The set S(N) of all natural numbers is generally believed to have infinite cardinality (ie S(N) has an infinite number of members) and yet every member of the set is believed to be finite. Infinite natural numbers are by convention "not allowed".(adsbygoogle = window.adsbygoogle || []).push({});

This leads to a contradiction, as follows :

The set S(N) of natural numbers is closed under the operation of addition, which basically means that the sum of any two (or more) natural numbers is also a natural number, and must be included in the set S(N).

Therefore, the arithmetic sum of ALL members of the set S(N) must also be a natural number, included in S(N).

However, if S(N) has infinite cardinality (as is generally supposed), then it follows that the process of summation of all of the members of S(N) (summation of an infinite number of finite natural numbers, all but one of which is greater than or equal to 1) must produce an infinite result. This would mean that the summation of all the members of S(N) is an infinite natural number - a contradiction.

On the other hand, if we insist that the result of summation of all of the members of S(N) must produce a finite natural number, then this can only come about if there are a finite number of members of S(N), ie S(N) has finite (not infinite) cardinality.

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# Summation of S(N)?

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