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A physicist is an atom's way of knowing about atoms. -- George Wald

tech / sci.math / THE PROOF |R|=|N|

THE PROOF |R|=|N|

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https://www.novabbs.com/tech/article-flat.php?id=127559&group=sci.math#127559

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Subject: THE PROOF |R|=|N|
From: grahamco...@gmail.com (Graham Cooper)
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by: Graham Cooper - Fri, 17 Feb 2023 13:19 UTC

Any enumeration of a powerset(N) is incomplete

let LIST(f(x y z..)) be some list of some or all of the elements of f(x y z..)

POWERSET PROOF
-----------------------------
ALL L
L = LIST( {T F}^N )
->
| {T F}^N - L | > 0

HOWEVER

EXIST L
L = LIST( {T F ?}^N)
^ {T F ?}^N = L

The Total Powerset can be listed

eg PRIMES
{1? 2 3 5 7...} e {T F ?}^N

| {T F ?}^N - {T F}^N | > 0

{T F}^N C {T F ?}^N

There is an incompatibility with POWERSET2 and LIST
But thats because {0 1}^N is just a subset of the TOTAL POWERSET {0 1 ?}^N

So all that has been proven is a subset of the TOTAL POWERSET is incompletely listable