Theme

What a system cannot say — page 4

Rules asked a question about themselves, and an answer that is provably not available from inside — which is a different kind of limit from not knowing yet.

A diagonal walk through countably many listed sets. A grid of six rows and seven columns, each cell numbered by Cantor's diagonal order, with the path of the first twenty-one cells drawn.

Logic

The choice inside a countable union

A countable union of countable sets is countable: list each set, then walk the grid of all their members along its diagonals. The proof is two lines and every student meets it early. It also makes infinitely many arbitrary choices at once — one listing for each set — and without the axiom of choice the theorem can fail: there are consistent worlds in which the real numbers are a countable union of countable sets, and worlds in which countably many pairs of socks cannot be counted.

Building a spanning tree by keeping every edge that closes no loop. Four stages of a greedy pass over the fourteen edges of an eight-point graph, ending with a spanning tree of 7 edges.

Logic

A spanning tree for every graph

Every connected graph has a spanning tree: a set of its edges that joins every point and closes no loop. For a finite graph the proof is a greedy pass over the edges. For an infinite graph the greedy pass has to keep going past the end of every list, and the statement turns out to be exactly as strong as the axiom of choice — Zorn's lemma supplies the tree, and the existence of spanning trees in every graph gives back the whole axiom.

Euclid's argument run twenty times. The first 20 terms of the Euclid–Mullin sequence: 2, 3, 7, 43, 13, 53, 5, 6221671, 38709183810571, 139, 2801, 11, 17, 5471, 52662739, 23003, 30693651606209, 37, 1741, 1313797957.

Number

Euclid's proof run as a machine

Euclid proved there is no last prime by multiplying the primes on any list, adding one, and noting that the result has a prime factor not on the list. Run the proof as a machine — start from 2, and each time take the smallest prime factor of one more than the product so far — and it produces 2, 3, 7, 43, 13, 53, 5, 6221671, … a sequence that never repeats, that reaches small primes late and large ones early, and that nobody can prove reaches every prime.

All themes