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first-order and higher-order logic, model theory, set theory, proof theory, computability theory, formal languages, definability, interplay of syntax and semantics, constructive logic, intuitionism, philosophical logic, modal logic, completeness, Gödel incompleteness, decidability, undecidability, theories of truth, truth revision, consistency.
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What is the canonical way to extend Peano's axioms to the set of all integers?
This is a rambling collection of thoughts which are tangential to your question--I am posting in the hope that they might interest you. Let me begin by recalling that the reals can be charaterised a …