Analysis of Penrose’s Second Argument Formalised in DTK System

This article aims to examine Koellner’s reconstruction of Penrose’s second argument – a reconstruction that uses the DTK system to deal with Gödel’s disjunction issues. Koellner states that Penrose’s argument is unsound, because it contains two illegitimate steps. He contends that the formulas to which the T-intro and K-intro rules apply are both indeterminate. However, we intend to show that we can correctly interpret the formulas on the set of arithmetic formulas, and that, as a consequence, the two steps become legitimate. Nevertheless, the argument remains partially inconclusive. More precisely, the argument does not reach a result that shows there is no formalism capable of deriving all the true arithmetic propositions known to man. Instead, it shows that, if such formalism exists, there is at least one true non-arithmetic proposition known to the human mind that we cannot derive from the formalism in question. Finally, we reflect on the idealised character of the DTK system. These reflections highlight the limits of human knowledge, and, at the same time, its irreducibility to computation.