2025-05-21

the consistency of arithmetic is unknowable

introduction

We are concerned with answering whether it is possible for us to know that arithmetic is consistent, i.e. whether we can know the consistency of a theory of true arithmetic. I will argue that:

1. If we want our true theory of arithmetic to be axiomatizable and consistent.
2. And if we want provability in a formal system as our standard for knowledge concerning facts about arithmetic.
3. Then, for all intents and purposes, we cannot satisfiably know that arithmetic is consistent.

I will argue that because of Gödel’s second incompleteness theorem, we cannot construct a theory of true arithmetic that can satisfy both our conditions (i.e. 1. and 2.). Largely because Gödel’s theorem restricts the ability of consistent, axiomatizable theories of a certain strength from internally proving their own consistency by way of a canonical consistency sentence. Forcing us, as a result, to compromise either on the properties we would want of a theory of true arithmetic (i.e. 1.) or, on our standard for knowledge concerning the consistency of theories (i.e. 2.) – neither of which is desirable/satisfying.

I will begin with a brief overview of theories of arithmetic and how they relate to Gödel’s second incompleteness theorem. Throughout, I will motivate the desire for the properties of consistency and axiomatizability in a theory of true arithmetic. Before highlighting how Gödel’s theorem kills the possibility for sufficiently strong, axiomatizable, consistent theories of arithmetic to prove their own consistency. Offering a defence of internal provability in a formal system as the highest standard for knowledge concerning facts about arithmetic in the process. Finally, I will very briefly address some objections relating to the potential for weak theories of true arithmetic and our capacity to know their consistency through other means, such as model-theoretic and proof-theoretic ones. Touching on broader relative consistency arguments in the process.

arithmetic theories & Gödel’s second incompleteness theorem

arithmetic theories

For us to even discuss whether arithmetic can be known to be consistent or not, we first need to appeal to some external/meta-theoretic notion of consistency. Intuitively, a collection of claims can be understood to be consistent if they don’t entail a contradiction. That is, if by reasoning about our initial claims via approved rules of inference we don’t arrive at a particular conclusion and that same conclusion’s negation.

Conveniently, first-order languages capture this idea well. We can formalise our initial claims as a set of formal sentences \(\Gamma\), and we can talk about all the claims that can be inferred from them as a theory \(\T\). Note that \(\T\) will include each of our initial sentences \(\Gamma\), seeing as a sentence infers/entails itself. In this way, we can also understand our theory \(\T\), as a set of first-order sentences closed under entailment.

From here, we can start to talk about a theory of arithmetic. More specifically, an ideal theory of true arithmetic that captures all true arithmetic statements. That is, the set of formalised first-order sentences of true arithmetic claims that is closed under entailment – call this theory \(\bold{TA}\) for “True Arithmetic”.

Now, we can consider some properties we would like such an ideal theory to have for us to recognise it as an accurate representation of what we understand arithmetic to be.

consistency

Arguably, at the top of this list lies the property of consistency. If our theory \(\TrueA\) were inconsistent, by explosion, it would entail, and therefore contain, by a theory’s closure under entailment, all first-order sentences. In other words, an inconsistent theory of arithmetic would entail everything and anything in the language of arithmetic, failing to pick out only the true statements of arithmetic as intended. This makes consistency strictly necessary to our theory, its omission renders our whole attempt at formalisation futile otherwise.

axiomatizability

Second to consistency, is probably the property of axiomatizability, i.e. the property that our theory \(\TrueA\) can be axiomatized by a decidable set of axioms \(\Gamma\). In other words, that there exists a decidable set of sentences \(\Gamma\) that entail all the sentences \(A \in \TrueA\).

At first glance, our reasons for wanting axiomatizability are not so obvious, the property doesn’t seem as necessary to a theory of true arithmetic as consistency.

However, axiomatizability is not only desirable but largely necessary.

Firstly, axiomatizability enables a theory of arithmetic to capture the feature of intersubjective robustness we understand arithmetic to have. It empowers a theory with the capacity to check and generate proofs by finite means, granting, in turn, any person the ability to verify any arithmetic claim.

What’s more, to leave out axiomatizability would be self-sabotaging in our case. Without it, the process of checking proofs would carry the risk of being unbounded, leaving the truth of certain arithmetic claims to be undecidable within our theory. Thus, such claims would be, for all intents and purposes, “uncaptured” by our theory, working explicitly against our attempt to capture arithmetic in a formal theory to begin with. Put differently, a formal theory of arithmetic with proofs that cannot be checked by finite/bounded means simply fails to fulfil our intended goal of formalising arithmetic in the first place. After all, what’s the point in formalising arithmetic and leveraging formal proof only to be left with proofs that can’t be checked?

the trouble: Gödel’s incompleteness

While other properties of theories exist, axiomatizability and consistency suffice to run us into trouble formalising an ideal theory of true arithmetic whose consistency we can satisfiably know.

The trouble takes the form of Gödel’s second incompleteness theorem which states that no theory \(\T\) can exist s.t:

1. \(\T\) is consistent;
2. \(\T\) is axiomatizable;
3. \(\T \supseteq \ISig\); and
4. \(\T \vdash \Con_{T}\)

More concretely, it tells us that if we want to our ideal theory of true arithmetic \(\TrueA\) to be consistent and axiomatizable. Then, for it to also prove its own canonical consistency sentence \(\Con_{\TrueA}\), it must not be stronger than the theory \(\ISig\). That, or we must accept that our theory cannot prove its own consistency via its canonical consistency sentence – an undesirable option.

This is bad news for the simple reason that \(\ISig\) is an already incredibly weak theory. It’s a theory that extends the theory of Robinson Arithmetic, denoted \(\Q\), with a weak induction principle. A theory (\(\Q\)) so weak as to be incapable of proving the trivial sentence \(\forall x 0 + x = x\).

What’s more, despite extending \(\Q\), \(\ISig\) is still strictly weaker than the theory of Peano Arithmetic, denoted \(\PA\); a more promising theory strong enough to at least prove \(\forall x 0 + x = x\).

So, under Gödel’s theorem, to keep hold of consistency, axiomatizability, and an internal consistency proof (via a canonical consistency sentence), our ideal theory \(\TrueA\) would have to be strictly weaker than a theory only marginally stronger than a theory so weak as to be incapable of proving trivialities.

It almost goes without saying: that doesn’t give us much room. And while I cannot prove nor definitively claim that no such theory can occupy that narrow gap, i.e. between \(\Q\) and \(\ISig\)¹, in the same way I cannot prove a teapot isn’t orbiting the sun, I argue the burden of proof as to whether such a theory exists lies with those that claim it does. Hence, until shown otherwise, given the obvious difficulty of the constraints of fitting between \(\Q\) and \(\ISig\), I contend that it is perfectly justified to believe and act as if no such theory exists.

knowledge as proof: consistency is unknowable

At this stage, all we’ve shown is that we’re justified in accepting that there is no room for a theory of true arithmetic with strength between \(\Q\) and \(\ISig\), that is also: consistent, axiomatizable, and capable of proving its own canonical consistency sentence.

But in accepting such a claim we must also accept that we cannot know the consistency of arithmetic.

Because, if we want the highest possible degree of certainty in arithmetical knowledge. Then we must commit ourselves to the highest possible standard of knowledge. So, if we wish to equate knowledge with formal proof, as we do in attempting to formalise arithmetic in the first place, then knowing consistency can only be satisfiably achieved by way of an internal formal proof of a theory’s own canonical consistency sentence – there is no higher standard.

Therefore, in accepting that no such theory exists, we have, by extension, also accepted that no such proof exists. Hence, we cannot know the consistency of arithmetic to our desired standard.

A standard, I maintain, is perfectly warranted to be upheld as the ideal, highest standard of knowledge we can have concerning facts of arithmetic.

An internal proof of a theory’s own canonical consistency sentence renders a theory perfectly self-sufficient. It makes it epistemically autonomous, carrying the knowledge of its own consistency within itself, with no appeal to external or meta principles. In this way, such a theory is also perfectly transparent, its proofs, including that of its own consistency, are a pure product of its axioms and rules of inference.

objections

While I doubt many would argue that there exists a higher standard of knowledge to the one just laid out, some might argue that this standard is too high, unnecessarily so. And in that vein, they point to the possibility of other theories whose consistency can be established, and therefore known, by other means than an internal proof of a canonical consistency sentence.

Some may suggest that there are theories of arithmetic weaker than \(\Q\). And in spite of their inability to represent provability, their consistency can be known by proof-theoretic or model-theoretic means, breathing hope into the idea that the consistency of arithmetic can still be known.

The trouble here is twofold.

Firstly, proof-theoretic and model-theorectic methods of establishing consistency are undeniably weaker standards of knowledge to that of internal proofs of canonical consistency sentences. Model-theoretic approaches rely on models typically constructed with methods that lie outside of the theory they’re trying to establish the consistency of. Similarly, proof-theoretic approaches typically appeal to (meta-)principles outside the theory under inspection, like in the case of the principle of transfinite induction in Gentzen’s theorem.

In either case, their demonstrations of a theory’s consistency simply make them contingent on external notions and structures that then require their own justifications/validation. They replace our ideal, self-sufficient standard with a lesser, dependent one.

Overall, it’s a form of relative consistency that succumbs to the same regress as the more direct approach of relative consistency that involves demonstrating a theory’s consistency with a proof in a background theory.

Secondly, theories weaker than \(\Q\) will fail to capture trivial sentences of arithmetic, making them unsatisfying from the outset. There simply isn’t much merit to formal theories that can’t capture trivialities.

conclusion

Knowing the consistency of a theory of true arithmetic rests on two conditions:

1. The properties we want such a theory to have.
2. The standard of knowledge for arithmetic we want to impose on ourselves.

The requirements of consistency and axiomatizability are wholly justified properties of a theory of true arithmetic. The former being strictly necessary, and the latter being self-defeating to leave out. Similarly, formal proof, particularly the internal proof of a theory’s canonical consistency sentence, is undoubtedly the highest standard of knowledge to which we can hold ourselves concerning facts about arithmetic, namely that of its consistency. As such, Gödel’s second theorem restricts us to theories strictly weaker than \(\ISig\) but stronger than \(\Q\). A gap that is practically inconceivable to fill with a satisfying theory of true arithmetic. Therefore, for all intents and purposes, we cannot know the consistency of arithmetic.