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

From: |-|erc (h_at_r.c)
Date: 01/21/05 

Date: Fri, 21 Jan 2005 13:04:36 +1000

"The Ghost In The Machine" <ewill@sirius.athghost7038suus.net> wrote in

> >> 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 (nat(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?

can you come up with a new natural, or did I cover every one?

>
> >
> >
> >
> >>
> >>
> >> > 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).
>

More importantly
(Aj)(Ei)(F(i,k) = AntiDiag(F,k) 0<=k<=j

but the assumption is F covers every possible finite sequence as
part of its infinite sequences.

This does not crop up anywhere in Cantor's proof.

There is no maximum to j.

There is no maximum to the amount of natural numbers
-> There are oo amount of natural numbers.

There is no maximum to the amount of flips of AD on F.
-> There are oo amount of flips of AD on F.

Herc

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