That is not computability, but provability, or inductive inference, which are indeed NOT universal. There are as many ways to get conclusion than there exist thinking creatures. That is why Church thesis is truly miraculous. Limiting us on the arithmetical reality, all theories gives different theorems, but for computability (on any effective domain) all
languages gives exactly the same class of computable functions.

JM: Please, forget now about 'provability' WITHIN mathematics- related theories.

OK. Those are indeed infinitely extendible.

My parenthesis (com-putare) refers to the language-origin of the word:

Which is very nice to remind us. It is a nice etymology, which unfortunately describe more the notion of proof than of computation.

PUT together AND THINK about it. That MAY include math, or other ways of thinking. Maybe ways we do not even know about at our present development.
(You basically seem to be open for such).

Yes. Even by staying with the computationalist hypothesis (with the sense of Church, Turing, Post, etc.), we cannot circumscribe the non enumerable ways for machines to get knowledge.

The more you understand machines/numbers, the more you get familiar with the idea that we really can only scratch the surface. Provably so if we are machine ourselves.

The universal machine is a born universal dissident. It eventually refutes all theories, making its learning ability without bounds.

If we are machines, we are bound to get an infinity of surprises (good or bad, this is part of the surprises).


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