Draft:Surreal number

This draft space is currently being used by User:Guy vandegrift. I will gladly make it a subpage if you want to create a draft here... Or, you can go to Surreal number and create subpage for your draft.

Surreal number (subpages/draftspace) · Draft subspace

See bottom of #An very brief overview of Surreal Numbers. It explores circular reasoning.

#MitCourse explains the 3 1/2 4 1/2 paradox. See p 28 of https://web.mit.edu/sp.268/www/2010/surrealSlides.pdf

tO DO: https://www.sciencedirect.com/science/article/pii/S2352711018302152

Wikipedia

In set theory, an ordinal number, or ordinal, is a generalization of ordinal numerals (first, second, nth, etc.) aimed to extend enumeration to infinite sets. Induction rule There is a generation S0 = { 0 }, in which 0 consists of the single form { | }. Given any ordinal number n, the generation Sn is the set of all surreal numbers that are generated by the construction rule from subsets of ${\textstyle \bigcup _{i<n}S_{i}}$. The base case is actually a special case of the induction rule, with 0 taken as a label for the "least ordinal". Since there exists no Si with i < 0, the expression ${\textstyle \bigcup _{i<0}S_{i}}$ is the empty set; the only subset of the empty set is the empty set, and therefore S0 consists of a single surreal form { | } lying in a single equivalence class 0. For every finite ordinal number n, Sn is well-ordered by the ordering induced by the comparison rule on the surreal numbers. The first iteration of the induction rule produces the three numeric forms { | 0 } < { | } < { 0 | } (the form { 0 | 0 } is non-numeric because 0 ≤ 0). The equivalence class containing { 0 | } is labeled 1 and the equivalence class containing { | 0 } is labeled −1. These three labels have a special significance in the axioms that define a ring; they are the additive identity (0), the multiplicative identity (1), and the additive inverse of 1 (−1). The arithmetic operations defined below are consistent with these labels. For every i < n, since every valid form in Si is also a valid form in Sn, all of the numbers in Si also appear in Sn (as supersets of their representation in Si). (The set union expression appears in our construction rule, rather than the simpler form Sn−1, so that the definition also makes sense when n is a limit ordinal.) Numbers in Sn that are a superset of some number in Si are said to have been inherited from generation i. The smallest value of α for which a given surreal number appears in Sα is called its birthday. For example, the birthday of 0 is 0, and the birthday of −1 is 1. A second iteration of the construction rule yields the following ordering of equivalence classes: { | −1 } = { | −1, 0 } = { | −1, 1 } = { | −1, 0, 1 } < { | 0 } = { | 0, 1 } < { −1 | 0 } = { −1 | 0, 1 } < { | } = { −1 | } = { | 1 } = { −1 | 1 } < { 0 | 1 } = { −1, 0 | 1 } < { 0 | } = { −1, 0 | } < { 1 | } = { 0, 1 | } = { −1, 1 | } = { −1, 0, 1 | } Comparison of these equivalence classes is consistent, irrespective of the choice of form. Three observations follow: S2 contains four new surreal numbers. Two contain extremal forms: { | −1, 0, 1 } contains all numbers from previous generations in its right set, and { −1, 0, 1 | } contains all numbers from previous generations in its left set. The others have a form that partitions all numbers from previous generations into two non-empty sets. Every surreal number x that existed in the previous "generation" exists also in this generation, and includes at least one new form: a partition of all numbers other than x from previous generations into a left set (all numbers less than x) and a right set (all numbers greater than x). The equivalence class of a number depends only on the maximal element of its left set and the minimal element of the right set. The informal interpretations of { 1 | } and { | −1 } are "the number just after 1" and "the number just before −1" respectively; their equivalence classes are labeled 2 and −2. The informal interpretations of { 0 | 1 } and { −1 | 0 } are "the number halfway between 0 and 1" and "the number halfway between −1 and 0" respectively; their equivalence classes are labeled 1/2 and −1/2. These labels will also be justified by the rules for surreal addition and multiplication below. The equivalence classes at each stage n of induction may be characterized by their n-complete forms (each containing as many elements as possible of previous generations in its left and right sets). Either this complete form contains every number from previous generations in its left or right set, in which case this is the first generation in which this number occurs; or it contains all numbers from previous generations but one, in which case it is a new form of this one number. We retain the labels from the previous generation for these "old" numbers, and write the ordering above using the old and new labels: −2 < −1 < −1/2 < 0 < 1/2 < 1 < 2. The third observation extends to all surreal numbers with finite left and right sets. (For infinite left or right sets, this is valid in an altered form, since infinite sets might not contain a maximal or minimal element.) The number { 1, 2 | 5, 8 } is therefore equivalent to { 2 | 5 }; one can establish that these are forms of 3 by using the birthday property, which is a consequence of the rules above. Birthday property. A form x = { L | R } occurring in generation n represents a number inherited from an earlier generation i < n if and only if there is some number in Si that is greater than all elements of L and less than all elements of the R. (In other words, if L and R are already separated by a number created at an earlier stage, then x does not represent a new number but one already constructed.) If x represents a number from any generation earlier than n, there is a least such generation i, and exactly one number c with this least i as its birthday that lies between L and R; x is a form of this c. In other words, it lies in the equivalence class in Sn that is a superset of the representation of c in generation. Taken from w:Surreal number

My summary:

When discussing the surreals, Arabic numerals ( ±0 ±1, ±1/2, ±2, ...) might represent two entirely different entities, namely (1) actual numbers and (2) the surreal numbers.

Even crows understand "addition" in the sense that "★" plus "★★" equals "★★★", but do not associate "1" with "★" or "2" with "★★". And do not associate the crow's understanding of "plus" with the binary operation "+". formally associated with "numbers" ( ±0 ±1, ±1/2, ±2, ...), we include two Greek letters (ω, ε) to this list of what one might call the "surreals".

• Binary operations (+, −, ÷, ×) will have new definitions for the surreals. While is true that, x+y=y+x, among the surreals, it is necessary to prove commutativity for + and ×.
• The relational operations (=, >, <, ≤, ≥) obey the usual rules when acting as surreal entities. As with ordinary numbers, the same surreal can be expressed in many forms (11/2=3/2=6/4), except the situation is far more bewildering for the surreals. This is because the surreals are defined as a pair of entities that superficially represent sets. However these sets are created in such a way that Russel's paradox cannot be excluded.

The symbols L and R are used to define a pair of sets called the "left" and "right" sets, respectively.

Both sets, called the { L | R } represents some number intermediate in value between members of L and R.

After infinite subsets become available, the reals may be defined. Also { 0, 1, 2, 3, ... | } = ω and { 0 | 1, 1/2, 1/4, 1/8, ... } = ε.

A form is a pair of sets of surreal numbers, called its left set and its right set. A form with left set L and right set R is written { L | R }. When L and R are given as lists of elements, the braces around them are omitted. Either or both of the left and right set may be the empty set.

STOP HERE Numeric forms and their equivalence classes

Construction rule

A form { L | R } is numeric if the intersection of L and R is the empty set and each element of R is greater than every element of L, according to the order relation ≤ given by the comparison rule below.

Equivalence rule

Two numeric forms x and y are forms of the same number (lie in the same equivalence class) if and only if both x ≤ y and y ≤ x.

??? An ordering relationship must be antisymmetric, i.e., it must have the property that x = y (i. e., x ≤ y and y ≤ x are both true) only when x and y are the same object. This is not the case for surreal number forms, but is true by construction for surreal numbers (equivalence classes).

The equivalence class containing { | } is labeled 0; in other words, { | } is a form of the surreal number 0.

Order

Given numeric forms x = { X_L | X_R } and y = { Y_L | Y_R }, x ≤ y if and only if both:

• There is no x_L ∈ X_L such that y ≤ x_L. That is, every element in the left part of x is strictly smaller than y.
• There is no y_R ∈ Y_R such that y_R ≤ x. That is, every element in the right part of y is strictly larger than x.

Induction rule

• There is a generation S₀ = { 0 }, in which 0 consists of the single form { | }.
• Given any ordinal number n, the generation S_n is the set of all surreal numbers that are generated by the construction rule from subsets of ${\textstyle \bigcup _{i<n}S_{i}}$.

The base case is actually a special case of the induction rule, with 0 taken as a label for the "least ordinal". Since there exists no S_i with i < 0, the expression ${\textstyle \bigcup _{i<0}S_{i}}$ is the empty set; the only subset of the empty set is the empty set, and therefore S₀ consists of a single surreal form { | } lying in a single equivalence class 0.

For every finite ordinal number n, S_n is well-ordered by the ordering induced by the comparison rule on the surreal numbers.

For every i < n, since every valid form in S_i is also a valid form in S_n, all of the numbers in S_i also appear in S_n (as supersets of their representation in S_i). (The set union expression appears in our construction rule, rather than the simpler form S_n−1, so that the definition also makes sense when n is a limit ordinal.) Numbers in S_n that are a superset of some number in S_i are said to have been inherited from generation i. The smallest value of α for which a given surreal number appears in S_α is called its birthday. For example, the birthday of 0 is 0, and the birthday of −1 is 1.

A second iteration of the construction rule yields the following ordering of equivalence classes:

{ | −1 } = { | −1, 0 } = { | −1, 1 } = { | −1, 0, 1 }

< { | 0 } = { | 0, 1 }

< { −1 | 0 } = { −1 | 0, 1 }

< { | } = { −1 | } = { | 1 } = { −1 | 1 }

< { 0 | 1 } = { −1, 0 | 1 }

< { 0 | } = { −1, 0 | }

< { 1 | } = { 0, 1 | } = { −1, 1 | } = { −1, 0, 1 | }

Comparison of these equivalence classes is consistent, irrespective of the choice of form. Three observations follow:

1. S₂ contains four new surreal numbers. Two contain extremal forms: { | −1, 0, 1 } contains all numbers from previous generations in its right set, and { −1, 0, 1 | } contains all numbers from previous generations in its left set. The others have a form that partitions all numbers from previous generations into two non-empty sets.
2. Every surreal number x that existed in the previous "generation" exists also in this generation, and includes at least one new form: a partition of all numbers other than x from previous generations into a left set (all numbers less than x) and a right set (all numbers greater than x).
3. The equivalence class of a number depends only on the maximal element of its left set and the minimal element of the right set.

The informal interpretations of { 1 | } and { | −1 } are "the number just after 1" and "the number just before −1" respectively; their equivalence classes are labeled 2 and −2. The informal interpretations of { 0 | 1 } and { −1 | 0 } are "the number halfway between 0 and 1" and "the number halfway between −1 and 0" respectively; their equivalence classes are labeled 1/2 and −1/2. These labels will also be justified by the rules for surreal addition and multiplication below.

The equivalence classes at each stage n of induction may be characterized by their n-complete forms (each containing as many elements as possible of previous generations in its left and right sets). Either this complete form contains every number from previous generations in its left or right set, in which case this is the first generation in which this number occurs; or it contains all numbers from previous generations but one, in which case it is a new form of this one number. We retain the labels from the previous generation for these "old" numbers, and write the ordering above using the old and new labels:

−2 < −1 < −1/2 < 0 < 1/2 < 1 < 2.

The third observation extends to all surreal numbers with finite left and right sets. (For infinite left or right sets, this is valid in an altered form, since infinite sets might not contain a maximal or minimal element.) The number { 1, 2 | 5, 8 } is therefore equivalent to { 2 | 5 }; one can establish that these are forms of 3 by using the birthday property, which is a consequence of the rules above.

Birthday property

A form x = { L | R } occurring in generation n represents a number inherited from an earlier generation i < n if and only if there is some number in S_i that is greater than all elements of L and less than all elements of the R. (In other words, if L and R are already separated by a number created at an earlier stage, then x does not represent a new number but one already constructed.) If x represents a number from any generation earlier than n, there is a least such generation i, and exactly one number c with this least i as its birthday that lies between L and R; x is a form of this c. In other words, it lies in the equivalence class in S_n that is a superset of the representation of c in generation i.

Arithmetic

The addition, negation (additive inverse), and multiplication of surreal number forms x = { X_L | X_R } and y = { Y_L | Y_R } are defined by three recursive formulas.

Negation

Negation of a given number x = { X_L | X_R } is defined by

$-x=-\{X_{L}\mid X_{R}\}=\{-X_{R}\mid -X_{L}\},$

where the negation of a set S of numbers is given by the set of the negated elements of S:

$-S=\{-s:s\in S\}.$

This formula involves the negation of the surreal numbers appearing in the left and right sets of x, which is to be understood as the result of choosing a form of the number, evaluating the negation of this form, and taking the equivalence class of the resulting form. This only makes sense if the result is the same, irrespective of the choice of form of the operand. This can be proved inductively using the fact that the numbers occurring in X_L and X_R are drawn from generations earlier than that in which the form x first occurs, and observing the special case:

$-0=-\{{}\mid {}\}=\{{}\mid {}\}=0.$

Addition

The definition of addition is also a recursive formula:

$x+y=\{X_{L}\mid X_{R}\}+\{Y_{L}\mid Y_{R}\}=\{X_{L}+y,x+Y_{L}\mid X_{R}+y,x+Y_{R}\},$

where

$X+y=\{x+y:x\in X\},x+Y=\{x+y:y\in Y\}$.

This formula involves sums of one of the original operands and a surreal number drawn from the left or right set of the other. It can be proved inductively with the special cases:

$0+0=\{{}\mid {}\}+\{{}\mid {}\}=\{{}\mid {}\}=0$

$x+0=x+\{{}\mid {}\}=\{X_{L}+0\mid X_{R}+0\}=\{X_{L}\mid X_{R}\}=x$

$0+y=\{{}\mid {}\}+y=\{0+Y_{L}\mid 0+Y_{R}\}=\{Y_{L}\mid Y_{R}\}=y$

For example:

1/2 + 1/2 = { 0 | 1 } + { 0 | 1 } = { 1/2 | 3/2 },

which by the birthday property is a form of 1. This justifies the label used in the previous section.

Multiplication

Multiplication can be defined recursively as well, beginning from the special cases involving 0, the multiplicative identity 1, and its additive inverse −1:

${\begin{aligned}xy&=\{X_{L}\mid X_{R}\}\{Y_{L}\mid Y_{R}\}\\&=\left\{X_{L}y+xY_{L}-X_{L}Y_{L},X_{R}y+xY_{R}-X_{R}Y_{R}\mid X_{L}y+xY_{R}-X_{L}Y_{R},xY_{L}+X_{R}y-X_{R}Y_{L}\right\}\\\end{aligned}}$

The formula contains arithmetic expressions involving the operands and their left and right sets, such as the expression $X_{R}y+xY_{R}-X_{R}Y_{R}$ that appears in the left set of the product of x and y. This is understood as the set of numbers generated by picking all possible combinations of members of $X_{R}$ and $Y_{R}$, and substituting them into the expression.

For example, to show that the square of 1/2 is 1/4:

1/2 ⋅ 1/2 = { 0 | 1 } ⋅ { 0 | 1 } = { 0 | 1/2 } = 1/4.

Division

The definition of division is done in terms of the reciprocal and multiplication:

${\frac {x}{y}}=x\cdot {\frac {1}{y}}$

where

${\frac {1}{y}}=\left\{\left.0,{\frac {1+(y_{R}-y)\left({\frac {1}{y}}\right)_{L}}{y_{R}}},{\frac {1+\left(y_{L}-y\right)\left({\frac {1}{y}}\right)_{R}}{y_{L}}}\,\,\right|\,\,{\frac {1+(y_{L}-y)\left({\frac {1}{y}}\right)_{L}}{y_{L}}},{\frac {1+(y_{R}-y)\left({\frac {1}{y}}\right)_{R}}{y_{R}}}\right\}$

for positive y. Only positive y_L are permitted in the formula, with any nonpositive terms being ignored (and y_R are always positive). This formula involves not only recursion in terms of being able to divide by numbers from the left and right sets of y, but also recursion in that the members of the left and right sets of 1/y itself. 0 is always a member of the left set of 1/y, and that can be used to find more terms in a recursive fashion. For example, if y = 3 = { 2 | }, then we know a left term of 1/3 will be 0. This in turn means 1 + (2 − 3)0/2 = 1/2 is a right term. This means

${\frac {1+(2-3)\left({\frac {1}{2}}\right)}{2}}={\frac {1}{4}}$

is a left term. This means

${\frac {1+(2-3)\left({\frac {1}{4}}\right)}{2}}={\frac {3}{8}}$

will be a right term. Continuing, this gives

${\frac {1}{3}}=\left\{\left.0,{\frac {1}{4}},{\frac {5}{16}},\ldots \,\right|\,{\frac {1}{2}},{\frac {3}{8}},\ldots \right\}$

For negative y, 1/y is given by

${\frac {1}{y}}=-\left({\frac {1}{-y}}\right)$

If y = 0, then 1/y is undefined.

Consistency

It can be shown that the definitions of negation, addition and multiplication are consistent, in the sense that:

• Addition and negation are defined recursively in terms of "simpler" addition and negation steps, so that operations on numbers with birthday n will eventually be expressed entirely in terms of operations on numbers with birthdays less than n;
• Multiplication is defined recursively in terms of additions, negations, and "simpler" multiplication steps, so that the product of numbers with birthday n will eventually be expressed entirely in terms of sums and differences of products of numbers with birthdays less than n;
• As long as the operands are well-defined surreal number forms (each element of the left set is less than each element of the right set), the results are again well-defined surreal number forms;
• The operations can be extended to numbers (equivalence classes of forms): the result of negating x or adding or multiplying x and y will represent the same number regardless of the choice of form of x and y; and
• These operations obey the associativity, commutativity, additive inverse, and distributivity axioms in the definition of a field, with additive identity 0 = { | } and multiplicative identity 1 = { 0 | }.

With these rules one can now verify that the numbers found in the first few generations were properly labeled. The construction rule is repeated to obtain more generations of surreals:

S₀ = { 0 }

S₁ = { −1 < 0 < 1 }

S₂ = { −2 < −1 < −1/2 < 0 < 1/2 < 1 < 2}

S₃ = { −3 < −2 < −3/2 < −1 < −3/4 < −1/2 < −1/4 < 0 < 1/4 < 1/2 < 3/4 < 1 < 3/2 < 2 < 3 }

S₄ = { −4 < −3 < ... < −1/8 < 0 < 1/8 < 1/4 < 3/8 < 1/2 < 5/8 < 3/4 < 7/8 < 1 < 5/4 < 3/2 < 7/4 < 2 < 5/2 < 3 < 4 }

Arithmetic closure

For each natural number (finite ordinal) n, all numbers generated in S_n are dyadic fractions, i.e., can be written as an irreducible fraction a/2^b, where a and b are integers and 0 ≤ b < n.

The set of all surreal numbers that are generated in some S_n for finite n may be denoted as ${\textstyle S_{*}=\bigcup _{n\in N}S_{n}}$. One may form the three classes

${\begin{aligned}S_{0}&=\{0\}\\S_{+}&=\{x\in S_{*}:x>0\}\\S_{-}&=\{x\in S_{*}:x<0\}\end{aligned}}$

of which S_∗ is the union. No individual S_n is closed under addition and multiplication (except S₀), but S_∗ is; it is the subring of the rationals consisting of all dyadic fractions.

There are infinite ordinal numbers β for which the set of surreal numbers with birthday less than β is closed under the different arithmetic operations. For any ordinal α, the set of surreal numbers with birthday less than β = ω^α (using powers of ω) is closed under addition and forms a group; for birthday less than ω^ωα it is closed under multiplication and forms a ring; and for birthday less than an (ordinal) epsilon number ε_α it is closed under multiplicative inverse and forms a field. The latter sets are also closed under the exponential function as defined by Kruskal and Gonshor.

However, it is always possible to construct a surreal number that is greater than any member of a set of surreals (by including the set on the left side of the constructor) and thus the collection of surreal numbers is a proper class. With their ordering and algebraic operations they constitute an ordered field, with the caveat that they do not form a set. In fact it is the biggest ordered field, in that every ordered field is a subfield of the surreal numbers. The class of all surreal numbers is denoted by the symbol $\mathbb {No}$.

Infinity

Define S_ω as the set of all surreal numbers generated by the construction rule from subsets of S_∗. (This is the same inductive step as before, since the ordinal number ω is the smallest ordinal that is larger than all natural numbers; however, the set union appearing in the inductive step is now an infinite union of finite sets, and so this step can only be performed in a set theory that allows such a union.) A unique infinitely large positive number occurs in S_ω:

$\omega =\{S_{*}\mid {}\}=\{1,2,3,4,\ldots \mid {}\}.$

S_ω also contains objects that can be identified as the rational numbers. For example, the ω-complete form of the fraction 1/3 is given by:

${\tfrac {1}{3}}=\{y\in S_{*}:3y<1\mid y\in S_{*}:3y>1\}.$

The product of this form of 1/3 with any form of 3 is a form whose left set contains only numbers less than 1 and whose right set contains only numbers greater than 1; the birthday property implies that this product is a form of 1.

Not only do all the rest of the rational numbers appear in S_ω; the remaining finite real numbers do too. For example,

$\pi =\left\{3,{\tfrac {25}{8}},{\tfrac {201}{64}},\ldots \mid 4,{\tfrac {7}{2}},{\tfrac {13}{4}},{\tfrac {51}{16}},\ldots \right\}.$

The only infinities in S_ω are ω and −ω; but there are other non-real numbers in S_ω among the reals. Consider the smallest positive number in S_ω:

$\varepsilon =\{S_{-}\cup S_{0}\mid S_{+}\}=\left\{0\mid 1,{\tfrac {1}{2}},{\tfrac {1}{4}},{\tfrac {1}{8}},\ldots \right\}=\{0\mid y\in S_{*}:y>0\}$.

This number is larger than zero but less than all positive dyadic fractions. It is therefore an infinitesimal number, often labeled ε. The ω-complete form of ε (respectively −ε) is the same as the ω-complete form of 0, except that 0 is included in the left (respectively right) set. The only "pure" infinitesimals in S_ω are ε and its additive inverse −ε; adding them to any dyadic fraction y produces the numbers y ± ε, which also lie in S_ω.

One can determine the relationship between ω and ε by multiplying particular forms of them to obtain:

ω · ε = { ε · S₊ | ω · S₊ + S_∗ + ε · S_∗ }.

This expression is only well-defined in a set theory which permits transfinite induction up to S_ω². In such a system, one can demonstrate that all the elements of the left set of ωS_ω·S_ωε are positive infinitesimals and all the elements of the right set are positive infinities, and therefore ωS_ω·S_ωε is the oldest positive finite number, 1. Consequently, 1/ε = ω. Some authors systematically use ω⁻¹ in place of the symbol ε.

Contents of S_ω

Given any x = { L | R } in S_ω, exactly one of the following is true:

• L and R are both empty, in which case x = 0;
• R is empty and some integer n ≥ 0 is greater than every element of L, in which case x equals the smallest such integer n;
• R is empty and no integer n is greater than every element of L, in which case x equals +ω;
• L is empty and some integer n ≤ 0 is less than every element of R, in which case x equals the largest such integer n;
• L is empty and no integer n is less than every element of R, in which case x equals −ω;
• L and R are both non-empty, and:
  • Some dyadic fraction y is "strictly between" L and R (greater than all elements of L and less than all elements of R), in which case x equals the oldest such dyadic fraction y;
  • No dyadic fraction y lies strictly between L and R, but some dyadic fraction $y\in L$ is greater than or equal to all elements of L and less than all elements of R, in which case x equals y + ε;
  • No dyadic fraction y lies strictly between L and R, but some dyadic fraction $y\in R$ is greater than all elements of L and less than or equal to all elements of R, in which case x equals y − ε;
  • Every dyadic fraction is either greater than some element of R or less than some element of L, in which case x is some real number that has no representation as a dyadic fraction.

S_ω is not an algebraic field, because it is not closed under arithmetic operations; consider ω+1, whose form

$\omega +1=\{1,2,3,4,...\mid {}\}+\{0\mid {}\}=\{1,2,3,4,\ldots ,\omega \mid {}\}$

does not lie in any number in S_ω. The maximal subset of S_ω that is closed under (finite series of) arithmetic operations is the field of real numbers, obtained by leaving out the infinities ±ω, the infinitesimals ±ε, and the infinitesimal neighbors y ± ε of each nonzero dyadic fraction y.

This construction of the real numbers differs from the Dedekind cuts of standard analysis in that it starts from dyadic fractions rather than general rationals and naturally identifies each dyadic fraction in S_ω with its forms in previous generations. (The ω-complete forms of real elements of S_ω are in one-to-one correspondence with the reals obtained by Dedekind cuts, under the proviso that Dedekind reals corresponding to rational numbers are represented by the form in which the cut point is omitted from both left and right sets.) The rationals are not an identifiable stage in the surreal construction; they are merely the subset Q of S_ω containing all elements x such that x b = a for some a and some nonzero b, both drawn from S_∗. By demonstrating that Q is closed under individual repetitions of the surreal arithmetic operations, one can show that it is a field; and by showing that every element of Q is reachable from S_∗ by a finite series (no longer than two, actually) of arithmetic operations including multiplicative inversion, one can show that Q is strictly smaller than the subset of S_ω identified with the reals.

The set S_ω has the same cardinality as the real numbers R. This can be demonstrated by exhibiting surjective mappings from S_ω to the closed unit interval I of R and vice versa. Mapping S_ω onto I is routine; map numbers less than or equal to ε (including −ω) to 0, numbers greater than or equal to 1 − ε (including ω) to 1, and numbers between ε and 1 − ε to their equivalent in I (mapping the infinitesimal neighbors y±ε of each dyadic fraction y, along with y itself, to y). To map I onto S_ω, map the (open) central third (1/3, 2/3) of I onto { | } = 0; the central third (7/9, 8/9) of the upper third to { 0 | } = 1; and so forth. This maps a nonempty open interval of I onto each element of S_∗, monotonically. The residue of I consists of the Cantor set 2^ω, each point of which is uniquely identified by a partition of the central-third intervals into left and right sets, corresponding precisely to a form { L | R } in S_ω. This places the Cantor set in one-to-one correspondence with the set of surreal numbers with birthday ω.

Transfinite induction

Continuing to perform transfinite induction beyond S_ω produces more ordinal numbers α, each represented as the largest surreal number having birthday α. (This is essentially a definition of the ordinal numbers resulting from transfinite induction.) The first such ordinal is ω+1 = { ω | }. There is another positive infinite number in generation ω+1:

ω − 1 = { 1, 2, 3, 4, ... | ω }.

The surreal number ω − 1 is not an ordinal; the ordinal ω is not the successor of any ordinal. This is a surreal number with birthday ω+1, which is labeled ω − 1 on the basis that it coincides with the sum of ω = { 1, 2, 3, 4, ... | } and −1 = { | 0 }. Similarly, there are two new infinitesimal numbers in generation ω + 1:

2ε = ε + ε = { ε | 1 + ε, 1/2 + ε, 1/4 + ε, 1/8 + ε, ... } and

ε/2 = ε · 1/2 = { 0 | ε }.

At a later stage of transfinite induction, there is a number larger than ω + k for all natural numbers k:

2ω = ω + ω = { ω+1, ω+2, ω+3, ω+4, ... | }

This number may be labeled ω + ω both because its birthday is ω + ω (the first ordinal number not reachable from ω by the successor operation) and because it coincides with the surreal sum of ω and ω; it may also be labeled 2ω because it coincides with the product of ω = { 1, 2, 3, 4, ... | } and 2 = { 1 | }. It is the second limit ordinal; reaching it from ω via the construction step requires a transfinite induction on

$\bigcup _{k<\omega }S_{\omega +k}$

This involves an infinite union of infinite sets, which is a "stronger" set theoretic operation than the previous transfinite induction required.

Note that the conventional addition and multiplication of ordinals does not always coincide with these operations on their surreal representations. The sum of ordinals 1 + ω equals ω, but the surreal sum is commutative and produces 1 + ω = ω + 1 > ω. The addition and multiplication of the surreal numbers associated with ordinals coincides with the natural sum and natural product of ordinals.

Just as 2ω is bigger than ω + n for any natural number n, there is a surreal number ω/2 that is infinite but smaller than ω − n for any natural number n. That is, ω/2 is defined by

ω/2 = { S_∗ | ω − S_∗ }

where on the right hand side the notation x − Y is used to mean { x − y : y ∈ Y }. It can be identified as the product of ω and the form { 0 | 1 } of 1/2. The birthday of ω/2 is the limit ordinal ω2.

Powers of ω and the Conway normal form

To be continued.

An very brief overview of Surreal Numbers

• Steve Carlton https://guests.mpim-bonn.mpg.de/spc/talks/7_gandalf_surrealnumbers_notes.pdf

Definition 2.1. A surreal number x is a pair of sets, the left set XL and the right set XR, of previously created surreal numbers, such that no element r ∈ XR of the right set is to be ≤ any element ` ∈ XL of the left set, i.e. ¬ ∃` ∈ XL ∃r ∈ XR (r ≤ `). (This requirement means that x is well-formed.) Write x ≡ { XL | XR } to mean x is given by this particular presentation, for many different presentations can lead to the same surreal number value. And to make sense of this we need the definition of ≤, which is as follows

Definition 2.2. For two surreal numbers x ≡ { XL | XR } and y ≡ { YL | YR }, we say x ≤ y if

• y is not ≤ any element of XL, i.e. ¬ ∃x` ∈ XL (y ≤ xi), and

• no element of YR is ≤ x, i.e. ¬ ∃yr ∈ YR (yr ≤ x).

Theorem 5.1. On day n, we create n = { n − 1 | }, and −n = { | −(n − 1) }, and all midpoints between previously existing surreal numbers. This means on finite days we create only integers, and dyadic fractions. We can also see some structure on the values a surreal number represent. The number x ≡ { XL | XR } is always between all values in XL and XR. More precisely

Theorem 5.2. The value of x ≡ { XL | XR } is the earliest-created surreal y such that y < every element of XR, and every element of XL is < y.

Definition 6.1. For surreal numbers x ≡ { XL | XR } and y ≡ { YL | YR }, we define x + y ≡ { XL + y, x + YL | XR + y, x + YR } , where number + set means add number to each element of set.

Theorem 6.2. 1 + 1 = 2 Proof. { 0 | } + { 0 | } = { 0 + 1, 1 + 0 | ∅ + 1, 1 + ∅ } = { 1 | }, where we’ve made use of a simpler result that we should have proven beforehand 0 + 1 = 1 + 0 = 1. So what we labelled two is justified. Similarly we can show the name 1 2 is justified: Theorem 6.3. 1 2 + 1 2 = 1 Proof. { 0 | 1 }+{ 0 | 1 } = � 0 + 1 2 , 1 2 + 0��1 2 + 1, 1 + 1 2 . Both sets here contain simpler elements (the day sum is ≤ 3 compared to the original 4), so unspooling the definition further gives = � 1 2�� 1 1 2 . Now the earliest created surreal number between the left and right sets is 1.

I don't know when this happened, but I really liked this statement because it emphasizes that we can't do anything until all the dyadics are created. I think we used circular reasoning to create the dyadics: We assumed the meaning, and proved that they are self consistent.

Is it true that we create a system and use its own rules to prove it is valid? In other words, we assume something means 1/2 and then prove that 1/2 plus 1/2 equals one.

Once we move to allowing infinite sets, on day ω, we can write down 1/ 3 = � 0, 1/ 4 , 5/ 16 , 21/ 64 . . .����1, 1 /2 , 3 /8 , 11 /32 , . . . � The left hand set is all dyadic fractions < 1/ 3 , and the right hand set is all dyadic fractions > 1/ 3 . (This is very similar to the Dedekind cut definition of the real numbers, especially if we look for the surreal number with value π.)

SHOULD I INTRODUCE A COUNTER EXAMPLE: REPLACE {-1,0} = 1/3 AND SEE IF I GET 1/3 PLUS 1/3 EQUALS 2/3.

Notes

• See Surreal number/Infinity plus one
• w:Surreal_number#Addition adds 1/2+1/2. Introduces convention {a<b<c|C>B>A}
• https://www.m-a.org.uk/resources/downloads/4H-Jim-Simons-Meet-the-surreal-numbers.pdf Begins with "An ordinal is the set of all previously defined ordinals" 0={}, 1={0}, 2={0,1}. Awkard beginning.
• https://web.mit.edu/sp.268/www/2010/surreal.pdf Uses the left-right set notation to describe the hackenbush score! The base case is {∅|∅} which will be called the endgame and occurs when neither player has any moves left.
  • G = {0|}, since L can move to the 0 game and R has no moves.
  • Figure b is red over blue. Blue is left and red is right. SEE PAGE 7: It explains why {2 1/2 | 4 1/2} = 3 = {2|} =y.
    • We know that x={2 1/2 | 4 1/2} is less than or equal to {2|} because 2 1/2 is less than or equal to 3. We know that the right side of {2 1/2 | 4 1/2} cannot be less than the right side of 3 because the right side of 3 is empty. Therefore x \le y.
    • Now we need to prove that y \le x:

DMP Notes Surreal Numbers 2022/04/29 v0.3

• K. Sutner https://www.cs.cmu.edu/~sutner/dmp/surreals.pdf
• Google search: Comparison rule surreal numbers

x le y is equivalent to: ∀ u ∈ XL, v ∈ XR (v ̸≼ u) in other words:

x ≼ y ⇔ ∀ u ∈ XL, v ∈ YR (y ̸≼ u ∧ v ̸≼ x) x ̸≼ y ⇔ ∃ u ∈ XL, v ∈ YR (y ≼ u ∨ v ≼ x) ... same as Wikipedia's definition.

Journal of Logic & Analysis 6:5 (2014) 1–39

Analysis on Surreal Numbers, SIMON RUBINSTEIN-SALZEDO, ASHVIN SWAMINATHA http://logicandanalysis.org/index.php/jla/article/viewFile/210/97/

The “Conway-Norton” integral failed to have standard properties such as $\int _{a}^{b}f(x)dx=\int _{a-t}^{b-t}f(x+t)dx.$ Another problem:$\int _{0}^{\omega }f(x)dx$ should be, $e^{\omega }$, when it should be $e^{\omega }-1.$

MitCourse

Saved on laptop: "C:\Users\vande\OneDrive\Desktop\hackenbush\MIT COURSE"

Relies heavily on hackenbush. Used birthdays and the two-set model for surreal numbers. Emphasizes numeric and non-numeric pairs. Shows:

$\{2{\tfrac {1}{2}}|4{\tfrac {1}{2}}\}=3=\{2|\}\neq 3{\tfrac {1}{2}}$ (the latter is the average.)

• Essential: https://web.mit.edu/sp.268/www/2010/surrealSlides.pdf OR https://web.mit.edu/sp.268/www/surrealSlides.pdf
• Only 13 pages: https://web.mit.edu/sp.268/www/2010/surreal.pdf
• Nonessential: index.html, TOC, coursenotes.html, Nim and Sprague-Grundy, Various games

Feature Column (ams.org)

done

1-Introduction 2-Hex ... 7-References 3 Hackenbush 3-Hackenbush Defines "grass" as green line and "trees" as single base. Circuits are also possible (connects different trees). Left cuts blue, Right cuts red. Blue is positive (Red negative.) Introduces 0 +-1, +-2 as linear trees. My notation for linear trees: (RB, RB, B)=-1/2-1/2+1=0 The first letter represents "attached" to "grass" (ground). Attached letter denotes sign. First example is RRRBBRB =R@1R@2R@3B@4B@5R@6B@7 First three RRR denotes -3 After that we have the binary point B@4. After B@4 is BRB=101 Hence: -3 + 1*1/2 + 0*1/4 + 1*1/8 b/c B->1 and R->0 Then tack on a 1 (why?) to get: -3 + 1*1/2 + 0*1/4 + 1*1/8 + 1*1/16 = -3+11/16 = -2 + 5/16 3/8=0*1/2+1*1/4+1*1/8. Book claims this is BBBRRB. Check :+2 from the first three. Then comes the binary RRB->0*1/2+0*1/2+1*1/4 (tack on +1/8). Stop at: "In the games we have looked at previously we have insisted that features such as the number of edges or the number of stones in a pile be a finite number." (Also introduces ω , ω+1 , ω+2 , 4 Counting and sets 4-Counting and sets 5 Surreal numbers 5-Surreal numbers 6 Surreal numbers and games 6-Surreal numbers and games

Images

• ${\textstyle \sum _{n=0}^{\infty }{\big (}{\frac {3}{4}}{\big )}^{n}=4}$

expand
There is nothing surreal about this collection of Dyadic rational numbers. Image under construction.

Set theory

Power set. The power set of {A, B}, can be written as P(A, B) It is all possible subsets of {A, B} including the empty set. It is easy to see that if x=P(A, B), then:

P(A,B) = x = { { }, {A}, {B}, {A, B} }

An excellent introduction on Grim.pdf)

Gretchen Grimm https://www.whitman.edu/documents/Academics/Mathematics/Grimm.pdf Careful introduction, saved as Grimm.pdf

Theorem 2.1 (The Simplicity Theorem)

• Given any number y = {YL|YR}, if x is the first number created with the property that YL < x and x < YR, then x ≡ y.

Compare this with Wikipedia

• Birthday property. A form x = { L | R } occurring in generation n represents a number inherited from an earlier generation i < n if and only if there is some number in S_i that is greater than all elements of L and less than all elements of the R. (In other words, if L and R are already separated by a number created at an earlier stage, then x does not represent a new number but one already constructed.) If x represents a number from any generation earlier than n, there is a least such generation i, and exactly one number c with this least i as its birthday that lies between L and R; x is a form of this c. In other words, it lies in the equivalence class in S_n that is a superset of the representation of c in generation. Taken from w:Surreal number

Collection of online resources saved on disk

• EXCELLENT: Surreal Birthdays and Their Arithmetic by Matthew Roughan.
  • https://arxiv.org/pdf/1810.10373
  • Desktop/hackenbush/https://thehighergeometer.wordpress.com/2018/10/25/surreal-birthdays-and-their-arithmetic/
  • Has jokes. I like his careful notation. Defines Dali function for dyadic surreals. . It is (almost) a connected Directed Acyclic Graph (DAG) with links showing how each surreal is constructed from its parents. But a DAG, by itself, would loose information. The graph would only specify parents, not left and right parents. So in displaying the DAG, we show a box for each surreal number, with the value given in the top section, and the left and right sets shown in the bottom left and right sections, respectively. From each member of each set we show a link to its box, and its parents in turn: a red link indicates a left parent, and blue right. The advantage of the DAG is that it shows the whole recursive structure of a surreal. High level discussion of birthdays, inheritance, and equivalent forms. Lots of complicated graphs.

• POOR David Roberts. Surreal birthdays and their arithmetic – theHigherGeometer.pdf (Also published as Practically surreal: Surreal arithmetic in Julia, SoftwareX Volume 9, January–June 2019, Pages 293-298) (not much here)
  • https://thehighergeometer.wordpress.com/2018/10/25/surreal-birthdays-and-their-arithmetic/
  • Desktop/hackenbush/Surreal%20birthdays%20and%20their%20arithmetic%20–%20theHigherGeometer.pdf
  • DAG graphs (a way to depict equivalent forms via 'parents') No mention of birthday rule.

• EXCELLENT: Surreal number in nLab https://ncatlab.org/nlab/show/surreal+number
  • Website with links! Has interesting notation compatable but slightly different from Wikipedia.
  • https://docs.google.com/document/d/1REcnMPMQDmRvZxzpP-DJy75m1IhIkX-X4feotoeAf30/edit
  • Desktop/hackenbush/Surreal%20number-ncatlab.org%20-%20Google%20Docs.pdf

• EXCELLENT: https://www.tondering.dk/download/sur.pdf saved as "C:\Users\vande\OneDrive\Desktop\hackenbush\sur.pdf"
  • and I just skimmed it!

Simplicity and Birthday rules

In evaluating numbers and games we will use the Simplicity Rule, which says that out of all the numbers between the largest member of the left set and the smallest number of the right set, the surreal number value of the form is the simplest number that fits, where we use simplest as meaning the number born earliest. This is just either the smallest integer between the two, or else the fraction between them having the highest power of two in the denominator. https://web.mit.edu/sp.268/www/2010/surreal.pdf

• The simplicity rule and hackenbush are discussed at https://www.math.uchicago.edu/~may/VIGRE/VIGRE2006/PAPERS/Bartlett.pdf saved as Bartlett.pdf