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What a system cannot say

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.
S1S2S3S4S5S6S7S1S2S3S4S5S6S7Rthe sets that do not contain themselvesrow i, column j is marked when set i contains set j — the diagonal asks whether a set contains itselfthe row beneath is the complement of the diagonal, and it is not one of the rows above it Logic

A list that cannot contain itself

The set of all sets that do not contain themselves is not a set. The argument is the diagonal again, applied to a table whose rows and columns are the same objects, and it destroyed the foundations of mathematics in a postcard.

 φ10110101000φ21101010000φ30011111000φ41010100000φ50001001000φ60111110000φ71110011001φ8010100000110011001neweach row is a sentence, and whether it says yes to each numbered question; the markedsquares are the diagonalthe row underneath is the sentence that answers each question the opposite way to theone asked about it — so the answers cannot all be given by a sentence on the list Logic

The sentence that says it has no proof

Number every sentence and every proof, and a formal system can talk about itself. Then the diagonal is available one more time, and what it builds is a sentence that is true exactly when it is unprovable.

 basevaluein hereditary baseordinal242^2ω^ω3263^2·2 + 3·2 + 2ω^2·2 + ω·2 + 24414^2·2 + 4·2 + 1ω^2·2 + ω·2 + 15605^2·2 + 5·2ω^2·2 + ω·26836^2·2 + 6 + 5ω^2·2 + ω + 571097^2·2 + 7 + 4ω^2·2 + ω + 481398^2·2 + 8 + 3ω^2·2 + ω + 391739^2·2 + 9 + 2ω^2·2 + ω + 21021110^2·2 + 10 + 1ω^2·2 + ω + 11125311^2·2 + 11ω^2·2 + ω1229912^2·2 + 11ω^2·2 + 11G(4) written in hereditary base b, the base bumped to b+1, then one subtractedthe integers keep rising, and the ordinals fall at every single step — which is why it has to stop Logic

A sequence that explodes and still stops

Goodstein's sequence starting at 4 climbs past any number you care to name and reaches zero after about ten to the hundred and twenty million steps. The proof that it stops is a second sequence, running alongside it, that goes down.

 U¬UU ∪ ¬U¬¬UU is an interval with 1 point taken out; ¬U is the inside of what is leftU and ¬U together miss the endpoints, and ¬¬U hands them back — so U ∨ ¬U is not everything and ¬¬U is not U Logic

The middle that is not excluded

Either it is raining or it is not. Drop that as an axiom and what is left is still a logic — one with models made of open sets and of stages of knowledge, in which a set and its negation between them miss the boundary.

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