Logic — math, philosophy & computational aspects

logic, math, philosophy, math games, math help, mathematical logic, philosophy of education, math facts

Why can no one in sci.math understand my simple point?

Consider the list of increasing lengths of finite prefixes of pi

3
31
314
3141
….

Everyone agrees that:
this list contains every digit of pi   (1)

as pi is an infinite digit sequence, this means

this list contains every digit of an infinite digit sequence   (2)

similarly, as computable digit sequences contain increasing lengths of ALL possible finite prefixes

the list of computable reals contain every digit of ALL possible infinite sequences  (3)

OK does everyone get (1) (2) and (3).

There’s no need for bullying (George), it’s just a maths theory.  
Address the statements and questions and add your own.

Herc

If you ever rob someone, even to get your own stuff back, don’t use the phrase
"Nobody leave the room!" ~ OJ Simpson

Comments (19)

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Sad observation: Cantor/Godel/Turing/Tarski

I’m not going to name any names, but I recently found that I was
arguing the same arguments, in the same way, with the same people,
as I argued 4 or 5 years ago. Clearly, this is a pointless endeavor.
So I have kill-filed the people and the argument threads. This isn’t
because of any hostility towards those people, nor because of disgust
with the topics, but just as a self-discipline measure to try to keep
myself from staying in the same rut forever. I don’t want to be
like Sisyphus forever pushing a boulder up a hill, only to have it
roll back down again.


Daryl McCullough
Ithaca, NY

Comments (15)

Yet Another SD Rodrian Prediction True: Gravity is NOT an attractive force between bodies

Yet Another SD Rodrian Prediction True:
Gravity, as argued even by Einstein, is NOT
an attractive force between bodies.

  EINSTEIN WAS RIGHT
  – SPACE AND TIME BEND

  Ninety years after he
  expounded his famous theory,
  a $700m NASA probe has
  proved that the universe
  behaves as he said. [read
  the entire article below]

Actually, although what this experiment
PROVES is that "gravity is not an attractive
force between bodies" [merely], the results
are still misinterpreted here in the usual
nonsensical gibberish of conventional
"physics-talk" of the past century: In effect,
these interpreters speak of "space-time"
bending* because they are not yet aware
of the true evolution of the universe: They
have discovered that the universe behaves
as I describe it does [and which you can
read at: http://physics.sdrodrian.com ]
but they interpret it in their antediluvian
understanding (it’s as if the scientists who
once believed that the entire universe
revolved around the planet earth were
interpreting this discovery in terms of that
once so universally-held ancient superstition).

ONLY my description of a universe which
has evolved across immense amounts of
time from unimaginably vast volumes of
space could have produced/accumulated
the "energy" that today powers it. No other
explanation accounts for the prodigious
amounts of energy infused into every
last/smallest bit of the universe’s matter.
And not even the pretty fable of a magic
[Big Bang] bean bursting forth the whole
of "Creation" … for no reason at all.

But, of course, this remains for yet another
more impressive understanding still.

S D Rodrian
http://sdrodrian.com
http://physics.sdrodrian.com
http://mp3.sdrodrian.com

All religions are local.
Only science is universal.
.
* Space is strictly/only the absence of
anything between instances of something.
And Time is merely the human mind’s
attempt to synchronize one or more of
the universe’s unrelated motions with/to
one of more of the universe’s other
unrelated motion(s).     –SDR

  START QUOTE

 Now the race is on to show
 that the other half of
 relativity also works–
 for decades physicists have
 been asking the question:
 did Albert Einstein get it
 wrong? After half a century,
 seven cancellations and
 $700m, a mission to test
 his theory about the
 universe has finally confirmed
 that the man was a
 mastermind — or at
 least half proved it.

 The early results from Gravity
 Probe B, one of Nasa’s
 most complicated satellites,
 confirmed yesterday ‘to
 a precision of better than
 1 per cent’ the assertion
 Einstein made 90 years ago
 – that an object such as
 the Earth does indeed
 distort the fabric of space and
 time.

 But this — what is referred
 to as the ‘geodetic’
 effect — is only half of
 the theory. The other,
 ’frame-dragging’, stated
 that as the world spins it
 drags the fabric of the
 universe behind it.

 Francis Everitt, the Stanford
 University professor
 who has devoted his life to
 investigating Einstein’s
 theory of relativity, told
 scientists at the American
 Physical Society it would
 be another eight months
 before he could measure
 the ‘frame-dragging’ effect
 precisely.

 ’Understanding the details
 is a bit like an
 archeological dig,’ said
 William Bencze, programme
 manager for the mission.
 ’A scientist starts with a
 bulldozer, follows with
 a shovel, then finally uses
 dental picks and toothbrushes
 to clear the dust away.
 We’re passing out the
 toothbrushes now.’

 The Gravity Probe B
 project was conceived in the
 late 1950s but suffered
 decades of delays while other
 scientists ran tests
 corroborating Einstein’s theory.
 It was Everitt’s determination
 that stopped it being
 cancelled. The joint
 mission between Nasa and
 Stanford University uses
 four of the most perfect
 spheres — ultra precise
 gyroscopes — to detect
 minute distortions in th
 e fabric of the universe.
 Everitt’s aim was to prove
 to the highest precision
 yet if Einstein was correct
 in the way he described
 gravity.

 According to Einstein, in
 the same way that a large
 ball placed on a elasticated
 cloth stretches the
 fabric and causes it to
 sag, so planets and stars
 warp space-time. A marble
 moving along the sagging
 cloth will be drawn
 towards the ball, as the Earth is
 to the Sun, but not fall
 into it as long as it keeps
 moving at speed. Gravity,
 argued Einstein, was not an
 attractive force between
 bodies as had been
 previously thought.

 Few scientists need the
 final results, which will be
 revealed in December, to
 convince them of Einstein’s
 genius. ‘From the most
 esoteric aspects of time
 dilation through to the
 beautiful and simple
 equation, e=mc2, the vast
 bulk of Einstein’s ideas
 about the universe are
 standing up to the test of
 time,’ said Robert Massey,
 from the Royal
 Astronomical Society.

 He said the mission was
 ’legitimate science’ to test
 a theory and confirm its
 brilliance, but others have
 criticised the costs and
 length of the study,
 claiming that what was
 announced had already been
 shown. Sir Martin Rees,
 the Astronomer Royal, said
 the announcement would
 ’fork no lightning’.

     The theory explained

 When Einstein wrote his
 general theory of relativity
 in 1915, he found a new
way to describe gravity. It
 was not a force, as
 Sir Isaac Newton had supposed,
 but a consequence of the
 distortion of space and
 time, conceived together
 in his theory as
 ’space-time’. Any object
 distorts the fabric of
 space-time and the bigger
 it is, the greater the
 effect.

 Just as a bowling ball
 placed on a trampoline
 stretches the fabric and
 causes it to sag, so planets
 and stars warp space-time
 – a phenomenon known as
 the ‘geodetic effect’. A
 marble moving along the
 trampoline will be drawn
 inexorably towards the ball.

 Thus the planets orbiting
 the Sun are not being
 pulled by the Sun; they
 are following the curved
 space-time deformation
 caused by the Sun. The reason
 the planets never fall into
 the Sun is because of the
 speed at which they are travelling.

 According to the theory,
 matter and energy distort
 space-time, curving it
 around themselves. ‘Frame
 dragging’ theoretically occurs
 when the rotation of a
 large body ‘twists’ nearby
 space and time. It is this
 second part of Einstein’s
 theory that the Nasa
 mission has yet to corroborate.
 More at:
 http://www.guardian.co.uk/science/2007/apr/15/spaceexploration.universe

Comments (2)

Re: A Reformulation of Z-Reg.2

Theory A is the set of all sentences(entailed from FOL with identity
and membership) by the following non logical axioms:

(1) Comprehension: for n=0,1,2,3,…; if phi(y,w1,…,wn)
is a formula in which y,w1,…,wn *are* its sole free variables
and in which x is not free,then

Aw1…wn Ar As E!x Ay
( y e x <-> (( y c r \/ y c s ) /\ phi(y,w1,…,wn)))

were c is the subset relation.

(2) Union: Ar Ex Ayer(ycx)

were c is the subset relation.

(3) Infinity: as in Z.

/

Theorem schema of unique separation:

for n=1,2,3,…., if pi(y,w1,…,wn) is a formula in which
y,w1,…,wn *are* its sole free variables, and in which x is not free,
then

Aw1…wn E!x Ay ( y e x <-> ( y e w1 & pi(y,w1,…,wn) ) ).

Proof:

Let s=r in the comprehension schema, then for the same specifications
in comprehension we get:

Aw1…wn Ar E!x Ay
( y e x <-> (y c r  /\ phi(y,w1,…,wn)))

Let phi(y,w1,…,wn)<-> [y e w1 /\ pi(y,w1,...,wn)]

were pi(y,w1,…,wn) is a formula in which y,w1,…,wn *are* its sole
free variables and in which x is not free.

substitute in the above formula and we get:

Aw1…wn Ar E!x Ay
( y e x <-> (y c r  /\ y e w1 /\ pi(y,w1,…,wn)))

From union we have

Aw1 Ew* Ayew1(ycw*)

now w* is not unique, but we know that for every w1, at least one
w* must exist such that all members of w1 are subsets of w*.

Now instantiate any one of the w* sets for r in the above formula, we
get:

Aw1…wn E!x Ay
( y e x <-> (y c w*  /\ y e w1 /\ pi(y,w1,…,wn)))

Now since every member of w1 is subset of w* (by definition of w*
sets). then  we have

(y c w* /\ y e w1) -> y e w1

thus the above formula would be reduced to:

Aw1…wn E!x Ay
( y e x <-> (y e w1 /\ pi(y,w1,…,wn)))

Theorem schema of unique separation proved.

Zuhair

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Question for Aatu Koskensilta

What is "predicatively meaningful" reflection and how would you go
about justifying it?

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Can anyone spot the CONTRADICTION???

"Peter Webb" <webbfam…@DIESPAMDIEoptusnet.com.au> wrote

> Cantor said and proved that
> any purported list of all Reals cannot contain all Reals. His proof is
> simple and clear, provides an explicit construction of at least one
> missing Real.

Like so…

123
456
789

Diag = 159
Anti-diag = 260

Where are you getting a (missing real) ’260′ when

>the list of computable reals contain every digit of ALL possible infinite
>sequences  (3)

___________________________________________________________________

"Daryl McCullough" <stevendaryl3…@yahoo.com> wrote

>>the list of computable reals contain every digit of ALL possible infinite
>>sequences  (3)

>>OK does everyone get (1) (2) and (3).

> If you state it carefully, then yes, everyone gets it:

___________________________________________________________________

Herc

Comments (10)

Re: Adriana Xenides Dead

"Jesse F. Hughes" <je…@phiwumbda.org> wrote ..

- — -

> Herc7 <ozd…@australia.edu> writes:

>> On Jun 7, 6:56 pm, "Kelpie" <kelp…@gmail.com> wrote:
>>> "A Little Bit" <ti…@beerlover.com.au> wrote in messagenews:0l9p0696j5dql6jcuh953h1c4qg8teo6n3@4ax.com…

>>> > Adriana Xenides Dead of Wheel of Fortune fame has died after an operation
>>> > in hospital. She
>>> > was only 54.

>>> I hope she’s turning letters in the sky.

>> Xen dies

> X denies.

>> it appears I *have* disproven higher infinity

> No, you haven’t.  The mysterious value X denies the validity of your
> "proof".

Fraid so, when I booted Ullrich out of my office due to Principal Transfer, the
reason I’m here, he forgot to take his little doggy with him, skit boy!

Funny nobody on sci.math will answer this question on higher infinity, on the same day
Xendies dies, a favorite Australian model who spent a career revealing what label was on the box!

>>   Given a set of labeled boxes containing numbers inside them,
>>   can you possibly find a box containing all the label numbers of boxes
>>   that don’t contain their own label number?

Have a go mate!

Then if you’re the first sci.mather to admit I disproved Cantor, Halt, Turing, and Godel
in front of his peers you can be my Assistant Principal.

Herc

Comments (20)

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a problem in very very elementary set theory

I’m addressing this post to people who like to solve basic problems
related to foundational questions.

Start with ZFC. Now let ZF0 consist of the restriction of ZFC to the
following axioms:

Extensionality
Pair-set
Unions
Subsets

Given this theory, we can define the concept of an ordinal, and prove
that the ordinals are linearly ordered, and, in fact, well ordered.
The definition I am thinking of is:

x is an ordinal iff x if epsilon-transitive, linearly ordered by
epsilon, and if for any subset y of x y contains an epsilon-least
element.

You can also define finite ordinals and prove induction (mathematical
or transfinite) and show how to do recursive definitions (on finite
ordinals or in the form of transfinite recursion). (Well, actually,
transfinite recursion probably needs the Axiom of Replacement, so just
say recursion on finite ordinals.)

Now, let ZF00 be ZF0 without the Axiom of Subsets. It is easy to show
that there are models for ZF00 in which the ordinals, as just defined,
are not linearly ordered.

Note however, that the null-set — 0 — is provable to be an element
of any ordinal.

Now, here’s the problem: within ZF00 can you add a clause to the
ordinal definition such that you restrict attention to the usual
finite ordinals 0,1,2… and maybe transfinite ordinals, such that all
these more restricted "ordinals" — call these "really really
ordinals" — are linearly ordered? Definition has to admit 0,1,2….

I say you can’t, but I haven’t proven that.

BTW, if you add an appropriate formulation of the Axiom of Foundation
to ZF00, I believe you are able to do this. But I’d like to do this
without just the blanket assumption that all sets are well-founded.

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