A Calculus of Number Based on Spatial Forms - University of ...

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Using J, two concise and useful forms <strong>of</strong> inverse promotion become possible: the
inverse jumping inside <strong>of</strong> the instance boundary and the inverse jumping outside <strong>of</strong>
the abstract boundary:
=([])=(A[])=(AJ);
[]=[]=[A][]=[A]J:
6.2.2 Oscillation
The phase element has the property <strong>of</strong> being its own inverse. Only zero exhibits this
property in rational numbers.
Theorem 2 (J Cancellation) Jcancels with itself.
[][]=[]=[]=[]=
That J is its own inverse should be <strong>of</strong> no surprise, since the inverse boundary is
its own functional inverse and J embodies the inverse.
In light <strong>of</strong> this, J can be built into an oscillation function, osc(A) ! AJ. Every
second application causes the Js to cancel out, producing the sequence:
!J! !J! !J!:::
Wrapping the elements <strong>of</strong> this sequence with the instance boundary produces a
more familiar sequence. Instead <strong>of</strong> starting with void, this sequence starts with and
uses the oscillation function osc(A) ! ([A]J); producing this sequence:
!(J)!!(J)!!(J)!:::
Since (J)=; the function is just a multiplication by negative one. The sequence
translates to:
1!,1!1!,1!1!,1!:::