Later Gödel on the Formalization of Mathematics: A Comment on Hanna’s “Rational Human Mindedness and The Hierarchy of Increasingly Rich Formal Systems.”
By Carlo Cellucci
I have noticed that in your paper “Rational Human Mindedness…”[i] you do not consider [later] Gödel’s argument that, while by his first incompleteness theorem it is “impossible to formalize all of mathematics in a single formal system,” nevertheless “everything mathematical is formalizable” (Gödel 1986–2002, I, 389). It is formalizable not in a single formal system, but in “a sequence (continuable into the transfinite) of formal systems” (ibid., I, 237).
Since Gödel’s argument might be relevant to your paper, I copy below what I have observed in my forthcoming book The Making of Mathematics.[ii]
For Gödel’s argument to be credible, the transition from a formal system to the next one in the sequence of formal systems should itself be formal. For, if the transition is not formal and requires an appeal to intuition, it will be impossible to say that everything mathematical is formalizable, the appeal to intuition will lead beyond what is formalizable.
But, if the transition from a formal system to the next one in the sequence of formal systems is itself formal, then, as McCarthy argues, for the sequence of formal systems it will be possible to demonstrate a theorem that “is an exact analogue” of “Gödel’s first” incompleteness “theorem” (McCarthy 1994, 427). Therefore, not everything mathematical will be formalizable in the continuable sequence of formal systems.
Since not everything mathematical will be formalizable in the continuable sequence of formal systems, then, contrary to Gödel’s claim, mathematics cannot consist in the activity of “an idealized mathematician who entertains a sequence of successive” formal systems, and whose choices of formal systems “are effectively determined at each stage” (ibid., 444). Mathematics cannot consist in that, even if one identifies mathematics with the activity of “an idealized mathematician whose epistemic alternatives are effectively determined at each stage, but who may have a choice among these alternatives” (ibid., 446). Therefore, Gödel’s argument is invalid.
Gödel, Kurt. 1986–2002. Collected Works. Oxford: Oxford University Press.
McCarthy, Timothy G. 1994. “Self‑Reference and Incompleteness in a Non‑Monotonic Setting.” Journal of Philosophical Logic 23: 423–449.
[i] R. Hanna, “Rational Human Mindedness and The Hierarchy of Increasingly Rich Formal Systems,” Against Professional Philosophy (6 March 2022), available online HERE.
[ii] Carlo Cellucci, The Making of Mathematics: Heuristic Philosophy of Mathematics (Cham: Springer, 2022).
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