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Reflective oracles as a solution to the converse Lawvere problem — AI Alignment Forum

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(This post was originally published on Nov 30th 2017, and is 1 of 4 posts brought forwarded today as part of the AI Alignment Forum launch sequence on fixed points.) 1 Introduction Before the work of Turing, one could justifiably be skeptical of the idea of a universal computable function. After all, there is no computable function f : N × N → N such that for all computable g : N → N there is some index i g such that f ( i g , n ) = g ( n ) for all n . If there were, we could pick g ( n ) = f ( n , n ) + 1 , and then g ( i g ) = f ( i g , i g ) + 1 = g ( i g ) + 1 , a contradiction. Of course, universal Turing machines don't run into this obstacle; as Gödel put it, "By a kind of miracle it is not necessary to distinguish orders, and the diagonal procedure does not lead outside the defined notion." [1] The miracle of Turing machines is that there is a partial computable function f : N × N → N ∪ { ⊥ } such that for all partial computable g : N → N ∪ { ⊥ } there is an index

x Reflective oracles as a solution to the converse Lawvere problem — AI Alignment Forum Fixed Points Oracle AI Frontpage 19 Reflective oracles as a solution to the converse Lawvere problem by SamEisenstat 29th Nov 2018 9 min read 2 19 (This post was originally published on Nov 30th 2017, and is 1 of 4 posts brought forwarded today as part of the AI Alignment Forum launch sequence on fixed points.) 1 Introduction Before the work of Turing, one could justifiably be skeptical of the idea of a universal computable function. After all, there is no computable function f : N × N → N such that for all

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