Order types drawn on the line: ω, ω+1, ω·2, ω²
ordinal is one function. Everything below came out of it during this
build, at parameters taken from the essays rather than invented for this page — so a figure
here is the same figure a reader meets in an essay, and if the generator changes, this page
changes with it.
With nothing chosen
The Goodstein sequence from 3, with the ordinal beside each term
The fast-growing hierarchy at its first few ordinals
Limit ordinals and the sequences that approach them
The numbers to 18 in hereditary base 2, and their ordinals
Ordinal sums and products, in normal form
What it checks while it draws
Collected by running the family and recording what it asserted, not written here. The count is how many separate times the claim was put to the test while these drawings were made.
- the number of ticks per block is a whole number between 4 and 14 ×2
- a fundamental sequence belongs to a limit ordinal ×1
- a larger ordinal grows at least as fast at every argument ×1
- a limit mark is where the marks before it pile up ×1
- a successor is one step past the ordinal below it ×1
- an ordinal expression is built from whole numbers, w, +, * and ^ ×1
- and so do the two multiplications on the other side ×1
- and the terms strictly increase ×1
- at least one entry needs an exponent that is itself a power ×1
- at least one of the ordinals drawn is a limit ×1
- between one and five order types are drawn ×1
- between one and four pairs of expressions are compared ×1
- between three and six rates are drawn ×1
- between two and six ordinals are drawn ×1
- each number's ordinal is strictly larger than the one before it ×1
- each part of the expression is a number, w, or a bracket ×1
- each rate is increasing in its argument ×1
- every limit mark has marks before it ×1
- every limit mark has marks piling up before it ×1
- every term of the sequence is below the ordinal it approaches ×1
- F₀ adds one ×1
- F₁ doubles its argument ×1
- F₂ multiplies by two to the power of its argument ×1
- nothing sits between the pile-up and the limit it accumulates at ×1
- ordinal addition is associative on the drawn values ×1
- the arrow clears both the last term and the supremum it points at ×1
- the base the numbers are written in is a whole number between 2 and 4 ×1
- the brackets in the expression close ×1
- the drawn pair is an ordinal of the form ω·k + m with k between 1 and 3 ×1
- the exponents fall and the coefficients are positive ×1
- the hereditary form reads back as the number it came from ×1
- the index into a fundamental sequence is a positive whole number ×1
- the largest argument in the table is a whole number between 3 and 5 ×1
- the largest number in the table is a whole number between 6 and 34 ×1
- the marks along each line strictly increase ×1
- the number of steps shown is a whole number between 3 and 12 ×1
- the number of terms shown is a whole number between 3 and 7 ×1
- the ordinal beside each term is strictly smaller than the one before it ×1
- the ordinal view is one of goodstein, arith, cnf, fund, growth ×1
- the pair drawn on the lines is one of the pairs in the table, or null for none ×1
- the starting value is a whole number between 2 and 6 ×1
- the two additions agree ×1
- the two multiplications by a whole number agree ×1
- the whole expression was read ×1
- zero has nothing below it to approach ×1
Where it is called
Every figure on this list is drawn by the same rule, so a change to the rule changes all of them at once. That is why the list is published.
A game that decides what can be said
Two players take turns pointing at elements of two structures; if the second can survive k rounds, then no sentence with k quantifiers tells the structures apart — a statement about infinitely many formulas, settled by a finite search.
LogicA 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.
LogicAn ordinal as a growth rate
Index a family of functions by the ordinals, each one iterating the last, and the index becomes a measure of how fast a function grows. The point where the index leaves what arithmetic can prove is exactly where the Goodstein sequence became unprovable.
LogicEvery ordinal in base omega
Every ordinal below a certain point is a descending sum of powers of ω, in exactly one way. That notation makes comparison mechanical, it is what hereditary base notation becomes when the base is replaced, and it stops at the first ordinal it cannot name.
LogicOne step in front of infinitely many
Put one step before an infinite run of them and nothing has changed; put it after and something has. Ordinal addition records that difference, which is why it is not commutative — and why it keeps information that counting throws away.
LogicReached from below, or not at all
Every limit ordinal anybody meets is the end of an increasing sequence — ω, ω·2, ω^ω, all of them approached one step at a time. The first uncountable ordinal is not, and the reason it is not constrains the size of the continuum.
LogicThe arithmetic that loses subtraction
Adding one to an infinite collection changes nothing, and neither does doubling it, or squaring it. What that costs is the two operations that were doing the work — an equation between infinite sizes cannot be cancelled, and how many are left stops being a question.
LogicThe choice nobody can write down
Given finitely many pairs, picking one thing from each is a finite list of decisions and needs no justification. Given infinitely many, the list cannot be finished — and whether one exists anyway is an axiom, independent of everything else, whose consequences include a theorem most people refuse to believe.
LogicThe size that cannot be pinned down
There is no largest infinity, because no collection has as many members as it has sub-collections. What is not settled is whether anything sits between the first two — and that is not an open problem but a proved absence of an answer.