Thursday, August 9, 2007

in Peano's direction

Fact: The Successor function can only move away from the reference point.
Question: Does any aspect of the Peano system utilize the direction toward the reference point?

Now, if the successor function is not used for "going toward the reference point", then what is the mechanism that allows this directional procession? What allows you to go back like that? There is no other function defined and it does not appear to be coming from some feature "underneath" the formal framework of axiomatic systems. So, my guess is that the expressive capabilities of the axioms is what is being used to move backwards in such a case???

(If "moving backward" is not a notion you want to entertain, then perhaps an alternate view is that moving from "2" to S(1) is a symbol decoding function- something that decodes a symbol into it's appropriate parameterized successor-function "call". If so, then are the axioms creating this decoding function?)

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