The Strength of Abstraction with Predicative Comprehension

Sean Walsh

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arXiv · 1407.3860

The Strength of Abstraction with Predicative Comprehension

Sean Walsh

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Abstract

Frege's theorem says that second-order Peano arithmetic is interpretable in Hume's Principle and full impredicative comprehension. Hume's Principle is one example of an abstraction principle, while another paradigmatic example is Basic Law V from Frege's Grundgesetze. In this paper we study the strength of abstraction principles in the presence of predicative restrictions on the comprehension schema, and in particular we study a predicative Fregean theory which contains all the abstraction principles whose underlying equivalence relations can be proven to be equivalence relations in a weak background second-order logic. We show that this predicative Fregean theory interprets second-order Peano arithmetic.

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Sean Walsh. 2014-07-15. The Strength of Abstraction with Predicative Comprehension. https://arxiv.org/abs/1407.3860

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