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Seminar on surreal numbers and combinatorial games

Surreal numbers are a generalization of the real numbers that include infinitesimal and infinite numbers. In this series of about ten weekly meetings, approximately 60 minutes each, we will explore the connection between surreal numbers and combinatorial games. Our primary resource will be Conway's On Numbers and Games.

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Tentative Overview & Schedule

# Chapter Primary Subject Key Concepts
1 Ch. 0–1 The Axioms of Surreal Numbers Conway’s inductive construction \(x = \{L \mid R\}\), order, basic arithmetic
2 Ch. 2–3 Reals, Ordinals, and the Field \(\mathbf{No}\) Dyadic rationals, Dedekind vs. Conway cuts, embedding \(\mathbb{R}\) and \(\mathbf{On}\)
3 Ch. 4–6 Structure and Algebra of \(\mathbf{No}\) Conway normal form, sign sequences, field operations, \(x+y\) and \(x\cdot y\)
4 Ch. 7 Generalizing Numbers to Games Dropping \({x}_{L}<{x}_{R}\), Left/Right options, outcomes (\(P,N,L,R\))
5 Ch. 8–9 Game Algebra & Canonical Forms Equality/equivalence, domination, reversibility, simplifying games
6 Ch. 10 Impartial Games & Sprague-Grundy Impartial games as a sub-universe, nim-addition (\(\oplus\)), nim-values (\(*n\))
7 Ch. 12 Infinitesimal Partisan Games Number Avoidance Theorem, Star (\(*\)), Up/Down (\(\uparrow ,\downarrow\)), all-small games
8 Ch. 11 Temperature Theory & Thermography Mean values, cooling/warming, temperature, thermographs, stop values
9 Ch. 13, 15 Advanced Game Structures Atomic weight, tinies/minies, coin-turning/nim-like variations
10 Ch. 14, 16 Infinite Games & The Grand Synthesis Transfinite games, loopy games, \(\mathbf{No} \subset \mathbf{Pg}\) embedding theorem

Detailed Session Breakdown

Phase I: Surreal Numbers (\(\mathbf{No}\))

Session 1: The Construction of Surreal Numbers

  • ONAG Chapters: 0 & 1
  • Topics:
  • Conway’s single unified construction rule: \(x = \{L \mid R\}\) subject to \(x_L \nsim x_R\) (no left option is greater than or equal to any right option).
  • Inductive definition of the partial order \(\leq\) and equivalence \(=\).
  • Birthday construction: creation of numbers day by day starting from Day 0 (\(\{\emptyset \mid \emptyset\} = 0\)).
  • Constructing integers (\(\mathbb{Z}\)) and dyadic rationals (\(\mathbb{Z}[1/2]\)) on finite days.

Session 2: Embeddings of Reals (\(\mathbb{R}\)) and Ordinals (\(\mathbf{On}\))

  • ONAG Chapters: 2 & 3
  • Topics:
  • Transfinite days: infinite birthdays and the birth of \(\omega\), \(\epsilon =1/\omega\), \(\pi\), and \(\sqrt{2}\).
  • Conway cuts vs. Classical Dedekind cuts (why Conway cuts yield a field without equivalence classes).
  • Embedding the von Neumann ordinals \(\mathbf{On}\) directly into \(\mathbf{No}\).
  • Sign sequences (\(+/-\)) as an alternative representation for surreal numbers.

Session 3: Algebra and Arithmetic in \(\mathbf{No}\)

  • ONAG Chapters: 4, 5 & 6
  • Topics:
  • Inductive definitions of addition (\(x+y\)) and multiplication (\(x\cdot y\)).
  • Proof sketch: \(\mathbf{No}\) forms a proper class field containing both real numbers and ordinal arithmetic.
  • Conway Normal Form (expanding surreal numbers in terms of powers of \(\omega\)).
  • Overview of the surreal field extension \(O{n}_{2}\) (nim-arithmetic fields).

Phase II: The Transition from Numbers to Games (\(\mathbf{Pg}\))

Session 4: Dropping the Order Constraint

  • ONAG Chapter: 7
  • Topics:
  • Defining a general game \(G = \{G_L \mid G_R\}\) by dropping the condition \(x_L < x_R\).
  • Physical interpretation: Left moves to \(g_L \in G_L\), Right moves to \(g_R \in G_R\) under the normal play convention (last player to move wins).
  • The four fundamental outcome classes:
    • \(G>0\) (Left wins regardless of who goes first)
    • \(G<0\) (Right wins regardless of who goes first)
    • \(G=0\) (Second player wins, a \(\mathcal{P}\)-position)
    • \(G \parallel 0\) (First player wins, an \(\mathcal{N}\)-position / confused with zero)

Session 5: Equivalence and Canonical Forms

  • ONAG Chapters: 8 & 9
  • Topics:
  • Group structure of games: disjunctive sum \(G+H\) and negative \(-G\).
  • Partial order on games: \(G \ge H \iff G - H \ge 0\).
  • Simplification theorems:
    • Dominated options: deleting redundant moves.
    • Reversible moves: replacing moves that allow immediate counter-attacks.
  • Uniqueness of the canonical form for finite short games.

Phase III: Combinatorial Game Theory & Special Classes

Session 6: Impartial Games and Sprague-Grundy Theory

  • ONAG Chapter: 10
  • Topics:
  • Impartial games (\(G_L = G_R\) for all subpositions).
  • Nim and nim-values (\(\ast n = \{0, \ast 1, \dots, \ast(n-1) \mid 0, \ast 1, \dots, \ast(n-1)\}\)).
  • Sprague-Grundy Theorem embedded in Conway's framework: every finite impartial game is equivalent to a single nim-heap \(\ast n\).
  • Nim-addition (\(\oplus\)) via bitwise XOR.

Session 7: Partisan Infinitesimals & Numbers as Values

  • ONAG Chapter: 12
  • Topics:
  • Number Avoidance Theorem: why players avoid moving in positions that are true numbers unless forced.
  • Non-number games: Star (\(\ast = \{0 \mid 0\}\)), Up (\(\uparrow = \{0 \mid \ast\}\)), and Down (\(\downarrow = -\uparrow\)).
  • The order relationship: \(0<\uparrow <\frac{1}{{2}^{n}}\) (a strictly positive game that is smaller than every positive number).
  • Concept of all-small games.

Phase IV: Advanced Analysis & Structural Synthesis

Session 8: Temperature Theory and Thermography

  • ONAG Chapter: 11
  • Topics:
  • Motivation: evaluating positions in hot games where players want to move to gain advantage.
  • Cooling operator \(G_t\) and the definition of Temperature \(t(G)\) and Mean Value \(m(G)\).
  • Constructing and reading Thermographs.
  • The Thermographic Theorem and optimal play in disjunctive sums of hot games.

Session 9: Atomic Weight and Fine-Grained Infinitesimal Structure

  • ONAG Chapters: 13 & 15
  • Topics:
  • Comparing infinitesimals: why \(\uparrow\) dominates multiples of \(\ast\).
  • Definition of Atomic Weight \(aw(G)\) for analyzing games composed of up-switches and star-like components.
  • Tinies (\(+_x\)) and Minies (\(-_x\)): games that are smaller than all positive multiples of \(\uparrow\).
  • Applications to specific board games (e.g., Hackenbush, Domineering).

Session 10: Transfinite/Loopy Games & The Grand Synthesis

  • ONAG Chapters: 14 & 16
  • Topics:
  • Introduction to Infinite Games (transfinite play lengths) and Loopy Games (cycles, ties, stopper positions).
  • Synthesis: The precise embedding \(\mathbf{No} \subset \mathbf{Pg}\).
    • Numbers are total orders inside the game partial order.
    • Real numbers represent quiescent games, infinitesimals represent urgent local advantages, and non-numeric games represent tactical conflicts.
  • Open problems and modern extensions in CGT.