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> Unfortunately, mathematical systems without some mathematical (i.e. non-"logical") axioms are quite limited

What's an example without any axioms? It doesn't seem like you can do anything without say ∀x: x=x (which I believe is reciprocal identity.)

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I think Russell would consider that a logical axiom, and it commonly is an axiom of predicate logic with equality. BTW, by "logic", he didn't mean just a specific set of familiar axioms, but any axiom which could be considered "obviously true" and didn't involve numbers specifically. https://en.wikipedia.org/wiki/Logicism

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