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A simple undiagonalisable list - ILLUSTRATED

A simple undiagonalisable list - ILLUSTRATED - Physics Forum

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  #1  
Old 05-13-2005, 09:54 PM
HERC777
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Default A simple undiagonalisable list - ILLUSTRATED



FIXED FONT

0.1-----1......1..........1.........1
0.|2....|..2......2......2......2....
0.|.3...|.3...3.......3......3.......
0.|..4..|...4....4.......4.....4.....
0.|...5.|..5........5.......5........
0.|....6|......6.......6..........6..
0.7-----7.....7...7..........7.......
0........8--8------8.........8.......
0........|9.....9..|.........9.......
0........|.1---1...|......1.....1....
0........|.|2..|...|2.........2......
0........|.|.3.|...|..3..........3...
0........|.|..4|...|.......4..4......
0........|.5---5...|...5....5........
0........|......6..|....6.......6....
0........|.......7.|......7......7...
0........|........8|...8..........8..
0........9---------9........9..9.....

4DP <=>
where w = the natural numbers
and D = the decimal digits {0..9},
AnEcAd1d2[ new & cew & d1eD & d2eD & (d1=/=d2) & (c=/=d) ]
-> L[n,n]=d1
& L[c,c]=d2
& L[n,c]=d1
& L[c,n]=d2



This is an illustration of what I call the Quad-Digit-Property, or 4DP.
It's just a typical diagonal where each diagonal digit is repeated
numerous times in that real and is part of numerous 'squares' with
corresponding diagonal digits. In fact there must be atleast 9 squares
for each diagonal digit, one for each of {0..9}.

~


WHAT IS HE TALKING ABOUT UNDIAGONALISABLE LIST?
I CAN STILL MAKE A DIAGONAL TO EVERY LIST HE MAKES!!!

Let's play a game called Cantors list. I tell you the diagonal of my
list and you tell me a real that's not on the my list. Want to play?

DIAG = 0.135791357913579...
What's your real?

YOU : 0.246802468024680..

OK, you win, that's not on my list. But this time, I'll tell you what
class of sets I'm using. All the members are fractions of ninths.

My list was actually
0.1111111..
0.3333333..
0.5555555..
0.7777777..
0.9999999..
0.1111111..

hence the diagonal was 0.1357913579...

Lets try again using only the ninths fractions.
DIAG = 0.12345671234567...
What's your real?

Rather than EXPLAIN EVERYTHING, using only a single DIAG->ANTIDIAG
function, what reals can you come up with that are not on my list?

Herc

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  #2  
Old 05-13-2005, 10:01 PM
Randy Poe
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Default A simple undiagonalisable list - ILLUSTRATED


HERC777 wrote:


Why do you insist on thinking there's only one choice
of what you call "anti-diagonal"?

What's wrong with this:
0.203684706895286...
or this:
0.4444444.....
or this:
0.626262626262...
or this:
0.00000000....
or any of the other uncountably many numbers which
differ in the n-th position from the n-th number of
your list, for all n?


No it's not. If the 4th number in your list is 0.7777...
then the 4th digit of the diagonal is not 4.

- Randy

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  #3  
Old 05-13-2005, 10:15 PM
HERC777
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Default A simple undiagonalisable list - ILLUSTRATED

for the purposes of illustration


This is for a new list (but of the same type, repeating digits only).

DIAG = 0.12345671234567...

Anyone ?

Herc

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  #4  
Old 05-13-2005, 10:35 PM
Randy Poe
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Default A simple undiagonalisable list - ILLUSTRATED


HERC777 wrote:

function Y = ANTIDIAG(DIAG)

Let X = decimal expansion of pi.

for n=1,2,3,.....
let x_n = n-th digit of X.
let d_n = n-th digit of DIAG.
if (x_n == d_n)
set y_n = (d_n + 1) mod 10
else
set y_n = x_n
endif
end

return Y.
------------------
For that matter, I don't need pi or the if test, I
was just being cute, so I'd generate a non-repeating
decimal from your repeating one.

Here's a simpler version:

function Y = ANTIDIAG(DIAG)

for n=1,2,3,.....
let d_n = n-th digit of DIAG.
set y_n = (d_n + 1) mod 10
end

return Y.

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

I guarantee that neither Y will be on your list.

- Randy

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  #5  
Old 05-13-2005, 10:40 PM
Patricia Shanahan
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Default A simple undiagonalisable list - ILLUSTRATED

HERC777 wrote:


I'm not clear about what proposition you are trying
to prove. Here are a couple of alternatives:

1. For every x in R, there exists a function f:N->R such
that x=f(n) for some n in N.

2. There exists a function f:N->R such that, for every x in
R, x=f(n) for some n in N.

If you are trying to prove one of these, please indicate
which. If you are trying to prove something else, please
state clearly what it is.

Thanks,

Patricia

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  #6  
Old 05-13-2005, 10:47 PM
Proginoskes
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Default A simple undiagonalisable list - ILLUSTRATED


HERC777 wrote:

0.2345678234567823456782345678...

That real number's not on your list.

If you reshuffle the reals on your list (k/9, where k is an integer and
where each k occurs an infinite amount of times), you can come up with
this diagonal (or any other, by using an appropriate subset), but that
doesn't change the fact that the actual number still isn't ON YOUR
(allegedly complete) LIST, and that's what Cantor's proof sets out to
do.

And isn't shuffling the items on the list like that game where you say
an integer out loud, then I say an integer out loud, and whoever says
the larger integer wins?

If you're not familiar with the subtleties of it, I'll give you the
courtesy of moving first. What is your integer? (Reply to this post
instead of saying it out loud, so that we can have an undisputable
record.)

--- Christopher Heckman

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  #7  
Old 05-13-2005, 10:56 PM
HERC777
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Default A simple undiagonalisable list - ILLUSTRATED

I am proving that, given our current knowledge,

~proof(~[2])

Herc

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  #8  
Old 05-13-2005, 11:03 PM
HERC777
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Default A simple undiagonalisable list - ILLUSTRATED

Step right up young man, the rules of the game are...

RULE 1 Only use the diag function anti(digit) = digit+1
(if digit>9 then digit=0)



[Herc]
The list only contains repeating digits


[Proginoskes]
0.2345678234567823456782345678*...
That real number's not on your list.


Right! And now for the major prize, while still abiding by the rules,
can you tell me another?
(hint : what is the list?)

Herc

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  #9  
Old 05-13-2005, 11:53 PM
HERC777
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Default A simple undiagonalisable list - ILLUSTRATED

Looks like he's bolted again!

Don't be shy, say you there! Walk Right Up Walk RIght Up see if you
can tell me TWO antidiagonals given the diagonal
0.12345671234567... using only the one antidiag function!

anyone?

Herc

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  #10  
Old 05-14-2005, 01:17 AM
HERC777
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Default A simple undiagonalisable list - ILLUSTRATED

=> Here's a simpler version:
=>
=> function Y = ANTIDIAG(DIAG)
=> for n=1,2,3,.....
=> let d_n = n-th digit of DIAG.
=> set y_n = (d_n + 1) mod 10
=> end
=>
=> return Y.




I didn't notice that, the return is outside of the functions scope.

Its still computable using lazy parameter evaluation, it just gives the
required digits as they are needed by whatever function calls them.

defun y (stream) (cons (add (head stream) 1) (y (tail stream)))

first ( 5 (y "123456789" )) = "23456"

Herc

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