On 05 Jan 2015, at 19:54, meekerdb wrote:
On 1/5/2015 1:07 AM, 'Chris de Morsella' via Everything List wrote:
0={0} and then onward to: 0={0}= {0}+{0} = {{0}, {0}+{0}} etc.
There's your problem: "etc"
It gives Cantor ordinals, which can be made precise in some set theory
(like ZF).
0, 1, 2, 3, ..., omega, omega+1, ... epsilon_0
(omega^omega^omega^...), epsilon_0 + 1, .... Gamma_0, ...
... omega_1^CK (least non constructive ordinal CK=Church-Kleene), ....
aleph_one (first non countable ordinals) ... aleph_2, ...,
aleph_omega, .... beth_0, ... kappa, ...
Below omega_1^CK, you can name the ordinals in the language of a
universal machine, above we can't do that. below, they corresponds to
the recursive ordering (in which a < b is recursive). Above
omega_1^CK, a < b is no more recursive (algorithmically decidable in
general).
This thread is a thread on set theory, but with computationalism, we
don't need to assume sets (which assumes much more than the natural
numbers). We can assume sets, but it lakes the theory more complex,
without changing the physical laws. ZF is just a living number, whose
existence follows from RA. RA can emulate ZF, in the same sense that I
can emulate Einstein brain (I remain different from Einstein, like RA
is different and much weaker (proving less theorem) than RA.
Bruno
Brent
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