Integland practice

Let me jot things here before a (hopefully) smoother version will go to wlod.net. Too bad I don’t know how to use css at wordpress. It’d be nice to use the same style sheets as in wlod.net.

Integers are more elegant than natural numbers but they are elusive in that it’s harder to define them. I’ve searched for an elegant, direct definition of integers but never got anything aesthetically satisfying. Even a few days ago (but never in the past), after years of dealing with the problem (not too intensively but nevertheless), I made an embarrassing error of wishful thinking, of forgetting to exclude finite (cyclic) models. My present (final :-)) definition is fine but far from breath taking, too bad.

DEFINITION 0   Integland  is an ordered triple

Z  :=  (Z – 1)

where  Z  is a set,  – : Z2 → Z  is a binary operation,  1 ∈ Z,  and the following axioms are satisfied:

  1. x-(y-z)  =  z-(y-x)
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  3. x-x  =  y-y
  4. x-(x-x)  =  x
  5. 1-x  ≠ x
  6. if  1-x = x-1  then x=1
  7. ((1 ∈ A ⊆ Z) & (∀a b ∈ A  a-b ∈ A))   ⇒   A = Z

for arbitrary  x y z ∈ Z.   END of DEFINITION

(Let’s see how it looks. I’ll write a series of short posts instead of suffering editing a long one).

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