Can anybody point me to a simple way to show that the three axioms for a group (associativity, neutral element, symmetrical element) do not form a complete theory
2026-04-01 05:11:44.1775020304
Group axioms are not a complete theory
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There are a three first order properties undecided by the theory of groups:
Commutativity.
Presence of $p$ torsion.
Divisibility by $p$.
If we decide these to form the theory of torsion free divisible abelian groups we get a theory that is complete. Although there are many other ways to get complete theories.