MF99: "Infinite sequences": what are they?





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Published on Jul 3, 2012

This lecture tries to clarify the big gap between the (finite) sequences we introduced in the last lecture, and "infinite" or "ongoing sequences" (we introduce the term "on-sequence") as are found in Sloane's Online Encyclopedia of Integer Sequences. We concentrate discussion on three such: the basic on-sequence of natural numbers, the on-sequence of prime numbers, and the on-sequence of Catalan numbers.

We lay out Euclid's well-known proof that given any finite list of primes we can find a prime not in it, and suggest some computational snags that perhaps we might worry about. The Catalan sequence of combinatorics fame has many possible definitions and formulas, illustrating the crucial distinction: finite sequences are specified by the elements of the sequence themselves, while "on-sequences'' can only be specified more indirectly, and generally non-uniquely.

Along the way we point out that the grammatical closeness of the terms finite and infinite suggest a dichotomy which is not really born out in experience.

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