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Birkhoff's axioms (4 axioms) Hilbert's axioms (20 axioms) Tarski's axioms (10 axioms and 1 schema) Other axioms. Axiom of Archimedes (real number)
Axioms (1) and (2) govern the distinguished element 0. (3) assures that S is an injection. Axioms (4) and (5) are the standard recursive definition of addition; (6) and (7) do the same for multiplication. Robinson arithmetic can be thought of as Peano arithmetic without induction. Q is a weak theory for which Gödel's incompleteness theorem ...
Given the existence of a Godlike object in one world, proven above, we may conclude that there is a Godlike object in every possible world, as required (theorem 4). Besides axiom 1-5 and definition 1–3, a few other axioms from modal logic [clarification needed] were tacitly used in the proof.
Many different equivalent complete axiom systems have been formulated. They differ in the choice of basic connectives used, which in all cases have to be functionally complete (i.e. able to express by composition all n-ary truth tables), and in the exact complete choice of axioms over the chosen basis of connectives.
[1] [2] The precise definition varies across fields of study. In classic philosophy, an axiom is a statement that is so evident or well-established, that it is accepted without controversy or question. [3] In modern logic, an axiom is a premise or starting point for reasoning. [4] In mathematics, an axiom may be a "logical axiom" or a "non ...
Interpreting Euclid's axioms in the spirit of this more modern approach, axioms 1–4 are consistent with either infinite or finite space (as in elliptic geometry), and all five axioms are consistent with a variety of topologies (e.g., a plane, a cylinder, or a torus for two-dimensional Euclidean geometry).
This implies that if a formula is a logical consequence of an infinite set of first-order axioms, then it is a logical consequence of some finite number of those axioms. This theorem was proved first by Kurt Gödel as a consequence of the completeness theorem, but many additional proofs have been obtained over time.
Tarski stated, without proof, that these axioms turn the relation < into a total ordering.The missing component was supplied in 2008 by Stefanie Ucsnay. [2]The axioms then imply that R is a linearly ordered abelian group under addition with distinguished positive element 1, and that this group is Dedekind-complete, divisible, and Archimedean.