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Above we have focused on an example deterministic machine.
It is straight-forward to extend this
to computers whose actions are computed in
probabilistic fashion, given the current state.
Then the expectation calculus
used for probabilistic aspects of the environment
simply has to be extended to the hardware itself,
and the mechanism for verifying proofs has to
take into account that there is no such thing as
a certain theorem--at best there are formal statements
which are true with such and such probability.
In fact, this may be the most realistic approach
as any physical hardware is error-prone,
which should be taken into account by
realistic probabilistic Gödel machines.
Probabilistic Gödel Machine Hardware
Probabilistic settings also automatically avoid certain
issues of axiomatic consistency. For example, predictions
proven to come true with
probability less than 1.0 do not necessarily cause contradictions
even when they do not match the observations.