Given that we know of no computable function that isn't Turing computable, and the set of Turing computable functions is known to be equivalent to the lambda calculus and equivalent to the set of general recursive functions, what is an immensely large hurdle would be to show even a single example of a computable function that is not Turing computable.
If you can do so, you'd have proven Turing, Kleen, Church, Goedel wrong, and disproven the Church-Turing thesis.
No such example is known to exist, and no such function is thought to be possible.
> Turing machines (and equivalents) are predicated on a finite alphabet / state space, which seems woefully inadequate to fully describe our clearly infinitary reality.
1/3 symbolically represents an infinite process. The notion that a finite alphabet can't describe inifity is trivially flawed.
Function != Computable Function / general recursive function.
That's my point - computable functions are a [vanishingly] small subset of all functions.
For example (and close to our hearts!), the Halting Problem. There is a function from valid programs to halt/not-halt. This is clearly a function, as it has a well defined domain and co-domain, and produces the same output for the same input. However it is not computable!
For sure a finite alphabet can describe an infinity as you show - but not all infinity. For example almost all Real numbers cannot be defined/described with a finite string in a finite alphabet (they can of course be defined with countably infinite strings in a finite alphabet).
Non-computable functions are not relevant to this discussion, though, because humans can't compute them either, and so inherently an AGI need not be able to compute them.
The point remains that we know of no function that is computable to humans that is not in the Turing computable / general recursive function / lambda calculus set, and absent any indication that any such function is even possible, much less an example, it is no more reasonable to believe humans exceed the Turing computable than that we're surrounded by invisible pink unicorns, and the evidence would need to be equally extraordinary for there to be any reason to entertain the idea.
Humans do a lot of stuff that is hard to 'functionalise', computable or otherwise, so I'd say the burden of proof is on you. What's the function for creating a work of art? Or driving a car?
You clearly don't understand what a function means in this context, as the word function is not used in this thread in the way you appear to think it is used.
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