In which I attempt to answer the question "What even is a number?" for a friend. https://notebook.drmaciver.com/posts/2019-02-18-08:58.html …
I agree! The footnote is intended as satire, ridiculing logicism as an account of mathematical foundations.
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I do think it’s inaccurate to say Presburger arithmetic is provably consistent and Peano arithmetic is not. If that’s a joke it’s likely to mislead unless explained. (Also, Presberger is decidable and complete; first-order arithmetic is recursively enumerable and complete.)
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And whether you should *believe* the times table is about whether the times table is *sound* on a domain (which must include a correspondence theory!) and this semantic property is mathematically and practically a very different concept from consistency, a syntactic property.
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