37 (circular numbers) Redesign the axioms for the extended number system to make it circular, so that +∞ = –∞ . Be careful with the transitivity of < .
§ I add axiom
∞ = –∞
Now I show that there's an inconsistency by proving ⊥ .
direction transitivity usethenewaxiom double negation
= ¬¬(∞<∞) irreflexivity
= ¬⊤ evaluation rule
=⊥
Whenever there's an inconsistency, we have to weaken or withdraw one or more of the axioms used in the proof of inconsistency, so that the proof can no longer be made. The question asks us to keep the new axiom. I propose that we begin by weakening the direction law to just
0< 1
and not say how –∞ and ∞ compare with finite numbers and with each other. That saves us from the above proof, but now we need to make other changes. We need to delete the extremes law
⊤
= –∞ < 0 < 1 < ∞ ⇒ –∞<∞
= ∞ < ∞
–∞ ≤ x ≤ ∞
And in any law having the antecedent that compares ∞ or –∞ , for example or x<∞ or –∞