Hi, infinite-set readers. Set theory is back.
The previous post is on this link. Please read in advance to easily understand this post. Today we will discuss about the addition operation on natural numbers. Happy reading.
The consequence of the Finite Recursion Theorem raises a theorem. The theorem is also called Finite Recursion Theorem because the same in the rationale.
Theorem : (Finite Recursion Theorem)
Given any and the function , then there is a single function such that :
Based on the above hypothesis, it can be formed :
with is fuction from to . Obtained, is a binary predicate. With the previous proof of Finite Recursion Theorem in the previous post, then this theorem is proved.
(x) Theorem :
For any there is a single function such that :
By taking , , , and is function from to with for all then by Finite Recursion Theorem, this theorem is proven.
The above Theorem (x) then becomes a tool for defining the sum of natural numbers.
Given any and are functions defined in the Theorem (x). Defined with .
For all and are functions defined in the Theorem (x) then apply that for all .
Will be proved using mathematical induction. For then is true. Suppose that is true for then apply . Consequently, for then .
In the next post will be given the definition of multiplication and power. See you in the next post. Thanks for reading. May be useful.
If there is any criticism and suggestion, please comment below or email me. Hope you like this ‘story’.