lattice
lattice 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.
At its defaults
show: "cantor"
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.
- each named target is a subset index is a whole number between 0 and 7 ×1
- each subset is joined to the ones one element larger ×1
- no element is sent to the built set ×1
- the built set disagrees with f(k) about whether k belongs ×1
- the map names one subset per element ×1
- the number of subsets is two to the power of the set's size ×1
- the size of the underlying set is a whole number between 2 and 4 ×1
- the two ends of an edge differ by exactly one element ×1
- there are more subsets than elements ×1
Where it is called
Changing this generator changes every figure on this list. That is what makes the list worth publishing rather than keeping in a check script.
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.
LogicThe row that is not on the list
Write down a list of infinite sequences, any list at all, and there is a rule that builds a sequence missing from it. The rule reads one entry from each row, and it is the single most reused argument in this field.
LogicTwo injections make a bijection
If each of two collections fits inside the other without collisions, they are the same size. That sounds obvious and is not, because neither injection needs to be onto — and the proof is a rule for deciding which of the two to follow, one chain at a time.