Definition:Minimal Arithmetic
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Definition
Minimal arithmetic is the set $Q$ of theorems of the recursive set of sentences in the language of arithmetic containing exactly:
| \((\text M 1)\) | $:$ | \(\ds \forall x:\) | \(\ds \map s x \ne 0 \) | ||||||
| \((\text M 2)\) | $:$ | \(\ds \forall x, y:\) | \(\ds \map s x = \map s y \implies x = y \) | ||||||
| \((\text M 3)\) | $:$ | \(\ds \forall x:\) | \(\ds x + 0 = x \) | ||||||
| \((\text M 4)\) | $:$ | \(\ds \forall x, y:\) | \(\ds x + \map s y = \map s {x + y} \) | ||||||
| \((\text M 5)\) | $:$ | \(\ds \forall x:\) | \(\ds x \cdot 0 = 0 \) | ||||||
| \((\text M 6)\) | $:$ | \(\ds \forall x, y:\) | \(\ds x \cdot \map s y = \paren {x \cdot y} + x \) | ||||||
| \((\text M 7)\) | $:$ | \(\ds \forall x:\) | \(\ds \neg x < 0 \) | ||||||
| \((\text M 8)\) | $:$ | \(\ds \forall x, y:\) | \(\ds x < \map s y \iff \paren {x < y \lor x = y} \) | ||||||
| \((\text M 9)\) | $:$ | \(\ds \forall x:\) | \(\ds 0 < x \iff x \ne 0 \) | ||||||
| \((\text M 10)\) | $:$ | \(\ds \forall x, y:\) | \(\ds \map s x < y \iff \paren {x < y \land y \ne \map s x} \) |
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These are just the usual axioms of arithmetic, except for the inductive axioms.
Note in particular that this is a finite list.
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