This paper by Normann and Sanders apparently triggered a stir within the reverse arithmetic group when it got here out a pair years in the past. It says that Cousin’s lemma, which is an extension of the Heine-Borel theorem, requires the total energy of second-order arithmetic (SOA) to show. It additionally says this lemma is useful in mathematically justifying the Feynman path integral. So that is an obvious counterexample to each
- a) The reverse arithmetic principle that theorems of classical evaluation can (often?) be proved utilizing one of many “Large 5” subsystems of SOA, with the energy of subsystem required forming a helpful classification of such theorems; and
- b) Solomon Feferman’s argument that scientifically helpful arithmetic can usually be dealt with by comparatively weak axioms, usually not stronger than PA / RCA0.
I do not precisely have a mathematical query in regards to the Normann-Sanders paper, however wish to know if it has impacted the reverse arithmetic program, and what its significance is seen as. Might path integrals actually require such highly effective axioms?
Additionally, Cousins’ lemma is historically pretty simply proved utilizing the completeness property of the true numbers. The difficulty is that the completeness property is a second-order property of the reals (i.e. it makes use of a set quantifier), and SOA is a first-order principle of the reals, that does not have units of reals. In classical evaluation although, the completeness axiom actually does consult with units of reals, and this consequence reveals that changing a completeness-based proof to a first-order proof is not really easy (I have not learn the paper intently and do not know proper now the best way to show Cousin’s lemma in SOA). Is that vital?
I can perceive that the (second order) induction axiom from the Peano axioms interprets naturally to the induction schema in first order PA, making induction proofs work about the identical approach as earlier than. I might have an interest to know why evaluation is recognized with SOA as a substitute of one thing that enables units of reals (wanted for features anyway), since there’s not such an easy translation of the completeness axiom. Evaluation=SOA goes again a great distance, for the reason that Hilbert program aimed to show CON(SOA) as soon as it was completed with the consistency of arithmetic. Reverse arithmetic got here a lot later.