my assessment. So which of 2) or
3) do you think it is?
-Rotwang
.
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