Ideas of Giuseppe Peano, by Theme
[Italian, 1858 - 1932, Professor at Turin.]
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6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / a. Axioms for numbers
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Numbers have been defined in terms of 'successors' to the concept of 'zero' [Blackburn]
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6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / d. Peano arithmetic
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All models of Peano axioms are isomorphic, so the models all seem equally good for natural numbers [Cartwright,R]
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PA concerns any entities which satisfy the axioms [Bostock]
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17634
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Peano axioms not only support arithmetic, but are also fairly obvious [Russell]
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5897
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0 is a non-successor number, all successors are numbers, successors can't duplicate, if P(n) and P(n+1) then P(all-n) [Flew]
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6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / g. Incompleteness of Arithmetic
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We can add Reflexion Principles to Peano Arithmetic, which assert its consistency or soundness [Halbach]
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6. Mathematics / C. Sources of Mathematics / 6. Logicism / a. Early logicism
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Arithmetic can have even simpler logical premises than the Peano Axioms [Russell]
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