Pile Space or Numbers-As-Points on a Line

Some clickable line art

Pspace.PNG

to modify my earlier description of bootstrapping a PSystem, which you can find here.

We start with Stathis' first axiom again.

We need two Pile_Objects to define the origin(s) of our two-dimensional Pile Space (short: PSpace). As result, there are two Pile_Objects sitting on a line (the Diagonal of our 2D Cartesian PSpace): #0 and #1. The Combinative pointers Cp2 are points on the Diagonal line, i.e. zeros (or origins) — a sort of membrane between two dimensions — and nothing else yet.

P_Object #1 stretches a new partition, i.e. its own infinite 2D Cartesian coordinates systems (but in the same PSpace!), with the Combinative pointer Cp0 as y-axis (Associative) and the Cp1 as x-axis (Normative). #1 is yet undefined. We need two more Pile_Objects to define #1. Their Cp2s point to Cp0 and Cp1 of #1, molding an identity, i.e. there is an Ui of #1 now.

Let's identify a Pile_Object #2 as e.g. the right child of #1 (we could have choosen the left child as well). To identify a Pile_Object we need what? Right, two other partly defined Pile_Objects, i.e. we only care about their Cp2s. They also strech new partitions.

The moment the two Cp2s point at Cp0 and Cp1 of #2, #2 is identified by an Ui and jumps onto the Diagonal molding a new zero or origin of its 2D Cartesian coordinates systems. The zero is the twin of infiniteness.

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