No known unknowns in Peano arithmetic

Reports that say that something hasn’t happened are always interesting to me, because as we know, there are known knowns; there are things we know we know. We also know there are known unknowns; that is to say we know there are some things we do not know. But there are also unknown unknowns—the ones we don’t know we don’t know. And if one looks throughout the history of our country and other free countries, it is the latter category that tend to be the difficult ones.

Donald Rumsfeld

Rumsfeld’s statement may have been correct about American politics, but he was wrong in the context of Peano arithmetic. In Peano arithmetic, every known is a known known, and every unknown is an unknown unknown. Let me explain.

A paragraph for background: PA expresses enough arithmetic to allow it to encode sentences like “φ is provable from the axiom set T” as arithmetic properties of the natural numbers. But Peano arithmetic also has nonstandard models containing objects that aren’t natural numbers. It’s possible for a sentence about arithmetic to be true of these nonstandard models and false of the standard models. And in particular, the sentences that Gödel-encode proofs of other sentences can take on a different meaning in nonstandard models. For instance, the Gödel encoding of a sentence like “PA is consistent” looks like “There is no number with the property that it encodes a proof of 0=1 from the axioms of PA”. This might be true of the standard natural numbers, but false in some nonstandard model. There might be some nonstandard number that satisfies Gödel’s arithmetic formula for a proof of 0=1 from the axioms of PA, but which doesn’t actually encode any such proof (as only standard natural numbers Gödel-encode valid proofs). Gödel encodings are only logically equivalent to the statements they encode in the standard model of the natural numbers.

Of all the sentences of the form “φ is provable from the axiom set T”, which are provable from the axioms of Peano arithmetic? How much does Peano arithmetic know about what arbitrary formal theories prove? For all of the following results, I will assume that PA is consistent.

Our first result: If PA proves “φ is provable from T”, then φ really is provable from T.

This follows from the soundness of first-order logic. If PA proves “φ is provable from T”, then this sentence must be true in all models. In particular, it’s true in the standard model. And if it’s true in the standard model, then it must actually be true that φ is provable from T.

Second result: If φ is provable from T, then PA proves “φ is provable from T.”

This follows from three facts: (1) that the standard natural numbers are a subset of every nonstandard model, (2) that the sentence “φ is provable from T” is a statement of the form “There exists a number with some property”, and (3) that first-order logic is complete.

The Gödel encoding of “φ is provable from T” is something like “there’s some number n with the property that n encodes a proof of φ from T.” Now, suppose that φ is provable from T. Then there’s a standard natural number that encodes this proof. And since every nonstandard model contains every natural number, this number exists in all these models as well! So the statement is true in all models. So by completeness, it’s provable from PA.

So far we have that PA proves “φ is provable from T” if and only if φ is actually provable from T. Loosely, if a theory can prove something, then Peano arithmetic knows this. Even though ZFC is a stronger theory than Peano arithmetic, there’s nothing that ZFC can prove that PA doesn’t know it can prove. In particular, ZFC proves the consistency of PA, so PA proves that ZFC proves the consistency of PA! This of course doesn’t mean that PA proves its own consistency, because PA doesn’t trust ZFC to be true (it doesn’t accept the axioms of ZFC).

Furthermore, for every sentence that PA can prove, PA can prove that PA can prove it. In this sense, everything that PA knows is a known known. What we’ll see next is that for PA, every unknown is an unknown unknown. For Peano arithmetic, and in fact for ANY consistent theory that expresses enough arithmetic to be subject to Gödel’s theorems, there are no known unknowns.

Third result: If φ is not provable from PA, then “φ is not provable from PA” is not provable from PA.

This follows from Gödel’s second incompleteness theorem and the principle of explosion.

Now, the proof of our third result. Suppose “φ is not provable from PA” is provable from PA. By the principle of explosion, if PA was consistent then EVERYTHING would be provable from it. So by finding something that isn’t provable from its axioms, PA is able to prove its own consistency! But then, by Gödel’s second incompleteness theorem, PA must be inconsistent. This contradicts our background assumption that PA is consistent.

Note that the sentence “φ is not provable from PA” isn’t the same type of sentence as “φ is provable from PA”. As we saw a moment ago, the second is a sentence of the form “there’s some number n with a particular property,” and all of these sentences are provable from PA if and only if they’re true of the standard naturals. But the first sentence is of the form “there’s no number n with a particular property”, which is not the same! It’s possible for this sentence to be true of the standards but false of the nonstandards. All we need is for there to be no standard numbers and at least one nonstandard number with that property.

So if PA is consistent, then it can never prove that it doesn’t prove something. Now, notice that all that I’ve said applies more generally than just Peano arithmetic. For any consistent mathematical theory that express sufficient arithmetic for Gödel’s incompleteness theorems to apply, every known is a known known and every unknown is an unknown unknown.

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