Combinatorial Game Theory
who wins, and by how much
Ladder

Infinitesimals — the ladder

2 distinct arguments against one idea, from the one that introduces it to the one that assumes the rest.
  1. ↑{0 | {0 | 0}}> 0< 1/1024outcome L⇑{0 | {{0 | 0}, 0 | 0}}> 0< 1/1024outcome L↑∗{{0 | 0}, 0 | 0}‖ 0< 1/1024outcome N∗{0 | 0}‖ 0< 1/1024outcome N↓{{0 | 0} | 0}< 0< 1/1024outcome Rvaluecanonical formagainst 0against a thousandth↑ is positive and smaller than every positive number — which no real number is∗ is none of greater, smaller or equal — the order is partial, and that is the point

    Infinitesimals

    Some positions are positive — Left wins them whoever moves first — and smaller than every positive number, including a millionth and a millionth of that. They are the values that decide close games, and the smallest of them is a single move's worth of nothing.

    rung 1 · values
  2. −4↑−3↑−2↑−1↑01↑2↑3↑4↑↑↑1pinned exactly⇑⇑2pinned exactly∗∗04 ups of slack↑∗↑∗14 ups of slack⇑∗2·↑∗24 ups of slack↓↓-1pinned exactly⇓⇓-2pinned exactlythe largest n with G ≥ n·↑, and the smallest with G ≤ n·↑ — both found by testinga position with no star in it is pinned to a single multiple of upand one with a star is not — comparison cannot see past it, which is why the calculus exists

    How many ups

    When every component of a position is smaller than every positive number, no number can decide it. What decides it is a count of ups — and comparison can pin that count down exactly, except when a star is present, when it cannot.

    rung 2 · sums

All ladders