Godel theorem: confusion on "PM is consistent -> PM's consistency is not PM-provable"

Question

I came to a good summary of Godel's incompleteness theorem proof (http://www.jamieyorkpress.com/wp-content/uploads/2012/04/G%C3%B6dels-proof-summary.pdf) but I'm confused on one step which leads to the conclusion that "PM is consistent -> PM's consistency is not PM-provable." Would appreciate someone can enlighten me. Because I haven't finished reading the original paper, I based my understanding on this summary. I could be confused because of this summary is wrong.

As the summary says, assuming PM is consistent, then the formula G "G is not PM-provable" is not PM-provable. Then meta-mathematically, it's clear that G is true, therefore PM is incomplete.

This conclusion: "PM is consistent -> PM is incomplete" (*) is then used as a theorem in a series of steps that leads to a contradiction which proves that "PM is consistent -> PM's consistency is not PM-provable". Basically:

"PM is consistent and such consistency is PM-provable" ->   
     (using (*) here.  This is the step I'm confused about)
"PM's incompleteness is PM-provable" ->
"G is PM-provable" -> contradiction. 

My confusion is that (* ) is clearly the conclusion of a meta-mathematical proof. Can we use that as part of a PM-proof? Because the above proof uses (*), I wonder we therefore can't say G is PM-provable but just G is meta-mathematically-provable, which doesn't lead to any contradiction with the conclusion that "G is not PM-provable"?

Hope I've explained my confusion clearly. Would appreciate any kind of replies. Thanks!

Answer 1

This is a very good question! Many treatments of Goedel's theorem gloss over exactly where the relevant statements are being proved.

Here's what's going on:

• Goedel's argument, though on the face of it a metamathematical one, can be formulated in a sufficiently powerful theory. PA, for instance, is more than powerful enough for the task! That is, the theory PA itself proves, "If PA is consistent, then PA is incomplete."

Actually, this is not quite due to Goedel, but rather Rosser's improvement of Goedel's argument; Goedel original showed a somewhat weaker fact.

• Now, PA can go on to prove, via the usual argument, the statement: "If PA is consistent, then PA does not prove its own consistency."

However, note the constant hypothesis "If PA is consistent." The unqualified claim $$\mbox{PA is incomplete}$$ is not provable in PA alone; rather a stronger theory, capable of proving PA's consistency, is needed. Metamathematically, this corresponds to our accepting that PA is consistent. (Goedel's original argument had even more metamathematical baggage: it required that we accept that PA is true, or at least correct about a certain class of propositions. Rosser's improvement was to reduce the assumption needed to exactly "PA is consistent.")