Several questions about the incompleteness theorem

Question

I’ve read quite a few pop math books over the years with descriptions of the incompleteness theorem and I think I understand most of the broad details, but some still elude me. Specifically, these three below. I’m using Peano Arithmetic like most popularizations do.

As I understand it, the incompleteness theorem works in part because one can interpret the sentences of PA as talking about (or modeling) natural numbers (which I will call model A) and that is obviously the intended interpretation. However, as Gödel famously demonstrated, PA can also be interpreted as proving truths about strings of PA, which I will call the meta-model M.

Finally as I understand it, the completeness of First Order Logic means that any sentence which is true in every model is provable. So in other words, every model of the axioms need to agree on the truth value of the provable sentences. However, they do not have to agree on the unprovable sentences because those unprovable sentences are formally independent.

So here are my questions:

1. From what I’ve read, I think I understand why G is unprovable and why it must be true when interpreted with M. What I still don’t get is why does it also have to be true when interpreted about A? Could it not be the case that G is true under only the first model but not the second?

2. Assuming that the above question has a good answer (and I’m sure it does) does that means that assuming PA is consistent, couldn’t we mine it for arithmetical truths? We could construct a Gödel sentence and just look at it’s interpretation under A, then construct a new Gödel sentence by formalizing it a different way and that would represent another truth. I’m sure there is something fundamentally wrong with this idea but I don’t know what.

3. In regards to the proof predicate Prf(x, y), what exactly is that? Is it an equality like x=y (with a lot more constants I’m sure), is it an implication such as x y, or something else?

Keep in mind that I’m not a mathematician or a logician, I’m just a guy who is fascinated by logic but never learned the technical details. If you wish to reply, I’ve pasted an example I found on Peter Smith’s free online book Gödel Without (Too Many) Tears for your copying and pasting convenience:

Defn. 47 U(y) =def ∀x¬Gdl(x, y).

Now we diagonalize U, to give

G =def U(U) = ∀x¬Gdl(x,U). https://www.logicmatters.net/resources/pdfs/gwt/GWT2f.pdf (page 72)

Answer 1

Some comments...

one can interpret the sentences of $\mathsf {PA}$ as talking about (or modeling) natural numbers (which I will call model $\mathcal A$) and that is obviously the intended interpretation. However, as Gödel famously demonstrated, $\mathsf {PA}$ can also be interpreted as proving truths about strings of $\mathsf {PA}$, which I will call the meta-model $\mathcal M$.

Not exactly; the language of $\mathsf {PA}$ "speaks of" numbers.

Gödel's technique of arithmetization encodes expressions (strings) and sequences of expressions into numbers and sequences of numbers.

In this way, syntactical properties and relations of $\mathsf {PA}$ are translated into arithmetical properties and relations.

You can see this post for an "exercise in encoding"; following that encoding-schema, the number $10$ encodes the formula $0=0$.

$10$ is a number that we are using, in the context of arithmetization of syntax, to express a fact about syntactical stuff.

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Having said that, the unprovable statement $G$ [a statement of $\mathsf {PA}$ such that $\mathsf {PA} \nvdash G$, provided that $\mathsf {PA}$ is consistent] is a statement expressing a "facts" about numbers.

Why we assert that it is true ? Because when "decoded" it asserts also a relation between syntactical stuff, and more precisely it asserts that for a certain sentence $G$ of $\mathsf {PA}$ there is no sequence of formulas that is a derivation of it from the axioms.

This result is based on the definition of the relation :

$\text {Prf}(x,y)$.

It is a binary relation that reads :

"the number $x$ encodes (is the Gödel number of) a proof (a sequence of formulas) of the formula encoded by (with the Gödel number) $y$”.

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The definition of $\text {Prf}(x,y)$ is not intuitive, but it is manufactured step-by-step from very simple building blocks, based on the fact that the syntactical specifications of the language are mechanical.

We start encoding the basic symbols (very few) and then encoding finite sequences of symbols, in a way that we can mechanically compute the code corresponding to an expression of the language and decode a number to recover the corresponding expression.

In this way, we can define relations reflecting into arithmetic the syntactical relations and operations, like e.g. :

$\text{neg}(x)$, that is the function that maps the code of a formula $A$ into the code of the formula $\lnot A$; and similarly for $\text {to}(x, y)$, that is the function which maps the Gödel numbers of a pair $A$ and $B$ of formulas to the Gödel number of the formula $A \to B$; and so on.

Then we go on to encode axioms and rules of inference and finally sequences of formulas (i.e. derivation in the system, i.e. proof).

In this way we arrive at the $\text {Prf}(x,y)$ formula and its "twin", the provability predicate : $\text {Prov}(x)$, defined simply as :

$∃x \ \text {Prf}(x, y)$.

The previous one reads : "the number $x$ is the Gödel number of a proof of the formula with the Gödel number $y$”.

Thus, the new one is : "there is a proof of the formula with the Gödel number $y$”, i.e. :

"the formula with the Gödel number $y$ is provable (in the system)".

From it we arrive at the "encoding" of unprovability : $\lnot ∃x \ \text {Prf}(x, y)$ or, equivalently : $∀x \lnot \text {Prf}(x, y)$.