Truth and Provability—A Comment on Redhead

In the first part of this investigation we highlight two, seemingly irreconcilable, beliefs that suggest an impending crisis in the teaching, research, and practice of mathematics: (1) the belief, with increasing, essentially faith-based, conviction and authority amongst academics that a first-order Set Theory such as ZFC can be treated as the lingua franca of mathematics, since its theorems-even when unfalsifiable-can be treated as 'knowledge' because they are finite proof sequences which are entailed finitarily by self-evidently Justified True Beliefs; and (2) the gradually emerging, but powerfully rooted in economic imperatives, belief that only those Justified True Beliefs can be treated as 'knowledge' which are, further, categorically communicable as Factually Grounded Beliefs-hence as comprehensible pre-formal 'mathematical truths'-by any intelligence that lays claims to a mathematical lingua franca. We argue that this seeming irreconcilability merely reflects a widening chasm between an increasing underestimation of the value to society of 'semantic arithmetical truth', and an increasing over-estimation of the value to society of 'syntactic set-theoretical provability'; a chasm which must be narrowed and bridged explicitly to avoid lending an illusory legitimacy-by-omission to the perilous concept of 'alternative facts'. We thus proffer a Complementarity Thesis which seeks to recognize that mathematical 'provability' and mathematical 'truth' need to be interdependent and complementary, 'evidence-based', assignments-by-convention to the formulas of a formal mathematical language; where (a) the goal of mathematical 'proofs' may be viewed as seeking to ensure that any mathematical language intended to formally represent our pre-formal conceptual metaphors and their inter-relatedness is unambiguous, and free from contradiction; whilst (b) the goal of mathematical 'truths' must be viewed as seeking to 'validate' that any such representation does, indeed, uniquely identify and adequately represent such metaphors and their inter-relatedness. Our thesis is that, by universally ignoring the need to introduce goal (b) in our curriculum, the teaching of, and research in, mathematics at the postgraduate level cannnot justify its value to society beyond the mere intellectual stimulation of individual minds. In the second part we appeal to some recent-and hitherto unsuspected-unarguably constructive Tarskian interpretations, of the first-order Peano Arithmetic PA, which establish PA as both finitarily consistent, and categorical. Since we also show that the second order subsystem ACA 0 of Peano Arithmetic and PA cannot both be assumed provably consistent, we conclude that there can be no mathematical, or meta-mathematical, proof of consistency for Set Theory. Hence the theorems of any Set Theory are not sufficient for distinguishing between (i) what we can believe to be a 'mathematical truth'; (ii) what we can evidence as a 'mathematical truth'; and (iii) what we ought not to believe is a 'mathematical truth'; whilst the theorems of the first-order Peano Arithmetic PA are sufficient for distinguishing between (i), (ii) and (iii). We conclude from this that the holy grail of mathematics ought to be 'arithmetical truth', not 'set-theoretical proof'.

2013

We show that if the standard interpretation of PA is sound, then Gödel’s arithmetical representation of any recursive relation yields an undecidable PA formula. This Gödelian characteristic is merely a reflection of the fact that, by the instantiational nature of their constructive definition in terms of Gödel’s β-function, such formulas are designed to be algorithmically verifiable, but not algorithmically computable, under the standard interpretation of PA.

Theoretical Computer Science, 1983

We develop the proof theory of Hoare's logic for the partial correctness of whileprograms applied to arithmetic as it is defined by Peano's axioms. By representing the strongest postcondition calculus in Peano arithmetic PA, we are able to show that Hoare's logic over PA is equivalent to PA itself.

STUDIA LOGICA, vol.108(4), pp 857-875, 2020

In a classical 1976 paper, Solovay proved the arithmetical completeness of the modal logic GL; provability of a formula in GL coincides with provability of its arithmetical interpretations in Peano Arithmetic. In that paper, he also provided an axiomatic system GLS and proved arithmetical completeness for GLS; provability of a formula in GLS coincides with truth of its arithmetical interpretations in the standard model of arithmetic. Proof theory for GL has been studied intensively up to the present day. However, it might sound somewhat strange that no proof theory for GLS was ever developed nor even suggested thus far, except for the axiomatic system offered by Solovay. In this paper, we develop a proof theory for GLS based on the sequent calculus method. We provide a sequent calculus for GLS and prove the cut- elimination and some standard consequences of it: the interpolation and de- finability theorems. As another consequence of cut-elimination, we also prove the equivalence of GL and GLS with respect to a special form of formulas which we call G ̈odel sentences, using a purely proof-theoretical method.

Analysis, 2003

As the first illustration of a potential satisfier for the 'platitudes for truth' in the appendix to his engaging recent discussion of the concept of truth (Wright 1999), Crispin Wright has proposed a notion of 'truth conceived as coherence' for arithmetic.

Arxiv preprint math/0507044, 2005

Analysis, 2014

Tarski's Undefinability of Truth Theorem comes in two versions: that no consistent theory which interprets Robinson's Arithmetic (Q) can prove all instances of the T-Scheme and hence define truth; and that no such theory, if sound, can even express truth. In this note, I prove corresponding limitative results for validity. While Peano Arithmetic already has the resources to define a predicate expressing logical validity, as Jeff Ketland (2012) has recently pointed out, no theory which interprets Q closed under the standard structural rules can define nor express validity, on pain of triviality. The results put pressure on the widespread view that there is an asymmetry between truth and validity, viz. that while the former cannot be defined within the language, the latter can. I argue that Vann McGee's and Hartry Field's arguments for the asymmetry view are problematic.

Grazer Philosophische Studien, 1998

Lucas and Redhead ([2007]) announce that they will defend the views of Redhead ([2004]) against the argument by Panu Raatikainen ([2005]). They certainly re-state the main claims of Redhead ([2004]), but they do not give any real arguments in their favour, and do not provide anything that would save Redhead's argument from the serious problems pointed out in ). Instead, Lucas and Redhead make a number of seemingly irrelevant points, perhaps indicating a failure to understand the logico-mathematical points at issue.

2019

Hilbert's program of establishing consistency of theories like Peano arithmetic PA using only finitary tools has long been considered impossible. The standard reference here is Goedel's Second Incompleteness Theorem by which a theory T, if consistent, cannot prove the arithmetical formula ConT, 'for all x, x is not a code of a proof of a contradiction in T.' We argue that such arithmetization of consistency distorts the problem. ConT is stronger than the original notion of consistency, hence Goedel's theorem does not yield impossibility of proving consistency by finitary tools. We consider consistency in its standard form 'no sequence of formulas S is a derivation of a contradiction.' Using partial truth definitions, for each derivation S in PA we construct a finitary proof that S does not contain 0=1. This establishes consistency for PA by finitary means and vindicates, to some extent, Hilbert's consistency program. This also suggests that in the ari...