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Full Idea
The axiom of Peano which states that no two numbers have the same successor requires the Axiom of Infinity for its proof.
Gist of Idea
No two numbers having the same successor relies on the Axiom of Infinity
Source
Alan Musgrave (Logicism Revisited [1977], §4 n)
Book Ref
-: 'British Soc for the Philosophy of Science' [-], p.112
A Reaction
[He refers to Russell 1919:131-2] The Axiom of Infinity is controversial and non-logical.
| 10049 | Logical truths may contain non-logical notions, as in 'all men are men' [Musgrave] |
| 10050 | A statement is logically true if it comes out true in all interpretations in all (non-empty) domains [Musgrave] |
| 10060 | Logical positivists adopted an If-thenist version of logicism about numbers [Musgrave] |
| 10058 | No two numbers having the same successor relies on the Axiom of Infinity [Musgrave] |
| 10062 | Formalism seems to exclude all creative, growing mathematics [Musgrave] |
| 10063 | Formalism is a bulwark of logical positivism [Musgrave] |
| 10061 | The If-thenist view only seems to work for the axiomatised portions of mathematics [Musgrave] |
| 10065 | Perhaps If-thenism survives in mathematics if we stick to first-order logic [Musgrave] |