"NEVER AN OPTION":

But both Turing's and Gödel's results share a similarity in the way their theories seem to extend their scope beyond the originally intended domains. Gödel found that he could encode arbitrary number-theoretical statements into numbers; Turing found that his 'A-Machines' were themselves objects of the theory of computation he intended them to represent. For Turing Machines, this property is called universality: they are themselves objects of their own 'universe'--like the dreamer inhabiting their own dream. It is this property that enables the success of modern computers: whatever their design, they can all perform the same computations--if need be, by explicit simulation, running, say, a virtual Windows machine on a Mac.

If we leave this universe, we also break free of the paradox: suppose we have access to an 'oracle' that can tell us whether a given program halts. Giving a TM access to this oracle, we obtain what Turing called an 'O-Machine'. As this now can perform computations that no TM can perform, it is no longer a member of their universe, but inhabits some separated realm. As long as it only takes TMs as its object, no paradoxical consequences ensue.

But now suppose we 'enlarge' the universe to include O-Machines, and ask for a solution to their halting problem: again, we find the same old troubles resurface, and undecidability return once more--the O-Machine halting problem is undecidable for O-Machines.

Thus, undecidability seems to emerge whenever a theory's substrate--the axioms of number theory, Turing's A- and O-Machines--becomes the theory's own object. The analogy to physics is immediate. Many theories are such that they apply to a limited set of phenomena. Maxwell's theory of electromagnetism describes phenomena concerning the behavior of charges in electric and magnetic fields--a comprehensive, but clearly delineated subset of the (physical) universe. These can be studied 'from the outside': the theory has a limited scope.

But any putatively fundamental theory, claimed to apply to literally everything in the universe, leaves no 'outside' from which its phenomena can be appraised--it leaves no room for a 'detached observer': any possible observer must themselves be an object of the theory. Thus, once a physical theory is universal in this sense, we are in a situation much like that of Gödel and Turing. It is then only natural to look for analogous consequences, as well.

This circumstance has been noticed several times in modern physics. One of the earliest examples, in fact, is due to philosopher Karl Popper, originator of the falsificationist approach to the philosophy of science. In 1950, Popper, in a two-part paper entitled Indeterminism in Quantum Physics and in Classical Physics, considers 'near-Gödelian questions', and proposes that they imply 'the physical impossibility of predicting [...] certain physical events; or [...] an indeterminism of a kind somewhat similar to the one implied by quantum physics'.

However, the task Popper has in mind really is that of self-prediction: asking an agent about their future state. Important work along related lines was done by the eminent quantum logician Maria Luisa Dalla Chiara, who in 1977 explicitly considered the famous 'measurement problem' of quantum mechanics in light of the Gödelian results, by the Austrian mathematician Thomas Breuer, and the Viennese physicist Karl Svozil.

These results largely concern Gödelian phenomena as applied to quantum theory. Wheeler's intuition, however, was quite different: that undecidability might be a foundational principle of quantum theory. That, in other words, quantum theory is quantum because of undecidability. This is a more daring possibility: if it is correct, then it would imply that classical physics was never an option--that the impossibility of predicting certain outcomes is, in fact, a necessary feature of any putatively 'universal' theory.

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