If sentences proved from first-order arithmetic axioms are 'true' in non-standard models containing infinite natural numbers, then 'true in all models' conflates formal truth-in-a-structure with arithmetical truth about the natural numbers.
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When a statement is logically valid according to the rules of a particular mathematical system, regardless of whether it matches reality.
non-standard models(as used in mathematical logic)
Alternative mathematical structures that follow the same formal rules as the standard system but contain different kinds of objects (like infinite numbers that don't exist in regular arithmetic).