Hilbert's distinction between formal systems and their intended domains supports the view that the standard model N is fixed by our pre-formal grasp of the natural numbers, not by any axiom set.

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Formal system(as used in logic and mathematics): A set of rules and symbols (like mathematical axioms) that you use to prove whether statements are true or false, similar to how a chess game has specific rules that determine what moves are legal.
Hilbert: # Hilbert David Hilbert was an influential German mathematician (1862-1943) who made groundbreaking contributions to many areas of mathematics and helped shape how mathematicians think about solving problems. He's famous for proposing a list of 23 major unsolved math problems in 1900, which guided mathematical research for decades and demonstrated the power of identifying important questions. His work emphasized the importance of rigorous proof and formal logical systems, influencing everything from geometry to quantum mechanics.
Intended domain(what a formal system is meant to represent): The real-world things or concepts that a formal system is supposed to describe or apply to—for example, the intended domain of arithmetic rules is actual numbers.
Pre-formal grasp(source of our understanding of natural numbers)

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