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p4wnc6 comments on Unconditionally Convergent Expected Utility - Less Wrong Discussion
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I'm aware of this, but fail to see how it would change the ability to make probability distributions over the space of utility functions and then take expectations there. Sure, you'd be doing it over equivalence classes of functions, but that's hardly any difficulty. What I am saying is you can assign utility to choices of utility functions: utility functions must inherently be recursive in practice. And so their non-summability (or other technical difficulties) causes immediate problems.
Utility functions are not primitive. They are constructed using an algorithm specified by vN&M (or Savage, or A&A). Constructed from preferences over lotteries over outcomes. Preferences are primitive. Priors over states of nature are primitive. Utility functions are constructs. They are not arbitrary.
As has been mentioned, if you constrain preferences using one of the standard vN&M axioms, and if you assume that you can construct a lottery leading to any outcome, then you can prove that outcome utilities are bounded.
I think that the OP needs to be seen as a proposal for constraining the freedom to construct arbitrary lottery-probes. And, if the constraint is properly defined, we can have an algorithm that generates unbounded utilities, but not poorly behaved utilities - utilities which cannot be used to construct expectations that are not unconditionally convergent.