tree
tree 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: "konig"
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.
- every leaf of the formula is a variable ×1
- every level has a node that reaches the bottom ×1
- every opening bracket in the formula is closed ×1
- no node has more children than the branching factor ×1
- the branching factor is a whole number between 2 and 3 ×1
- the depth drawn is a whole number between 3 and 7 ×1
- the formula and its disjunctive normal form agree on every row ×1
- the formula has between one and five variables ×1
- the formula is a non-empty string ×1
- the path passes through one node per level ×1
- the pruning parameter is a whole number between 2 and 9 ×1
- the table has one row per assignment ×1
- the tableau and the truth table reach the same verdict ×1
- the tableau stays small enough to draw ×1
- the tableau tests validity or satisfiability ×1
- the tree drawn actually reaches the bottom row ×1
- the walk always has a surviving child to step to ×1
- the whole formula is consumed by the parser ×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.
An infinite tree has an infinite path
A tree that goes on forever, in which every node has only finitely many children, must contain a single branch that goes on forever. The proof is a rule for walking, and the rule is the whole of why finite information can decide an infinite question.
LogicThe 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.
LogicThe tree that closes
To prove a formula, assume it false and take it apart. Every branch ends in a contradiction, or one of them describes exactly how it could have been false — and either way the tree is the answer, drawn.