Here is the question:

>What criteria of recursive definitions should a rational person accept as valid? For example, should I accept a recursive definition if it has an empty proven base?

Wikipedia describes recursion as follows:

>In mathematics and computer science, a class of objects or methods exhibits recursive behavior when it can be defined by two properties:
>
>1. A simple base case (or cases)—a terminating scenario that does not use recursion to produce an answer  
>2. A set of rules that reduces all other cases toward the base case

A rational person should accept a recursive definition if it has both of these properties.

Two examples were provided. 

>A person A is sane if another sane person B considers A's thoughts/actions rational

This example lacks both properties. There is no simple base case upon which one can determine sanity without reference to someone else. And the reference to others does not reduce to anything that could be used to define a base case. What makes the reduction possible for natural numbers is their ordering on a finite set of natural numbers less that the one being considered.

>A deity is almighty if it can make anything, including something it can't make;

This example also lacks both properties and so it is not a recursive definition. It contains an inconsistent definition of "making" which is easy to fix. Define "almighty", at least provisionally until another inconsistency is observed, as being able to make anything *makable*. Even this is not a recursive definition because it does not rely on a "simple base case" nor a "set of rules that reduces all other cases toward the base case".



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Wikipedia contributors. (2019, June 10). Recursion. In Wikipedia, The Free Encyclopedia. Retrieved 17:11, July 10, 2019, from https://en.wikipedia.org/w/index.php?title=Recursion&oldid=901212161