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
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-
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
thinking. Maybe ways we do not even know about at our present
(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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