Does set theory really ground arithmetic truth?

Por • 30 nov, 2019 • Sección: Filosofía

Alfredo Roque Freire

 

This article starts criticizing the understanding that finite set theory (ZFHf) is the set theoretic equivalent of arithmetic (PA). We argue that a version of set theory should be a subtheory of our axioms for set theory. However, we prove that no subtheory of any extension of Zermelo set theory is bi-interpretable with any extension of PA. Further, we show that, for every well-founded interpretation of recursive extensions of PA in extensions of ZF, the interpreted version of arithmetic has more theorems than the original. This theorem expansion is not complete however. We continue by defining the coordination problema. In summary, we consider two independent communities of mathematicians responsible for deciding over new axioms for ZF and PA. How likely are they to be coordinated regarding PAs interpretation in ZF? We prove that it is possible to have extensions of PA not interpretable in a given set theory ST. We further show that a random extension of arithmetic to be interpretable in ST is zero

arXiv:1911.10628v1 [math.LO]

Logic (math.LO)

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