Re: How many flips of DIAG are on the infintie list of infinite con flippers ?

From: The Ghost In The Machine (ewill_at_sirius.athghost7038suus.net)
Date: 01/20/05 

Date: Thu, 20 Jan 2005 16:01:48 GMT

In sci.logic, |-|erc
<h@r.c>
 wrote
on Thu, 20 Jan 2005 16:42:35 +1000
<35925sF4dhuntU1@individual.net>:
> "george" <greeneg@cs.unc.edu> wrote in message
>> >
>> In English, that sentence suffers from the syntax
>> error known as a comma splice. Not only is it ungrammatical,
>> it is just incoherent. I can always come up with a new
>> anything, period, if there are more than a finite number
>> of that kind of thing. You can't prove that they've all
>> been thought of already.
>
>
> Sure I can. Think of a natural number not on this list.
>
> defun nats (nat(0))
>
> defun nat(n) (cons n (nats(plus ( n 1 ))))
>
> nats
> <1 2 3 4 5 6 7 8..>
>

All natural numbers are on that list, by definition. Did
you have a point here?

>
>
>
>>
>>
>> > AntiDiag = <HHHHTTTTHHHHTTTTHHHHTTTT..>
>> > |<------ How Many flips ? ------->|
>>
>> Your calling these "flips" is stupid.
>> They are just letters. This is just a string.

They are also flips. A coin flip can be modeled as
letter strings, binary digit sequences (0101010101...),
raw bits (which are hard to represent in ASCII directly;
one usually uses letters or binary digit sequences),
photon/non-photon, red/green, pointer at 100% / pointer at 0%
on a hypothetical dial, current pulse/absence, +5V/-5V, etc.

Of course it would help if |-|erc gave us a complete
specification for these flips, which implies a function
spec -- and here's where it gets interesting.

If we define F(i,j) as a function defining |-|erc's flips
(domain: N x N, range, boolean), one can define an antidiagonal
function AntiDiag(F,i)
(domain: {functions with domain NxN and range boolean} x N,
range, boolean).

It is clear that AntiDiag cannot be on F's list, because
of a simple issue with typing (they aren't compatible).
One can attempt to fix this by asking the more intelligent
question

   for what i is F(i,*) = AntiDiag(F,*) [*]

or asking whether

   (Ei)(Aj)(F(i,j) = AntiDiag(F,j))

is true. For most F this will obviously be false. In
fact, for all F this is provably false, since one can
derive:

1. (Ei)(Aj)(F(i,j) = AntiDiag(F,j))
2. (Aj)(F(k,j) = AntiDiag(F,j)) [1, EI]
3. (F(k,k) = AntiDiag(F,k)) [2, UI]

which is clearly false because of the construction
method of AntiDiag(F,j), which is defined as

AntiDiag(F,j) = !(F(j,j))

Of course this is not to be confused with the
proposition

   (Aj)(Ei)(F(i,j) = AntiDiag(F,j))

which for most F is true. |-|erc, you've had this problem before.

[rest snipped]

[*] the notation is borrowed from the Illiac IV.
    Basically, if f : N x N -> Boolean, then
    g_i = f(i,*) is a function mapping N to Boolean,
    such that g_i(j) = f(i,j). 

-- 
#191, ewill3@earthlink.net
It's still legal to go .sigless.

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