Forgive me if this question sounds naive or somewhat poorly constructed due to my nebulous understanding of the concepts involved...
I have recently been reading about the nature of PA, and how, being a first-order theory, it leads to the existence of multiple non-standard models. On the other hand, while second-order logic provides a single model of the natural numbers, it has problems in that the proof system is not complete. So it seems that a first-order theory is the best we have.
So my question is, how does the existence of non-standard models, which I believe entails the existence of numbers 'beyond' the natural numbers, affect the working mathematician in real life, if at all? Does it materially affect any field of mathematics outside of mathematical logic? Can we just ignore the existence of such rogue elements, or can they in some way 'interfere' with the properties and workings of the natural numbers and other number systems derived from them?
Again, if any of my assertions or assumptions are false, please let me know.