Re: infinity

Tony Orlow wrote:
> [...]. If Virgil had actually
> tried to remember TO's position, Virgil might have recalled that 111...111 is
> the largest whole number (in binary), but that the largest countable real
> number is 111...111.111...111, approximately 1 greater in quantity. He might
> have recalled my revelation that this extra 1 explains the addition of 1 when
> finding the negative of a number in 2's complement.

Ha ha! That's a good one.

Oh, but you appear to be serious.

This brings to mind the proof that the algebraic Universe operates
on a two's-complement number system, posited by Bill Gosper in
HAKMEM item 154 (circa 1972):

By this strategy, consider the universe, or, more precisely,
algebra:

let X = the sum of many powers of two = ...111111
now add X to itself; X + X = ...111110
thus, 2X = X - 1 so X = -1
therefore algebra is run on a machine (the universe) which is
twos-complement.

Clever, yet fatally flawed. Gosper succeeds in proving that X
is an infinitely large positive number and, simultaneously, less
than zero. So much for attempting to do arithmetic with infinite
quantities.

See: http://www.inwap.com/pdp10/hbaker/hakmem/hacks.html
See: http://catb.org/~esr/jargon/html/H/HAKMEM.html


Your slip-up is claiming that the real number ...111.111... is
(almost) one greater than the integer ...111, when in fact they
are the same value (oo). The fraction .111... is exactly equal
to 1, and oo + 1 is simply oo.

-drt

.

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