You could try, but your internal view of "standardness" won't always coincide with the external view, just like with finiteness or countability. In fact, here's a proof that the attempt can never succeed:
Take any formal theory T that talks about natural numbers, like PA or ZF. We'll assume that T can be encoded as a computer program, printing out sentences about natural numbers in some simple language with arithmetic and quantifiers. We won't make any other assumptions about the theory's internal workings, it can use any axioms it likes.
Thanks to the miracle of Turing completeness, any facts about that computer program can also be encoded as sentences about natural numbers. So we can construct a sentence called Con(T) which says essentially "the program will never print 0=1", or equivalently "there's no natural number N encoding a proof in T that 0=1". That sentence encodes the idea that "T is a consistent theory" within T itself.
Godel's theorem says that T cannot prove Con(T). Therefore we can extend T with the negation of Con(T), and the resulting "self-hating" theory T+¬Con(T) will be as consistent as T itself. Like all consistent theories, that theory will have a model, which is also a model of the original theory T. In that model, Con(T) is axiomatically false, so the above-defined number N is asserted to exist. But if we assume that T was consistent to begin with, then there's actually no finite proof in T that 0=1, so N can't be a standard natural number. Therefore N is nonstandard, so T has a model with a nonstandard natural number. QED.