Fascinating, tortoise -- but might these remarks apply equally regardless of whether the universe is a "system" or an "organism?" I know you prefer the former model; and I the latter. Either type of universe might be said to be an entity of which nothing is "outside." For in a certain way, whichever you choose, can we not say that it exists in a non-countable space, and that this space is itself a "part of the system (organism)" -- because whichever model we choose, the universe would be directly contingent (i.e., utterly dependent) on it for its own existence?
It seems possible to me that humans do in fact ultimately live in an "uncountable space" and for the very reason you give: that "normal concepts of 'computation' are not even applicable to such spaces."
This seems to hold equally true, whether we opt for your idea of a system arising in a non-countable space, or an organism arising from Newton's concept of absolute space, which he terms the sensorium Dei.
I really like this: "Countability is a very deep mathematical assumption for mathematics applicable to our universe. And in fact, our universe is by all mathematical measures a classic example of an algorithmically finite system (a type of countable space). The type of space you are in determines what properties and capabilities will exist within that space -- subjectively you would find countable spaces to be "richer" than non-countable spaces. That might change if you were an aleph-n (n>0) computer though."
Please elaborate your point regarding the significance of the aleph-n (n>0) computer as it pertains to the present issues?
Thank you so much for your thought-provoking observations, tortoise!
Don't forget that the real numbers are catagorical. There is essentially only one model for the real numbers. The integers are more problematical. There is no unique model for the integers. There is no set of axioms that uniquely selects the integers from the reals. (There is a set that selects the integers from the rationals, however.)