Set-Theoretic Definition of Natural Numbers
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High Quality Content by WIKIPEDIA articles! Several ways have been proposed to define the natural numbers using set theory. A consequence of Kurt Gödel's work on incompleteness is that in any axiomatization of number theory (ie. one containing minimal arithmetic), there will be true statements of number theory which cannot be proven in that system. So trivially it follows that ZFC or any other formal system cannot capture entirely what a number is. Whether this is a problem or not depends on whether you were seeking a formal definition of the concept of number. For people such as Bertrand…mehr

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High Quality Content by WIKIPEDIA articles! Several ways have been proposed to define the natural numbers using set theory. A consequence of Kurt Gödel's work on incompleteness is that in any axiomatization of number theory (ie. one containing minimal arithmetic), there will be true statements of number theory which cannot be proven in that system. So trivially it follows that ZFC or any other formal system cannot capture entirely what a number is. Whether this is a problem or not depends on whether you were seeking a formal definition of the concept of number. For people such as Bertrand Russell (who thought number theory, and hence mathematics, was a branch of logic and number was something to be defined in terms of formal logic) it was an insurmountable problem. But if you take the concept of number as an absolutely fundamental and irreducible one, it is to be expected. After all, if any concept is to be left formally undefined in mathematics, it might as well be one which everyone understands.