Algebraic Recursion Theory by L.L. Ivanov

By L.L. Ivanov

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The above examples are important and will be discussed in more detail in the next section. We use the geodesic distance to express similarity. In [7], we argued that it is possible to view similarity (the geodesic distance) as a derived notion, the more basic concept being indistinguishability (the accessibility relation): Our idea rests on the following maxim: two objects are similar when there is a context under which they are indistinguishable. Therefore, similarity can be measured with degrees of distinguishability.

T > x∈T >d B Applying again NORM we get T > B ⊆ T > A. x∈T >d B This shows that x ∈ T > B which is a contradiction. Therefore, ¬A ∩ B=∅ x∈T >d B and let z ∈ T belong to the above set. Then we have that xRz and z ∈ ¬B which contradicts the initial hypothesis. The following two lemmas are useful for the main result Lemma 7. For all x ∈ T , A ⊆ T and n > 1, dF (x, A) = n iﬀ dF (x, ¬(T >d ¬A))) = n − 1. Conditioning by Minimizing Accessibility 27 Proof. Suppose that dF (x, ¬(T >d ¬A)) = n − 1. Then there exists y ∈ ¬(T >d ¬A) and z ∈ A such that dF (x, y) = n − 1 and dF (y, z) = 1.

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