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It is not the case that Anything that can be computed can also be computed by the universal Turing machine.
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Reasons For
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Reason for 1 of 2
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1.
Gödel argued that human mathematical intuition allows recognition of truths no formal system can prove, suggesting minds exceed Turing-machine capacities.
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2.
Lucas and Penrose extended this: if human cognition is not Turing-computable, then 'computable' in the claim smuggles in an unjustified restriction to formal-mechanical processes.
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3.
The claim is therefore either trivially true by definitional fiat or false if 'computable' tracks a mind-independent, broader notion of effective procedure.
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Reason for 2 of 2
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1.
Hypercomputation models (e.g., Zeno machines, oracle Turing machines) can solve the halting problem, which no standard Turing machine can.
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2.
If physically realizable processes exceed Turing-computable functions, the Church-Turing thesis describes a subset of computation, not its totality.
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Reasons Against
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Reason against
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1.
The universal Turing machine can compute what any other Turing machine computes.
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2.
The Turing machine notion fully captures computability (Church-Turing thesis).
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