The above proof of the intermediate value result can be r... — Carmelics

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The above proof of the intermediate value result can be read either as a syntactic derivation from the axioms or as a semantic argument

Philosophy of Language Proof of definition segments

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Reasons For

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Reason for

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1.On the surface the proof looks like a semantic argument
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2.Every step of the proof can be derived from the axioms
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3.Syntactic derivations in second-order logic based on the Comprehension Axiom Schema and Axioms of Choice are very much like syntactic derivations in set theory, and working mathematicians write both in shorthand
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Reasons Against

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Reason against 1 of 2

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1.Syntactic derivability in second-order logic is not recursively enumerable, so no finite proof system can capture all second-order validities.
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2.A proof that cannot be mechanically verified as syntactically complete conflates semantic entailment with derivability, undermining the syntactic reading.
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Reason against 2 of 2

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