Experimental Logics, Mechanism and Knowable Consistency

Abstract

In a paper published in 1975, Robert Jeroslow introduced the concept of an experimental logic as a generalization of ordinary formal systems such that theoremhood is a Π 2 0 -property (or in practice Δ 2 0 ) rather than Σ 1 0 . These systems can be viewed as (rather crude) representations of axiomatic theories evolving stepwise over time. Similar ideas can be found in papers by Putnam (1965) and McCarthy and Shapiro (1987).The topic of the present article is a discussion of a suggestion by Allen Hazen, that these experimental logics might provide an illuminating way of representing "the human mathematical mind". This is done in the context of the well-known Lucas-Penrose thesis. Though we agree that Jeroslow's model has some merit in this context, and that the Lucas-Penrose arguments certainly are less than persuasive, some semi-technical doubts are raised concerning the alleged impact of experimental logics on the question of knowable self-consistency.

Gödel’s second incompleteness theorem for Σn-definable theories

“…We strengthened this result by deleting the requirement of "explicit extension of PA" (Theorem 3). The optimality of this result, in a sense, follows from the fact that a complete ∆ n+1 -definable and Σ n−1sound theory (which is an explicit extension of PA) may prove its own Σ n−1 -soundness (Theorem 6-which generalizes an example of [6] reconstructed in [7]).…”

“…But if Pr T does not satisfy the derivability conditions, then Gödel's second incompleteness theorem may not hold anymore; see [6, pp. 263-264] for an example of a consistent ∆ 2 -definable extension of PA which proves its own (standard) consistency statement (Section 5 of [7] contains a more modern treatment). So, the derivability conditions may not hold for definable theories in general, even if they are sufficiently strong, e.g.…”

GENERALIZATIONS OF GÖDEL’S INCOMPLETENESS THEOREMS FOR ∑n-DEFINABLE THEORIES OF ARITHMETIC

“…1 -soundness should be satisfied by any theory which is intended to be a formalization of mathematics including arithmetic, and 2 -definable theory whose consistency is provable in it (see Feferman (1960)). It is an interesting problem to investigate theories which are out of the range of Gödel's theorems, and, for example, nonrecursively-enumerable theories which prove their own consistency are investigated in Niebergall (2005) and Kaså (2012).…”