How is Addition a Most Primitive Relation? Set Functions
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From: Charlie-Boo on 18 May 2010 14:25 The 3-place relation ADD defined by x+y=z seems primitive in many ways. We are able to define the recursive functions based on ADD and it has a short definition using the same primitives used to define the natural numbers. However, one can argue that many relations have short definitions and there are other ways to compute besides using addition of natural numbers (e.g. string manipulation functions.) Can we show that ADD is the first or smallest relation of a certain type? How can we formally derive ADD from nothing while showing that it if one of the first relations to be derived? Consider: Addition, Multiplication, Conjunction and Disjunction (Set Functions) are all examples of functions that can be applied to the elements of a set in any order and arrive at a unique value. Why is this property useful? It means that Set Functions truly apply to sets no order is needed we dont need a list, just a set. The map VALUE = > Is it an element? suffices to apply a Set Function, when we dont have NATURAL NUMBER => Element of the Set. [Mathematicians have discovered many of the properties of Set Functions, but not the concept of Set Function from which these properties are born.] How do we formally define a Set Function? How do we prove these 4 functions are Set Functions? How can we generate them what are the first or smallest ones? Which number is ADD? C-B |