I wonder what you mean by "what's wrong with an unwritable natural"... It is sound to assume one! And it doesn't make any true mathematical facts false. But of

I wonder what you mean by "what's wrong with an unwritable natural"... It is sound to assume one! And it doesn't make any true mathematical facts false. But of course it's sound to assume many silly things. But why assume those things? The reality is that mathematicians didn't first come up with the Peano axioms and then study the interesting consequences of them. Both as a matter of history and also as a matter of why most mathematicians do math, mathematicians first came up with numbers, and only later came up with axioms that allow them to do rigorous reasoning about them. The same is actually true of almost all math—real numbers were invented to do rigorous reasoning about calculus, which by that point had been around for a century plus; set theory likewise. In other words, if the goal is to come up with any old consistent set of axioms, you can assume an unwritable natural number, that's fine, go ahead. But if your goal is to study natural numbers—which, like, I have a lot of experience with natural numbers in my day to day life, I'm pretty sure I could write all of them if I had enough time and space and so on—then you want to study natural numbers, not some other weird things where there are unwritable weird things.