#488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins
In this Lex Fridman Podcast episode, mathematician Joel David Hamkins explores the evolution of mathematical thinking about infinity, from Aristotle's early distinctions to Cantor's revolutionary proof that not all infinities are equal. The discussion covers how set theory became the foundation of modern mathematics and examines Gödel's incompleteness theorems, which revealed fundamental limitations in mathematical systems.
The conversation delves into several key developments in mathematics, including Tarski's theory of truth and its implications for formal mathematical structures. Hamkins also discusses Conway's surreal number system and the mathematical properties of infinite chess, showing how these concepts connect to foundational ideas in set theory and computability. This episode bridges complex mathematical concepts while highlighting their interconnected nature and significance to our understanding of mathematical truth and infinity.
This is a preview of the Shortform summary of the Dec 31, 2025 episode of the Lex Fridman Podcast
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1-Page Summary
Development and Implications of Set Theory and Infinity
The understanding of infinity has evolved dramatically from ancient philosophy to modern mathematics. Aristotle first distinguished between potential infinity (numbers continuing indefinitely) and actual infinity (which he considered impossible). This view was challenged when Galileo observed that infinite sets could be equinumerous, showing that parts of infinity could be as numerous as the whole.
Georg Cantor revolutionized mathematical thinking by proving that not all infinities are equal, demonstrating that the set of real numbers is larger than the set of natural numbers. This groundbreaking work led to Zermelo's development of set theory, which eventually became the Zermelo-Fraenkel axioms - the foundation of modern mathematics.
Relationships Between Truth, Provability, and Computability
David Hilbert proposed an ambitious program to provide a solid foundation for all mathematics through a comprehensive axiomatic system. However, Kurt Gödel's incompleteness theorems showed that no such system could prove all mathematical truths or guarantee its own consistency.
Alfred Tarski's semantic theory of truth further separated the concept of truth from formal mathematical structures. This separation, combined with Gödel's work, revealed fundamental limitations in mathematical systems, including the undecidability of certain computational problems like the halting problem.
Novel Math Structures: Surreal Numbers, Infinite Chess
John Conway's surreal number system provides a unified framework encompassing various number systems, including natural numbers, integers, rationals, reals, and infinitesimals. Though fundamentally discontinuous, surreal numbers offer new ways to approach mathematical analysis.
In the realm of infinite chess, played on an unbounded board, Joel David Hamkins and Corey Evans demonstrated how every countable ordinal could arise as a game value. This exploration connects infinite chess to fundamental concepts in set theory and computability, illustrating the profound relationships between different areas of mathematics.
1-Page Summary
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Clarifications
- Potential infinity refers to a process that can continue indefinitely, like counting numbers without end. Actual infinity treats an infinite collection as a completed, whole entity, such as the entire set of natural numbers existing at once. Aristotle accepted potential infinity but rejected actual infinity as impossible or contradictory. Modern mathematics, especially set theory, accepts actual infinity as a valid concept.
- Two sets are equinumerous if there is a one-to-one correspondence between their elements. This means each element in one set pairs with exactly one element in the other, with no leftovers. Galileo showed that even infinite sets, like natural numbers and their squares, can be matched this way. This challenges the intuition that a part must be smaller than the whole.
- Some infinite sets can be put into a one-to-one correspondence with each other, meaning they have the same size or cardinality. The set of natural numbers is countably infinite because its elements can be listed in a sequence. The set of real numbers is uncountably infinite, meaning it cannot be listed in a sequence that pairs each real number with a natural number. Cantor's diagonal argument proves that the real numbers have a strictly greater cardinality than the natural numbers.
- The Zermelo-Fraenkel axioms are a set of rules that define how sets behave in mathematics. They provide a rigorous foundation to avoid paradoxes like Russell's paradox. These axioms include principles such as extensionality, pairing, union, and infinity, ensuring sets can be constructed and manipulated consistently. Together with the Axiom of Choice, they form the standard framework called ZFC used in most of modern set theory.
- Hilbert's program aimed to formalize all of mathematics using a finite, complete set of axioms and rules. The goal was to prove these axioms consistent using only finitary methods, ensuring no contradictions. This would provide a secure foundation for all mathematical truths. The program sought to eliminate uncertainty in mathematics by making it fully rigorous and mechanizable.
- Gödel's incompleteness theorems show that in any sufficiently powerful axiomatic system, there are true statements that cannot be proven within the system. They also prove that such a system cannot demonstrate its own consistency. This means no single formal system can capture all mathematical truths or guarantee it is free of contradictions. These results limit the scope of formal mathematical reasoning and highlight inherent incompleteness in mathematics.
- Alfred Tarski's semantic theory of truth defines truth in terms of a formal language's structure and its interpretation, rather than within the language itself. It uses the concept of satisfaction, where a statement is true if it corresponds to the facts or objects in a given model. This approach avoids self-referential paradoxes by distinguishing between the object language (the language being studied) and the metalanguage (the language used to talk about the object language). Thus, truth becomes a property defined externally, not derivable solely from the formal system's internal rules.
- Undecidability means there is no algorithm that can always give a correct yes-or-no answer for certain problems. The halting problem asks if a computer program will stop running or run forever on a given input. Alan Turing proved that no program can solve the halting problem for all possible program-input pairs. This result shows fundamental limits on what computers can decide or compute.
- Surreal numbers are a class of numbers created by recursively defining numbers from simpler numbers, starting from nothing. They include all real numbers, infinite numbers larger than any real, and infinitesimals smaller than any positive real. This construction allows them to represent and extend traditional number systems within a single, coherent framework. Their structure supports arithmetic operations and ordering, making them a rich and flexible number system.
- Surreal numbers form a vast class including infinitesimals and infinite quantities, constructed via a recursive process rather than as points on a continuous line. They do not fit the usual notion of continuity found in real numbers because their construction involves discrete steps and gaps. This "discontinuity" means there is no smooth progression between all surreal numbers as in the real number line. Instead, surreal numbers create a rich, ordered structure with many "jumps" and gaps.
- Infinite chess is a variant of chess played on an unbounded, infinite board with the same pieces and rules as standard chess. A game value, in this context, measures how many moves it takes for a player to force a win, extended into transfinite numbers called ordinals. Countable ordinals are a way to label positions with values that go beyond finite numbers but can still be listed in a sequence like natural numbers. This allows mathematicians to analyze the complexity and length of infinite chess games using set theory concepts.
- Infinite chess extends traditional chess to an unbounded board, allowing infinitely many moves and positions. Set theory provides tools to classify these positions using ordinals, which measure the complexity or "length" of potential game sequences. Computability theory studies which game outcomes or strategies can be effectively determined or predicted by algorithms. Together, they reveal deep links between infinite games, mathematical hierarchies, and algorithmic limits.
Counterarguments
- Aristotle's view on actual infinity being impossible is contested by modern mathematics, which comfortably uses the concept of actual infinity in various forms, such as in the analysis of infinite series or the concept of limit points.
- While Galileo's observation about equinumerous infinite sets was insightful, it was Cantor who rigorously defined and explored the concept of different sizes of infinity, which could be seen as a refinement and extension of Galileo's ideas.
- Cantor's proof that not all infinities are equal was revolutionary, but it also led to the "Cantor's paradox" and the "set of all sets" problem, which challenged the naive set theory and required the development of axiomatic set theory to resolve these issues.
- Zermelo's set theory and the Zermelo-Fraenkel axioms, while foundational, are not without their critics. Some mathematicians and philosophers argue that alternative set theories, such as New Foundations or Quine's set theory, offer different perspectives that could be more intuitive or avoid certain paradoxes.
- Hilbert's program was ambitious, but Gödel's incompleteness theorems showed that it could not be completed as intended. Some argue that this does not diminish the value of searching for more comprehensive axiomatic systems that can still provide deep insights into the nature of mathematics.
- Tarski's semantic theory of truth is one approach among several in the philosophy of mathematics. Some philosophers argue for deflationary theories of truth or other conceptions that do not rely on semantic notions.
- The undecidability of certain computational problems, such as the halting problem, is a fundamental result, but it does not preclude the possibility of solving many practical problems in computation. Some argue that focusing on these limitations overlooks the vast capabilities of computability and algorithmic processes.
- Conway's surreal numbers are a fascinating mathematical construction, but they are not without their critics. Some mathematicians question the practical applications of surreal numbers and their relevance to traditional mathematical analysis.
- The study of infinite chess and its connection to set theory and computability is an interesting niche, but some may argue that it is a mathematical curiosity rather than a field with significant practical applications or deep implications for the broader understanding of mathematics.
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Development and Implications of Set Theory and Infinity
The narrative of infinity has morphed dramatically from ancient philosophy to its profound impact on modern mathematics, fostering discussions around the very foundation of the mathematical reality.
Ideas on Infinity: Ancient to 19th Century Evolution
Aristotle's Distinction Between Potential and Actual Infinity
Our understanding of infinity has vastly improved from the days of Aristotle, who differentiated between potential infinity - the notion that numbers can continue indefinitely, and actual infinity - a quantity he deemed incoherent.
Galileo's Observations on Equinumerosity and Infinity
Galileo stirred the waters with his observations on equinumerosity. Hamkins recounts Galileo's example that the set of all numbers and the set of perfect squares are equinumerous, which suggests an apparent paradox where parts of infinity are as numerous as the whole. Galileo found that line segments of different lengths are equinumerous and that this equinumerosity also applies to circles, regardless of size or perfect squares against all natural numbers, introducing a stark contradiction to Euclid's principle.
Cantor's Discovery: Differences in Infinity Sizes and Their Impact on Math and Philosophy
Cantor revolutionized mathematics by revealing that not all infinities are created equal. He demonstrated that there are larger and smaller infinities, such as the uncountably infinite set of real numbers versus the countable infinity of natural numbers. His work, however, didn't just reshape mathematics; it also caused a theological controversy given infinity's link to the divine. Cantor’s establishment of different sizes of infinity was a precursor to the continuum hypothesis, which pondered whether there existed an infinity between the natural numbers and the real numbers. His contemplation of this hypothesis spanned his lifetime. Cantor's proof demonstrated that every closed set is either countable or equinumerous with the continuum, contributing to his groundbreaking concept of transfinite ordinals.
The Set Theoretic Foundations of Modern Mathematics
Zermelo's Set Theory and Development of Zermelo-Fraenkel Axioms
Zermelo's reaction to the foundational crises and paradoxes sparked by Cantor's work led to the development of Zermelo-Fraenkel set theory. This axiomatic system provided much-needed rigor and sought to put set theory and, by extension, all of mathematics, on stable ground. Zermelo-Fraenkel set theory, including the foundational Axiom of Extensionality and the Axiom of Choice, has since become the unifying language enabling mathematicians to work within a coherent and singular framework.
Zermelo's Original Theory and Subsequent Additions
Zermelo's theory was pressured into existence due to criticism of his proof of the well-ordering principle. Initially, his theory accounted for urelements—objects not considered as sets—but modern set theories typically exclude these based on structuralist philosophies.
Role of the Axiom of Choice and Controversies
The Axiom of Choice, essential to ZFC, allows for the selection of elements from different sets, a concept Hamkins finds natural yet acknowledges led to controversial outcomes, such as the well-ordering of real numbers. While ...
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Development and Implications of Set Theory and Infinity
Additional Materials
Clarifications
- Potential infinity refers to a process that can continue indefinitely, like counting numbers without end. Actual infinity treats infinity as a completed, fixed totality, such as an infinite set considered as a whole. Aristotle accepted potential infinity as meaningful but rejected actual infinity as contradictory or impossible. This distinction influenced how infinity was handled in philosophy and early mathematics.
- Equinumerosity means two sets have a one-to-one correspondence between their elements, showing they have the same size. For infinite sets, this can happen even if one set is a proper subset of the other, unlike finite sets. For example, the natural numbers and their squares can be paired exactly, so they are equinumerous. This challenges intuition because infinite sets can be "as large" as some of their parts.
- Euclid's principle suggests that a larger geometric object should contain more points than a smaller one. However, Galileo observed that the points on a line segment and a circle can be put into a one-to-one correspondence, meaning they have the same cardinality. This challenges the intuitive idea that bigger shapes have more points, revealing that infinite sets can be counterintuitive. It highlights that infinite sets can be "equinumerous" even if one seems larger geometrically.
- Countable infinity means a set's elements can be matched one-to-one with natural numbers, like the set of integers. Uncountable infinity means no such matching is possible, as the set is too large, like the real numbers between 0 and 1. This difference shows some infinities are strictly larger than others. Cantor's diagonal argument proves the uncountability of the real numbers.
- Cantor identified actual infinities in mathematics, which challenged traditional views linking infinity exclusively to God. Some theologians feared this diminished the divine uniqueness of infinity by treating it as a mathematical object. Cantor himself believed his work revealed aspects of the divine mind and sought to reconcile his findings with religious faith. This sparked debates on whether mathematical infinities could coexist with theological doctrines.
- The continuum hypothesis (CH) proposes there is no set size strictly between the size of the natural numbers and the real numbers. It addresses the question of whether an intermediate infinity exists between these two infinite sets. CH's significance lies in its independence from standard set theory axioms, meaning it can neither be proved nor disproved within that framework. This challenges the idea of a single, absolute mathematical truth about infinite sizes.
- A closed set in mathematics is a set that contains all its limit points, meaning it includes the boundary points where sequences within the set converge. Countability refers to a set having elements that can be listed in a sequence like natural numbers, while the continuum represents the size of the real numbers, which is uncountably infinite. Cantor showed that closed sets in the real numbers are either countable or have the size of the continuum, meaning no intermediate size exists for these sets. This result links topology (closed sets) with set theory (countability and cardinality).
- Transfinite ordinals extend the concept of counting beyond finite numbers to describe the order type of well-ordered infinite sets. They provide a way to index positions in infinite sequences, capturing different "sizes" and structures of infinity. Unlike finite ordinals, transfinite ordinals include a smallest infinite ordinal called omega (ω), representing the order type of natural numbers. These ordinals enable rigorous analysis of infinite processes and hierarchies in set theory.
- In the early 20th century, naive set theory led to paradoxes like Russell's paradox, which showed that some sets cannot consistently contain themselves. These paradoxes threatened the logical foundation of mathematics by revealing contradictions within set theory. Zermelo introduced axioms to restrict set formation, preventing such contradictions. His axiomatic system aimed to restore consistency and rigor to set theory.
- The Axiom of Extensionality states that two sets are identical if they have exactly the same elements, ensuring sets are defined solely by their members. The Axiom of Choice allows selecting one element from each set in a collection, even without a specific rule for selection. This axiom enables proofs of existence but can lead to counterintuitive results, like well-ordering any set. Together, they provide foundational rules that shape how sets behave and interact in set theory.
- Urelements are objects in a set theory that are not themselves sets and have no elements. They serve as basic, indivisible entities distinct from sets, allowing a richer ontology. Modern set theory typically excludes urelements to maintain a purely structural view where everything is built from sets alone. This exclusion simplifies the theory and aligns with the structuralist philosophy that mathematics studies abstract structures rather than objects with intrinsic identity.
- The Axiom of Choice allows selecting elements from infinitely many sets simultaneously, even without a specific rule. This leads to the well-ordering theorem, which states every set, including the real numbers, can be arranged in a sequence where every subset has a least element. Such a well-ordering of the real numbers is non-constructive and counterintuitive, as no explicit example can be given. This challenges traditional notions of order and measure, causing controversy among mathematicians.
- The constructible universe, denoted by L, is a class of sets built ...
Counterarguments
- Aristotle's view on actual infinity being incoherent is challenged by the development of modern mathematics, where actual infinity is a standard concept, particularly in set theory and calculus.
- Galileo's paradox of equinumerosity can be reconciled within the framework of set theory, which accepts that infinite sets can be put into a one-to-one correspondence with proper subsets of themselves without contradiction.
- Cantor's work on different sizes of infinity, while groundbreaking, is not without its critics. Some mathematicians, such as intuitionists and constructivists, reject the existence of actual infinities and thus Cantor's hierarchy of infinities.
- The theological controversy sparked by Cantor's work on infinity is not universally accepted as a valid criticism of his mathematical contributions, as many argue that mathematical concepts should be evaluated independently of religious or theological implications.
- The Zermelo-Fraenkel set theory, including the Axiom of Choice, is not universally accepted. Some mathematicians advocate for alternative set theories, such as New Foundations or Constructive Set Theory, which do not include the Axiom of Choice.
- The Axiom of Choice itself is controversial, with some mathematicians arguing that it leads to non-constructive and counterintuitive results, such as the Banach-Tarski paradox.
- Gödel's and Cohen's work on the independence of the Continuum Hypothesis and the Axiom of Choice from ZFC does not resolve the philosophical deba ...
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Relationships Between Truth, Provability, and Computability in Mathematics
The intricate relationships between truth, provability, and computability in mathematics are examined through the lenses of historical figures in the field, recent conversations, and theoretical implications.
Hilbert's Program and the Formalist Approach to Foundations
David Hilbert introduced a program as an ambitious goal towards providing a solid, formal foundation for all mathematics, which involved proving the consistency of a comprehensive axiomatic system.
Hilbert's Aim to Prove a Comprehensive Axiomatic System's Consistency
Hilbert supported the use of set theory as a foundation for mathematics and aimed to show that a powerful formal theory, such as set theory, could answer all mathematical questions and that its consistency could be proven within a purely finitary theory. He anticipated the development of a theorem enumeration machine capable of answering all mathematical questions, making the investigation a matter of rote computation. Hilbert viewed proofs as syntactic sequences of symbols following the rules of logical reasoning and was captivated by the power and unifying aspect of set theory in mathematics.
Gödel's Incompleteness Theorems and Formalist Limitations
However, Gödel’s incompleteness theorems provided a decisive refutation of Hilbert’s aspirations by showing no such strong and consistent axiomatic system exists that can prove all mathematical truths or guarantee its own consistency through finitary means. An inconsistent theory, Hamkins points out, could also "prove" its own consistency, undermining the reliability of self-verifying consistency. This revelation about the unattainability of Hilbert's program led to a broader understanding of the separation between truth and provability in mathematics.
The Distinction Between Mathematical Truth and Provability
The distinction between truth and provability transcends the concept of mathematical proof as a formal sequence of statements.
Tarski's Account of the Semantic Concept of Truth
Joel David Hamkins discusses Tarski's semantic theory of truth, which separates the concept of truth from the formal structure in which it is discussed. Tarski's discrotational take on truth, formalized in mathematics, allows for the definition of truth within any mathematical structure, but its ambiguity is highlighted unless specified within a particular structure, like the standard model of arithmetic.
Implications of Incompleteness Theorems For Provability
Gödel's incompleteness theorems imply that not all true statements in a sufficiently rich mathematical system can be proven, and no system can establish its own consistency. Theorems illustrate the limitations of provability, suggesting that there are true statements about the-arithmetic not provable within any given system, and that if a theory were complete and consistent, it could determine the outcome of any instance of the halting problem—an impossible feat due to the undecidability of the halting problem.
The Undecidability of Fundamental Computational Problems
Certain computati ...
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Relationships Between Truth, Provability, and Computability in Mathematics
Additional Materials
Clarifications
- Hilbert's program aimed to formalize all mathematics using a fixed set of axioms and rules. The goal was to prove that these axioms are free from contradictions, ensuring the system's reliability. This proof of consistency was intended to be carried out using only simple, finitary methods. Success would mean all mathematical truths could be derived mechanically within this system.
- A theorem enumeration machine is a hypothetical device that systematically generates all possible proofs in a formal system. It works by listing every sequence of symbols that could be a valid proof, eventually producing every provable theorem. This concept illustrates the idea that mathematical truths could be discovered through mechanical, algorithmic processes. Hilbert envisioned such a machine as a way to automate and complete mathematical reasoning.
- A "finitary theory" involves only finite, concrete mathematical objects and methods, avoiding infinite processes or completed infinities. "Finitary means" refer to proof techniques and reasoning that rely solely on finite, explicit steps understandable without invoking infinite totalities. Hilbert aimed to prove consistency using such finite, constructive methods to ensure absolute certainty. This contrasts with infinitary methods that allow reasoning about infinite sets or processes.
- Gödel’s incompleteness theorems show that in any sufficiently powerful formal system, there are true statements that cannot be proven within that system. They also prove that such a system cannot demonstrate its own consistency without external assumptions. This means no single formal system can capture all mathematical truths or guarantee its own reliability. These results reveal fundamental limits on what can be achieved through formal axiomatic methods.
- An inconsistent theory contains contradictions, meaning it can prove any statement, including false ones. Because it can prove everything, it can also "prove" its own consistency, even though it is actually inconsistent. This makes such a proof meaningless and unreliable. Thus, self-verifying consistency is only trustworthy in consistent systems.
- Mathematical truth refers to statements that accurately describe a mathematical structure, independent of any proof. Provability means that a statement can be derived using a formal system's rules and axioms. Some true statements cannot be proven within a given system due to inherent limitations of that system. This gap arises because formal systems are constrained by their axioms and inference rules, which may not capture all truths.
- Tarski's semantic theory of truth defines truth in terms of a formal language and a specific interpretation or model of that language. It uses the concept of satisfaction, where a statement is true if it accurately describes the elements and relations in the model. This approach avoids paradoxes by separating the language used to talk about truth from the language in which statements are made. Defining truth within a mathematical structure means specifying which statements correspond to actual facts about that structure under its interpretation.
- The standard model of arithmetic is the usual set of natural numbers (0, 1, 2, 3, ...) with their standard operations and order. It is the intended interpretation of arithmetic statements, where numbers behave as commonly understood. Non-standard models include "extra" elements beyond these natural numbers, which satisfy the same axioms but have unusual properties. Tarski's semantic truth is often defined relative to this standard model to avoid ambiguity.
- The halting problem asks whether there is a universal method to determine if any computer program will eventually stop or run forever. Alan Turing proved in 1936 that no such method can exist for all possible programs, making the problem undecidable. This is because if such a method existed, it could be used to create logical contradictions by analyzing programs that reference their own behavior. Thus, the halting problem reveals fundamental limits on what can be computed or predicted by algorithms.
- Gödel’s incompleteness theorems show that in any sufficiently powerful formal sy ...
Counterarguments
- Hilbert's program, while ambitious, may have been overly optimistic about the capabilities of formal systems and their ability to encapsulate all mathematical truths.
- The belief in the unifying power of set theory has been challenged by alternative foundational approaches, such as category theory, which some argue provides a more natural language for mathematics.
- Gödel's incompleteness theorems, while widely accepted, do not necessarily imply that all forms of mathematical reasoning or all mathematical questions are subject to the same limitations; there may be other forms of reasoning or areas of mathematics where these limitations do not apply.
- Tarski's semantic theory of truth, while influential, is not the only perspective on truth in mathematics; other philosophers and mathematicians may argue for different notions of truth that are not as closely tied to formal semantics.
- The practical effectiveness of probabilistic approaches to computationally hard problems does not negate the theoretical importance of undecidability; some might argue that understanding the theoretical limits remains crucial for the foundations of mathematics and computer science.
- The concept of a "black hole" in the decidability landscape, where almost ...
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Novel Math Structures: Surreal Numbers, Infinite Chess
Joel David Hamkins introduces the surreal number system—created by John Conway—as a beautiful mathematical system that unifies various other number systems and can construct all numbers. Fridman plans to further discuss the role and significance of surreal numbers and infinite chess.
The Surreal Number System and Its Unifying Properties
Conway's Construction of the Surreal Numbers From "Nothing"
Hamkins explains that surreal numbers are constructed from "nothing" using a transfinite sequence of stages, applying a specific rule that involves dividing existing numbers into two sets—a left set and a right set, ensuring each number in the left set is less than those in the right set. A new number is created to fit the gap between these two sets. The initial stage uses two empty sets to create zero, and the process continues to produce an ever-growing set of surreal numbers.
Surreal Numbers Encompassing and Extending Real and Other Number Systems
The surreal number system unifies numerous number systems, including natural numbers, integers, rationals, reals, ordinals, and infinitesimals. Notably, dyadic rationals are born at finite stages, while real numbers and additional numbers are birthed on day Omega.
Discontinuity of Surreal Numbers and Implications For Analysis
Surreal numbers are fundamentally discontinuous, lacking least upper bounds and convergent sequences. This makes conventional calculus methods based on limits and convergence unworkable. However, calculus can still be done with surreal numbers using non-standard methods based on infinitesimals.
The Mathematics of Infinite Chess
Infinite Chess Rules on an Unbounded Board
Hamkins explains that infinite chess is played on a chessboard extended infinitely in all directions. Pieces move according to standard chess patterns, but knights, rooks, bishops, and pawns, moving upwards or downwards depending on their color, move as far as the ...
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Novel Math Structures: Surreal Numbers, Infinite Chess
Additional Materials
Clarifications
- A transfinite sequence extends beyond finite steps into infinite ordinal numbers, indexing stages not just by natural numbers but by infinite "positions." "Day Omega" refers to the first infinite ordinal stage, representing the limit of all finite stages combined. At this stage, new surreal numbers emerge that cannot be formed at any finite step. This concept allows the construction process to continue through infinite steps, creating a richer number system.
- In Conway's construction, each surreal number is defined by two sets: a left set and a right set, containing previously created surreal numbers. Every number in the left set must be strictly less than every number in the right set. The new surreal number represents a value that lies between all numbers in the left set and all numbers in the right set. This rule ensures a well-defined ordering and allows the creation of all surreal numbers through successive stages.
- Dyadic rationals are fractions where the denominator is a power of two, like 1/2, 3/8, or 7/16. They appear at finite stages because the surreal number construction builds numbers step-by-step, and these fractions can be formed by repeatedly splitting intervals in halves. Each finite stage corresponds to a finite number of such splits, producing dyadic rationals. This makes dyadic rationals the simplest non-integer surreal numbers created early in the process.
- Ordinals extend natural numbers to describe positions in well-ordered sequences, including infinite ones, capturing the idea of "order type." Infinitesimals are numbers smaller than any positive real number but greater than zero, representing infinitely small quantities. In surreal numbers, ordinals appear as numbers representing infinite positions, while infinitesimals fill gaps smaller than any positive real number. This allows surreal numbers to include both infinitely large and infinitely small values in a unified system.
- Surreal numbers form a proper class, not a set, so they do not fit into the usual framework of real analysis. Their ordering is a proper class linear order, which means there is no smallest upper bound for every bounded subset. Convergent sequences rely on limits within a complete metric space, but surreal numbers lack such completeness. Instead, their structure allows infinitesimals and infinite numbers that break standard notions of limit and continuity.
- Non-standard analysis uses infinitesimals—numbers smaller than any positive real number but greater than zero—to rigorously define derivatives and integrals. It extends the real number system to include these infinitesimals, allowing calculus concepts to be expressed without limits. This approach mirrors intuitive ideas from the origins of calculus but with a solid logical foundation. In the surreal number system, infinitesimals enable similar non-standard calculus despite the lack of traditional limits.
- Infinite chess uses the same pieces and movement rules as standard chess but on a board that extends endlessly in all directions. There is no edge or boundary, so pieces can move indefinitely if unobstructed. This removes traditional constraints like checkmate by cornering, allowing for complex strategies involving infinite sequences of moves. The game explores new mathematical concepts, such as infinite game values and connections to set theory.
- Ordinal values in infinite chess measure the complexity of a position ...
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