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Gosh, sorry I killed the thread!
Maybe I can be indulged just once more. I'm just trying to understand if
operations on entities are to be defined per their type, the individual
facets of their type, or instead on the axioms of system.
I'm not sure there's a clear distinction between 'facet' and 'axiom'. I've
looked up the definitions, but it hasn't helped. So maybe I'll resort to
example:
Peanos axioms:
1) There is a natural number 0.
2) Every natural number a has a successor, denoted by a + 1.
3) There is no natural number whose successor is 0.
4) Distinct natural numbers have distinct successors: if a ≠ b, then a + 1
≠ b + 1.
5) If a property is possessed by 0 and also by the successor of every
natural number it is possessed by, then it is possessed by all natural
numbers.
Now, without axiom number 2, it would seem pretty useless to define the
addition or subtraction operators. On the other hand, one could say that
axiom 2 is a declaration of infinite cardinality, in which case is it a
facet on the natural number type!? Ay yi yi!
I realize that not all axiomatic systems are interesting or self-consistent.
I suppose there are, however, many that are, and some of those don't talk
each other. I'm specifically thinking of Euclidean vs. non-Euclidean
geometries; but I don't know offhand if the differences in axioms correspond
to differences in operations (or facets, for that matter).
If I'm wayyyy off in the weeds on this, forgive me. I expect I'll be hearing
the crickets.
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