Is there a mathematical theory postulating something like the existence of the maximum (natural) number?

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My math got pretty rusty since college, so please forgive the naivete and imprecise formulation of my question. I vaguely recall a mathematician telling me at a party something to the effect that there is a solid mathematical theory that posits the existence of the maximum number and then draws interesting inferences from this premise. Is there any truth to this? If so, what are the key words by which to Google it?