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.

Collection of online resources saved on disk

 * 4H-Jim-Simons-Meet-the-surreal-numbers 39 pages. Excellent introduction to the Ordinal numbers. Saved as hackenbush/4H-Jim-Simons-Meet-the-surreal-numbers.pdf
 * https://arxiv.org/pdf/2310.14879 saved as hackenbush/2310.14879. Also available on the website https://www.texmacs.org/joris/hyno/hyno.html. 62 pages. "Surreal numbers as hyperseries" (Vincent Bagayoko)


 * 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!

· Draft subspace

'''See bottom of. It explores circular reasoning.'''

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


 * 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.

An very brief overview of Surreal Numbers
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
 * Steve Carlton https://guests.mpim-bonn.mpg.de/spc/talks/7_gandalf_surrealnumbers_notes.pdf

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.

DMP Notes Surreal Numbers 2022/04/29 v0.3
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)
 * K. Sutner https://www.cs.cmu.edu/~sutner/dmp/surreals.pdf saved as hackenbush/surreals.pdf
 * Google search: Comparison rule surreal numbers

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/ saved as hackenbush/210-758-1-PB.pdf

The “Conway-Norton” integral failed to have standard properties such as $$\int_a^bf(x)dx = \int_{a-t}^{b-t}f(x +t)dx.$$ 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:

$$\{2\tfrac 1 2|4\tfrac 1 2\}=3=\{2|\}\ne 3\tfrac 1 2$$ (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)

 * 1-Introduction 2-Hex ... 7-References

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.)
 * 3-Hackenbush

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



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 &omega;, &omega;+1 , &omega;+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

 * $\sum_{n=0}^{\infty} \big(\frac{3}{4}\big)^n =4$

Set theory
Power set. The power set of {A, B}, can be written as It is all possible subsets of {A, B} including the empty set. It is easy to see that if, 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 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 Surreal number

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