At the suggestion of someone on this forum, I'm reading "Forcing for Mathematicians" by Nik Weaver. I'm on Chapter 2 and the third exercise asks:
Give an informal argument that if ZFC is inconsistent then there is a proof in PA that ZFC is inconsistent.
I've tried to do this and here is my argument. Can people advice if: (a) it is correct (insofar as it goes); and (b) if there is something I'm missing that I should know at this point? I've pointed out where I feel like I'm making leaps of faith. So far in the book, Chapter 1 is on PA and Chapter 2 is on ZFC.
(1) ZFC can be written using a finite character set (call it $\Sigma$) [1]
(2) There is a one-to-one function $N:\Sigma^*\rightarrow\mathbb{N}_0$ (i.e. different stings map to different number). [2]
(3) A proof is just a string in $\Sigma^*$ so for any proof there is a unique number associated with it (given by $N$ above) and the numbers associated with proofs all have a certain numerical property.
(4) If ZFC is inconsistent, there is a proof in ZFC of a theorem of the form $A \wedge \neg A$ for some $A$ and such the numbers associated with such proofs have a different numerical property.
(5) Leap of faith 1: there are formula schemas, $\phi(n)$ and $\psi(n)$ which can capture numerical properties associated with proofs in ZFC and the last line is of the form $A \wedge \neg A$ for some $A$ and these formula schemas can written in PA.
(6) As such, ZFC is inconsistent if there is a number, $n^*$ such that both $\phi(n^*)$ and $\psi(n^*)$ hold.
(7) Leap of faith 2: If such a number exists, there is a proof of it in PA.
(8) Thus, if ZFC is inconsistent, there is a proof of it in PA.
[1] In my mind, I include something like a "new line" character so that, in proofs, we can separate each line of reasoning from the next. Not sure if this is needed.
[2] I'm using $\Sigma^*$ to mean all strings that can be written with the character set $\Sigma$.
To fill in the leaps of faith, I feel like I'd need to get much more formal and specify $N$ and then figure out the details of $\phi$ and $\psi$.
Thanks in advance!