My beliefs about the integers are a little fuzzy. I believe the things that ZFC can prove about the integers, but there seems to be more than that. In particular, I intuitively believe that "my beliefs about the integers are consistent, because the integers exist". That's an uncomfortable situation to be in, because we know that a consistent theory can't assert its own consistency.
Should I conclude that my beliefs about the integers can't be covered by any single formal theory? That's a tempting line of thought, but it reminds me of all these people claiming that the human mind is uncomputable, or that humans will always be smarter than machines. It feels like being on the wrong side of history.
It's also dangerous to believe that "the integers exist" due to my having clear intuitions about them, because humans sometimes make mistakes. Before Russell's paradox, someone could be forgiven for thinking that the objects of naive set theory "exist" because they have clear intuitions about sets, but they would be wrong nonetheless.
Let's explore the other direction instead. What if there was some way to extrapolate my fuzzy beliefs about the integers? In full generality, the outcome of such a process should be a Turing machine that prints sentences about integers which I believe in. Such a machine would encode some effectively generated theory about the integers, which we know cannot assert its own consistency and be consistent at the same time.
So it seems that in the process of extracting my "consistent extrapolated beliefs", something has to give. At some point, my belief in my own consistency has to go, if I want the final result to be consistent.
But if I already know that much about the outcome, it might make sense for me to change my beliefs now, and end up with something like this: "All my beliefs about the integers follow from some specific formal theory that I don't know yet. In particular, I don't believe that my beliefs about the integers are consistent."
I'm not sure if there are gaps in the above reasoning, and I don't know if using probabilistic reflection changes the conclusions any. What do you think?
if you don't mean any ontological import you might want to phare it as "integers is system that works". The negation would be things that dont' work. For example a triangle with a right corner with angles summing over 180 degrees. But even then you have to specify the background assumptions as those kinds of triangles actually work out. Usually a mind defaults to a euclidean mindset while the applied concepts could apply to non-euclidean context too.
That integers exist could mean a number of things. Like that x + y = y + x for every x and y that is a value. However the logic of non-commuting values has been figured out. Therefore that sentence would be false. There are things that don't fall under this rule meaning this is not an universal rule. Having some assumtion that you need not have violated in your life doesn't mean such a violation would be impossible. The only way back would be to explicitly declare the delineation of that context ie the various properties needed. But then "integers exists" becomes just "assuming integers integers is all there is" which isn't very surprising or would need any explanation.