|-|ERCULES 4 - CANTOR 0 - GODEL 0 - HALT 0 - TURING 1/2

Herc

Graham Cooper (BInfTech University Of Queensland)

http://tinyurl.com/AMORTIZEDANALYSIS

and then

http://tinyurl.com/CANTORPROOF

-----8<-------------------------------------------------------
According to 99.99% of mathematicians,

DIAGONALIZATION WORKS ON THE INFINITE CASE
DESPITE THE EQUIVALENT COUNTER EXAMPLE FANTIDIAG
ONLY WORKS ON ALL FINITE CASES

but

ALL POSSIBLE DIGIT STRINGS THAT CAN BE LISTED ARE ONLY FINITE
BECAUSE THE EQUIVALENT COUNTER EXAMPLE ALL-FINITE-DIGIT-STRINGS
ONLY WORKS ON ALL FINITE CASES

FDP = Diagonalization for all finite cases
ADP = Diagonalization for the infinite case (sci.math)
FSP = All-Sequences for all finite cases (sci.math)
AAP = All-Sequences for the infinite case

So 99.99% of mathematicians are selective on whether INDUCTION
proves for all finite cases or the infinite case!

-----------------------------------------------8<-------------

THIS THREAD CONTAINING MY 4 DISPROOFS CAN BE REFERENCED VIA
http://tinyurl.com/GODELPROOF

Nope. Diagonalization works on the infinite case
*becuase* the FANTIDIAG proof works on all finite cases.

We need

i: For all n, the digits of FANTIDIAG-oo are the same as FANTIDIAG-n
up to digit n.

ii: For all n
LIST(n,n) =/= FANTIDIAG-n(n) [This is the FANTIDIAG proof]

The proof for FANTIDIAG-oo goes

From i we get for any n,
if LIST(n,n) =/= FANTIDIAG-n(n) then FANTIDIAG-oo
is not equal to row n

From this and ii we get

For all n [not inclusding n=oo, there is no row oo]

FANTIDIAG-oo is not equal to row n

We never use more than a finite number of digits of FANTIDIAG-oo

- William Hughes

Herc

What distiction. The point is that the proof for the "infinite"
case and the FANTIDIAG proof are essentially the same. If the
FANTIDIAG proof works so does the "infinite" proof.

- William Hughes

This distinction.

[HERC]

[WILL}

As stated, you are selective whether induction holds for the infinite
case or for all finite cases.

Pls don't tell me Cantor's proof AGAIN.

Herc

By definition, induction works for all finite cases. However, the
proof
of the "infinite case" never uses more than a finite number of digits.
It is essentially the FANTIDIAG proof.
So induction holds "for all finite cases" and "the infinite case".

- William Hughes

This does not hold in general. There are properties of the infinite
antidiagonal (e.g. the fact that it does not have a last digit) that
are not
provable by induction. However, the fact that: for all n, the
infinite antidiagonal
is not equal to row n; can be proved by induction.

So can 0.0.. be proven by induction to appear on a typical infinite
random plane!

Cantor proved exactly that the antidiag is different to all finite
subsets of the plane.

I won't be arguing the particular proofs any more as www.tinyurl.com/CANTORPROOF
is a meta-proof in which your argument is covered.

Herc

<snip>

And this[1] is enough. The point is that if the antidiag is not equal
to a finite part of a row, then the antidiag is not equal to the
whole
row.

- William Hughes

[1] Not quite, what Cantor showed is that for any row there is a
finite
prefix that differs from the antidiag.

The point is you snip one example of induction to infinity and not the
other.

Anyway if you insist on carrying on in my index thread to my set of
proofs I can't stop you.

Herc

No. All of these proofs are equivalent. A claim that induction
works on any of them is a claim that induction works on all of them.

FSP proves that 0.33333300000... is on the list of all finite digit
sequences, for all finite lengths of consecutive 3's.

AAP proves that 0.333... is on most typical random infinite planes.

Herc

OK. I misinterpreted your nomenclature. These are not equivalent.

The point remains. You have claimed

From this it follows without induction

rows of the plane

I'm not concerned with such deductions, merely that a diag argument
for all finite subsets exists, similarly a finite argument exists for
all possible digit sequences on a list (which is trivially incomplete)

It's no use trying to convince me of the existence of transfinite
sets, I'm not going to suddenly see the light you are blinded by.

Herc

<snip>

Oh. I see the stick your fingers in your ears and say "Nyah Nyah
Nyah"
stategy.

Don't make me laugh! You avoid 9 out of 10 of my points every post.

I'm one man arguing against the establishment with 100 qualified
mathematicians against me. I have to abide by every protocol or be
shot down, but sci.math can use every deceitful tactic possible. You
resorting to ad homs because you can't settle an argument is 1 of 100
illogical techniques used by sci.math.

I'm just posting my theory, not a formal proof. I can't stop you
seeing a valid use of antidiagonals and such an approach is
counterproductive to my meta-proof.

I unsubscribed from sci.math, this thread was just going to be a log
of my updates.

Herc

<<<<<YOU ARE HERE>>>>>
TINYURL.COM/GODELPROOF

GODEL: There are well founded specified formula that no computer can
prove!

HERC: Like what?

GODEL: Statements that cannot be proven by any formal system!

HERC: Give me one such statement?

GODEL: Statement X cannot be proven by any computer.

HERC: What's statement X?

GODEL: That WAS statement X! Statement X says statement X cannot be
proven by any computer!

HERC: No sh$

________________________________________

[BYRON]

It should be noted that Godels first incompleteness theorem is invalid
as Godel used impredicative definitions – and as we have seen above
many
mathematicians and philosophers say these lead to paradox and must be
outlawed from mathematics

http://en.wikipedia.org/wiki/Vicious_circle_principle

Many early 20th century researchers including Bertrand Russell and
Henri Poincaré. Frank P. Ramsey and Rudolf Carnap accepted the ban on
explicit circularity.

The vicious circle principle is a principle that was endorsed by many
predicativist mathematicians in the early 20th century to prevent
contradictions. The principle states that no object or property may be
introduced by a definition that depends on that object or property
itself. In addition to ruling out definitions that are explicitly
circular (like "an object has property P iff it is not next to
anything that has property P"), this principle rules out definitions
that quantify over domains including the entity being defined.

_______________________________________

[HERC]

Yes but no mathematician will recognize the distinction between
theoretical circular arguments
and practical constructs.

But there is a trivial proof that GODELS PROOF holds no water.

Every mathematician is *indoctrinated* with the following faulty
logic.

OLD GODEL BELIEVER: Let G = G has no proof.

OLD GODEL BELIEVER: Does G have a proof?

NEW GODEL DISCIPLE: No, that would make it true, and force a
contradiction.

OLD GODEL BELIEVER: Does G state that it has no proof?

NEW GODEL DISCIPLE: Yes, G is true.

OLD GODEL BELIEVER: So G is true and it has no proof?

NEW GODEL BELIEVER: *Light bulb* Ahhh, true statements exist without
proof!

------------------------

Now, a smarter disciple enters the classroom.

OLD GODEL BELIEVER: Let G = G has no proof.

OLD GODEL BELIEVER: Does G have a proof?

BYRON: Let OGB = "OLD GODEL BELIEVER CANNOT PROVE THIS STATMENT IS
TRUE"

OLD GODEL BELIEVER: OK

BYRON: Can you write a proof for OGB?

OLD GODEL BELIEVER: No, that would make it true, and force a
contradiction

BYRON: Does OGB state that you cannot prove OGB?

OLD GODEL BELIEVER: Yes, OGB is true.

BYRON: Can you prove that?

OLD GODEL BELIEVER: *Light bulb* Ahhh, some facts are relative to the
agent,
Formal systems are no more limited to what they can prove than I am
limited in my
knowledge despite statements to the contrary!

GC

colin leslie dean has shown Godels second theorem ends in paradox
http://www.scribd.com/doc/32970323/Godels-incompleteness-theorem-invalid-illegitimate

http://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems#Proof_sketch_for_the_second_theorem
The following rephrasing of the second theorem is even more
unsettling to the foundations of mathematics:

If an axiomatic system can be proven to be consistent and complete
from within itself, then it is inconsistent.

now this theorem ends in self-contradiction

http://www.scribd.com/doc/32970323/Godels-incompleteness-theorem-invalid-illegitimate

But here is a contradiction Godel must prove that a system cannot
be proven to be consistent based upon the premise that the logic he
uses must be consistent . If the logic he uses is not consistent then
he cannot make a proof that is consistent. So he must assume that his
logic is consistent so he can make a proof of the impossibility of
proving a system to be consistent. But if his proof is true then he
has proved that the logic he uses to make the proof must be
consistent, but his proof proves that this cannot be done

godel is useing a a mathematical system
his theorem says a system cant be proven consistent
this must then apply to the system he used to create the theorem
thus his theorem applies to itself

thus paradox

if godels theorem is true within this system-or outside it
ie a system cannot be proven to be consistent
then his theorem is in paradox
as
it can only be proven if his logic is consistent within that system
if his theorem is true
then he has proven his logic is consistent within that system
but his theorem says this cannot be done

he actually did say that circa late 1999.

About November.
When asked why, the interview I read noted that he was joking. Given how
dull and serious he always appears, people took it as serious,.

At least this is funnier than Reagan's joke about bombing the Russians

"We are never more discontented with others
than when we are discontented with ourselves."

Henri-Frederic Amiel