Infection as Induction and vice versa

Instead of saying induction, we can say infection or even just spreads.   If 0 is infected, and it spreads to the next number, then all the numbers are infected.

Infected can mean some proposition is true.  The proposition can be membership in a set.

In definition by recursion, infected means defined.  So addition is defined by the right additive identities as

x+0 = x

x+y’ = (x+y)’

So we start at x as the offset for the count.  But we count the y, what we add, starting from 0.

x+0, means x is infected.  That is addition is defined for y=0.   x+y is defined to be x at y=0.

Suppose that x+y is infected, so x+y is defined for y, holding x fixed.  Then x+y’ is infected, and it is infected as (x+y)’.

So addition spreads in the y variable, the second addend.  We can call x the offset and y the increment.  So for a given offset, the sum spreads to neighboring y values, with the sum incrementing by one as the y increments by one.

 

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About New Math Done Right

Author of Pre-Algebra New Math Done Right Peano Axioms. A below college level self study book on the Peano Axioms and proofs of the associative and commutative laws of addition. President of Mathematical Finance Company. Provides economic scenario generators to financial institutions.
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