Let $\sf PA$ denote the theory of natural numbers with constants $(0, 1)$ and binary operators $(+,\times)$ based on the first-order predicate calculus with equality, having the following axioms, where the last one is the axiom schema of induction yielding an axiom for each wff $\Phi(a)$:


* $a+0=a$
* $a\times1=a$
* $a+(b+c)=(a+b)+c$
* $a\times(b+c)=a\times b+a\times c$
* $a+c=b+c\implies a=b$
* $a+1\ne0$
* $\forall a\,\left(\Phi(a)\implies\Phi(a+1)\right)\implies\left(\Phi(0)\implies\Phi(c)\right)$


Note that $\sf PA$ is powerful enough to introduce Gödel numbering for its own formulae and define the predicate for their provability in $\sf PA$. 


Let's use $\sf ZFC$ as a meta-theory to reason about $\sf PA$ and its extensions defined below, and furthermore assume $\sf ZFC$ is consistent.




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For any recursively axiomatizable theory $\sf T$, that contains $\sf PA$ as its fragment, define $\sf T^+$ to be a new theory obtained from $\sf T$ by adjoining the following axiom schema yielding an axiom for each wff $\Phi$:


* $\left(\Phi\ \text{is provable in}\ \sf T\right)\implies\Phi$


Note that $\sf T^+$ can prove consistency of $\sf T$, thus, if $\sf T$ is consistent, $\sf T^+$ is stronger than $\sf T$.




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Let $\alpha$ range over recursive ordinals, i.e. $\alpha\in\omega\_1^{CK}$. Define the countable transfinite sequence of theories $\sf PA\_\alpha$ such that:


* $\sf PA\_0$ is $\sf PA$
* $\sf PA\_{\alpha+1}$ is $\sf PA\_\alpha^+$
* for a limit ordinal $\alpha$, $\sf PA\_\alpha$ is the theory whose set of axioms is the union of sets of axioms of all $\sf PA\_\beta$, where $\beta<\alpha$


Apparently, each of $\sf PA\_\alpha$ is recursively axiomatizable. I also believe each of them is consistent, but do not yet see how to prove it.


*Question 1*: Can we prove it?


*Question 2*: Does any of $\sf PA\_\alpha$ contain a theorem that is not provable in $\sf ZFC$ (when properly translated to the language of set theory, with natural numbers represented as finite von Neumann ordinals, and operators $(+,\times)$ as ordinal addition and multiplication)? If so, what's the least $\alpha$ with this property?




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*Update*: As pointed out in the comments below, my "definition" of the transfinite sequence $\sf PA\_\alpha$ is not really a definition because we have some wiggle room in choosing a specific ordinal notation at limit points (I do not yet completely understand how exactly it can affect the strength of theories in the sequence, but I've started to read a book on this topic — Thanks!). But I believe we still can define the set $\mathcal T$ of all possible transfinite sequences constructed this way (although it is not a singleton set). So, each of my questions can be restated as *"Is it the case for at least one sequence in $\mathcal T?$ Is it the case for all sequences in $\mathcal T?$"*



Transfinitely extending $\sf PA$ — can we get stronger than $\sf ZFC$?
A stark demonstration of why precisely defining how you form $PA\_{\lambda +1}$ for $\lambda$ a limit ordinal: in 1939 Turing showed that if $\varphi$ is a true $\Pi^0\_1$ statement, there is a notation for $\omega+1$ according to which $PA\_{\omega+1}$ proves $\varphi$.


Less pathologically, I believe (although I can't at present find a reference) that there are "nice" paths through Kleene's $\mathcal{O}$ such that the corresponding sequence of theories gotten by adding successive consistency assumptions (so, in particular, weaker than your theories) proves every true $\Pi^0\_1$ sentence; so in particular, one of them proves a $\Pi^0\_1$ sentence not provable from ZFC.


On the other hand, Spector and Feferman (<http://www.jstor.org/stable/2964544>) showed that there are paths through $\mathcal{O}$ which *don't* give you every true $\Pi^0\_1$ sentence. I don't know whether their arguments or others let you control which $\Pi^0\_1$ sentences you do get, or whether they extend to your theories, but in principle there could be a path along which you didn't get $\Pi^0\_1$ sentences which ZFC doesn't prove. This seems extremely unlikely, though.


