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PhilGoetz comments on A simple counterexample to deBlanc 2007? - Less Wrong
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This is supposed to be a counterexample. A counterexample means you are allowed to select any possible case. This is one possible case. I am not "replacing S_I by something completely different." I am choosing one possible S_I. If you don't like my choice of S_I, you need to show that it is not a possible case. That would include showing that I have chosen an S_I that is inconsistent with my earlier choices. Try to do that.
If I had specified the set of observations I, then also specified S_I in a way not determined by I, then you could make that argument. As I have not specified I, you need to show that my choice of S_I is inconsistent with any possible set I in order to make the argument you're trying to make.
Let J be the constant set that I is currently equal to. Your agent's set of hypotheses then does not contain the computable function f(k) =
(2(1-1/2^k))^4 for all K in J
-k for all K not in J
By construction, this hypothesis must be compatible with the known input-output pairs, since it is indistinguishable from your f_4 given the observations in J. S_I must therefore contain f iff it contains f_4. Your S_I does not satisfy this, so it is not a counterexample.
I think you're right. You may have unravelled my counterexample. Tentative congratulations!
I now get to eat a piece of chocolate, as a reward for being proven wrong.
Suggestion: treat yourself to a second piece of chocolate and add an ETA at the top of the post - the parent comment is buried too deeply in the thread, for informational purposes.
In the original paper, S_I is determined by I. There is no I for which your set S_I is the set defined in the paper. It really is as simple as that.