Axiom
Early mathematicians regarded axiomatic geometry as a model of physical space, and obviously there could only be one such model. The idea that alternative mathematical systems might exist was very troubling to mathematicians of the 19th century and the developers of systems such as Boolean algebra made elaborate efforts to derive them from traditional arithmetic. Galois showed just before his untimely death that these efforts were largely wasted. Ultimately, the abstract parallels between algebraic systems were seen to be more important than the details and modern algebra was born. In the modern view we may take as axioms any set of formulas we like, as long as they are not known to be inconsistent.
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References
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Mendelson, Elliot (1987). Introduction to mathematical logic. Belmont, California: Wadsworth & Brooks. ISBN 0534066240
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Notes
- ^ Mendelson, "6. Other Axiomatizations" of Ch. 1
- ^ Mendelson, "3. First-Order Theories" of Ch. 2
- ^ Mendelson, "3. First-Order Theories: Proper Axioms" of Ch. 2
- ^ Mendelson, "5. The Fixed Point Theorem. Gödel's Incompleteness Theorem" of Ch. 2
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See also
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External links
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