TITLE: Clarification of Gödel's second incompleteness theorem
QUESTION [16 upvotes]: I am sorry for the following question, because the actual answer to this question is in the beautiful works of Feferman and Jeroslow, but, unfortunately, I havn't any time to go into that specific field right now and maybe some of you is aware of the answer.
The situation with Gödel's second incompleteness theorem is quite delicate. Let Pf(a, b) means that a is the Gödel number of proof of statement with Gödel number b and Neg(a, b) means that b is the Gödel number of negation of statement with Gödel number a. The main result of Gödel's work is representability of predicates Pf and Neg in formal arithmetic. Knowing that, we can define the Consistency of formal arithmetic in formal arithmetic as follows:
" it is not the case that there exist a statement A such that A and (not) A are provable in formal arithmetic " or in the language of formal arithmetic:
$$\forall x_1, x_2, x_3, x_4 [\neg (\operatorname{Pf}(x_1, x_3) \wedge \operatorname{Pf}(x_2, x_4) \wedge \operatorname{Neg}(x_3, x_4))].$$
Let's denote last proposition by W. Gödel's second incompleteness theorem says that W is not provable in formal arithmetic, i. e. the consistency of formal arithmetic is not provable in formal arithmetic.
But Solomon Feferman in his remarkable paper of 1960 "Arithmetization of metamathematics in general setting" have found other formalization of consistency of formal arithmetic W' that is provable in formal arithmetic (see notes by E. Mendelson to the second theorem in formal arithmetic chapter of his "Intoduction to Mathematical logic"). Jeroslow ("Consistency statements in formal theories"), thereafter, studied consistency statements in a broader way.
Can someone explain the nature of W'?
(obviously, $W\Leftrightarrow W'$ can not be proven in PA).
Can we interpret the provability of W' as the proof of consistency of formal arithmetic in formal arithmetic?
How well are various consistency statements grasped today?  
Thanks in advance.

REPLY [4 votes]: Reading between the lines, I think your question may be answered in David Auerbach's article "How to Say Things With Formalisms."  Regarding whether these exotic "consistency" statements may be interpreted as "expressing" consistency, I would say that the answer is no.  The crux of the matter may be put more simply.  Let's suppose we want to express "$x$ is even" in the first-order language of arithmetic.  A natural attempt is $\exists y: y+y=x$.  However, you could be sneaky and try something like $\exists y: [y+y=x \wedge \neg G(y,"1=0")]$ where $G(a,b)$ says "$a$ is not the Goedel number of a proof of $b$."  If the theory in question is consistent then your exotic evenness predicate will be true of precisely the even integers.  However, it's pretty clear that there's something fishy about this predicate.  It doesn't really say only that $x$ is even; it says that $x$ is even and $x/2$ doesn't witness a proof of a contradiction.  Similarly, the exotic consistency predicates don't just say that the theory is consistent.
The issue is necessarily somewhat philosophical and not strictly mathematical, because when deciding whether $\exists y: y+y=x$ correctly "expresses" the statement that $x$ is even, we have to resort to our informal understanding of what it really means for $x$ to be even.  But any reasonable person would agree that $\exists y: y+y=x$ is the right way to express "$x$ is even" and not the exotic alternative above.  Similarly, when deciding whether a consistency predicate correctly expresses what we really mean by consistency, we have to look at each step of the construction and check that all the formalizations are "reasonable."  If someone digs in their heels and insists that the exotic predicate is the reasonable one then it's impossible to prove them wrong; however, any reasonable person can see that the standard consistency predicate is the right one.