34 Reinterpreting theorem Chapter of the book Infinity Put to the Test by Antonio Le´on available HERE

Abstract.Riemann’s Series Theorem is reinterpreted as a proof of the inconsistency of the actual infinity hypothesis. Keywords: Riemann’s Series Theorem, conditionally , inconsistency of the actual infinity.

Definitions P621 Riemann’s Series Theorem states that it is possible to reorder the summands of a conditionally convergent series in such a way that it con- verges to any desired number or to (positive or negative) infinity. As we will see in this chapter, the theorem only applies if infinitely many terms are involved in the rearrangement. In those conditions, to converge and not converge to a given number could be reinterpreted as a contradiction derived from the inconsistency of the actual infinity.

∞ a P622 A series Pi=0 i is conditionally convergent if it is convergent but not absolutely convergent. Or in other terms if, and only if:

a) The series converges to a finite number L:

∞

l´ım ai = L (1) n→∞ X i=0

b) The series of its positive (negative) terms diverges to positive (nega- tive) infinite. ∞

l´ım |ai| = ∞ (2) n→∞ X i=0

P623 Riemann’s Series Theorem states that by the appropriate rearrange- ment of its terms, any conditionally convergent series can be made converge to any given finite number or to infinity.

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Discussion P624 We will exclusively deal with conditionally convergent series of real numbers that may converge to infinity or to different finite numbers by rearrangements based on the application of the associative, commutative and distributive properties of the elementary arithmetic operations in the field of the real numbers.

S ∞ a v P625 Let = Pi=1 i be any conditionally convergent series; any na- tural number; and Sv,O the sum of the first v summands of S ordered in a certain way denoted by O. Let us apply one time one of the properties associative, commutative or distributive to the summands of Sv,O so that O S we get a new ordering 1 of the initial summands. Being v,O1 the new S S sum, it will hold v,O1 = v,O, otherwise the applied property would not be satisfied in the field of the real numbers, which is not the case. Assume that for any n it is possible to apply n successive times the properties associative, commutative and distributive to the summands of Sv,O to get a new ordering On of the initial summands and so that, being

Sv,On the new sum, it holds: Sv,On = Sv,O. The properties associative, com- mutative or distributive can be applied one time again to the summands S O S S of v,On to get a new ordering n+1 of them, and so that v,On+1 = v,On , otherwise the applied property would not be satisfied in the field of the real numbers, which is not the case.

P626 From the above inductive argument P625, we conclude that for any finite natural number n it is possible to apply n times the properties associative, commutative and distributive to the summands of Sv,o to get n different arrangements of the summands while their sum is always the same.

P627 It holds, then, the following: Theorema) of the Consistent Reordering.-For any natural number v, the sum of first v terms of any conditionally convergent series is always the same, irrespective of the rearrangement of the involved summands. We can therefore assert that only when the number of summands is infinite the result of the sum depends on the rearrangement of the summands. Therefore, it is the assumed actual infinite number of summands that made it possible Riemann’s conclusion.

P628 According to Riemann series theorem, if S is any conditionally con- vergent series and r any , the sum of its infinitely many terms is and is not equal to r, depending on the order the terms of the series Discussion 3 are summed. This is the type of result one can expect if the hypothesis of the actual infinity were inconsistent. Riemann’s Series Theorem could, therefore, be reinterpreted as a proof of the inconsistency of the actual infinity hypothesis. And that possibility, as legitimate as any other, should be explicitly declared in the statement of the theorem. 4 Reinterpreting Riemann series theorem Chapter References

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