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 \(f\colon\mathbb{N}\times\mathbb{N}\to\mathbb{N}\) such that for all computable \(g\colon\mathbb{N}\to\mathbb{N}\) there is some index \(i_{g}\) such that \(f\left(i_{g},n\right)=g\left(n\right)\) for all \(n\). If there were, we could pick \(g\left(n\right)=f\left(n,n\right)+1\), and then \[g\left(i_{g}\right)=f\left(i_{g},i_{g}\right)+1=g\left(i_{g}\right)+1,\] 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 \(f\colon\mathbb{N}\times\mathbb{N}\to\mathbb{N}\cup\left\{ \bot\right\}\) such that for all partial computable \(g\colon\mathbb{N}\to\mathbb{N}\cup\left\{ \bot\right\}\) there is an index \(i\) such that \(f\left(i,n\right)=g\left(n\right)\) for all \(n\). Here, we look at a different “miracle”, that of reflective oracles [2,3]. As we will see in Theorem 1, given a reflective oracle \(O\), there is a (stochastic) \(O\)computable function \(f\colon\mathbb{N}\times\mathbb{N}\to\left\{ 0,1\right\}\) such that for any (stochastic) \(O\)computable function \(g\colon\mathbb{N}\to\left\{ 0,1\right\}\), there is some index \(i\) such that \(f\left(i,n\right)\) and \(g\left(n\right)\) have the same distribution for all \(n\). This existence theorem seems to skirt even closer to the contradiction mentioned above.
We use this idea to answer “in spirit” the converse Lawvere problem posed in [4]. These methods also generalize to prove a similar analogue of the ubiquitous converse Lawvere problem from [5]. The original questions, stated in terms of topology, remain open, but I find that the model proposed here, using computability, is equally satisfying from the point of view of studying reflective agents. Those references can be consulted for more motivation on these problems from the perspective of reflective agency.
Section 3 proves the main lemma, and proves the converse Lawvere theorem for reflective oracles. In section 4, we use that to give a (circular) proof of Brouwer’s fixed point theorem, as mentioned in [4]. In section 5, we prove the ubiquitous converse Lawvere theorem for reflective oracles.
