Apparent Paradox Between Set Theory and the Fibonacci Generating Function

I was reminded of a result I first saw in Mathematical Problems and Proofs, that shows that the generating function for the Fibonacci Sequence implies that the sum over all Fibonacci numbers is -1.

There’s a very good proof here in the first response to the question:

Simply set z = 1, and you have the sum in question, which is plainly -1.

Of course, the sum diverges, which initially lead me to view this result as a mere curiosity, though it just dawned on me, that there’s an additional problem, that I think is tantamount to a paradox:

Let S be the set produced by the union of disjoint sets of sizes F_1, F_2, \ldots, where F_i is the i-th Fibonacci number. It must be the case that S has a cardinality of \aleph_0. More troubling, addition between integers can be put into a one-to-one correspondence between unions over disjoint sets, and thus, we have an apparent paradox.

To be clear, this has nothing to do with convergence –

It should in fact sum to infinity, and it does not, whereas a perfectly corresponding union over sets does.

This example implies that the rules of algebra fail in some cases given an infinite number of terms, whereas set theory does not. One initial observation, addition is plainly not commutative with an infinite number of terms. For example, consider an alternating sum of (+1,-1,+1,-1,...). If you sum from left to right, you can cause the sum to oscillate near any finite value, or to diverge to positive or negative infinity, without changing the terms at all, simply changing their order. So as a consequence, it must be the case that the rules of algebra require reconsideration in the context of an infinite number of terms. I’m not suggesting that this is what’s driving the apparent paradox above, but rather pointing to the general issue that mechanical application of the rules of algebra to an infinite number of terms is not appropriate, and this example plainly demonstrates that fact.


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