By Gödel's incompleteness (which holds even for $Q$ as discussed here) we have that the Robinson arithmetic $Q$ is consistent iff $Q+\lnot Con(Q)$ is consistent.
As I understand it, without further assumptions, we cannot exclude the possibility that $Q\vdash \lnot Con(Q)$, which would imply the inconsistency of $PA$ and $ZFC$ since they both prove $Con(Q)$ but also prove every theorem of $Q$.
I want to picture the situation, where $Q\vdash \lnot Con(Q)$ but $Q$ is in fact consistent. Now my questions are the following.
I would like to claim that all models of natural numbers are therefore non-standard. Is there any mathematical way in which this can be formalized, or is this purely a philosophical statement under my assumptions? I guess the extreme weakness of $Q$ plays a role in this which I find hard to pin down.
Could we find some potentially consistent theories which still formalize infinitary mathematics such as real analysis and topology in satisfactory ways?
What are some combinatorial or other statements I would have to believe in, to also believe such a situation cannot occur?