Surreal number

• See also Surreal number, Set theory, and Infinity on Wikipedia, as well as Surreal Numbers and Games on Wikibooks.

This discussion mixes prerequisite knowledge required by the Wikipedia article with insights that might make surreal numbers more interesting. Our focus is on concepts involving to countability and infinity that are easy to explain. To this end, we employ the language of naive set theory in a way that cannot fully explain or describe surreal numbers. The current author of this resource is not an expert on this subject, and for that reason, corrections and elaborations are welcome.

Fig. 1: Surreal numbers in the interval [−1,4] created up to and including day 4 are shown: On day zero, "0" is created. On day one, "−1" & "+1" are created. Four more numbers are created on day two (not shown is −2.) Since the empty set Φ contains no elements, it is not unreasonable to say that all surreal numbers are created out of "nothing".

Important facts about surreal numbers:

1. The story begins with 0 and dyadic rational fractions, which are ratios are of the form p/q where p is an integer and q=2ⁿ, where n is a non-negative integer. These dyadic rationals are shown in figure 1 as as "0", "±1", "±½","±2",…. The quotation marks around the "numbers" will be explained later.
2. In a strange sort of way, these dyadic ratios can define the set of all rational and irrational numbers.
3. The surreal rational and irrational numbers are defined in the language of set theory in a way that has little to do with numbers as we know them.
4. A certain version of set theory can be used to create definitions that reminds one of entities such as 0/0 and ∞/∞, that virtually all textbooks dismiss as "indeterminant". This variation of set theory can also give meaning to an algebraic expression like, $\infty ^{2}-3\infty +1$ (except that it is customary to use $\omega$ instead of $\infty$ to represent one of the many (infinite) versions of infinity associated with surreal numbers.)

"Counting" the dyadic rationals

Fig. 2: Rulers often subdivide the inch into dyadic fractions.

Dyadic rationals are fractions where the denominators are powers of 2, i.e., fractions form p/qⁿ, where p is an integer and n is a positive integer, as shown in figure 2.

For reasons to be explained later, the counting is done in groups known as "birthays" (or simply days when each dyadic rational is "born".)

$0$ is born on day 0. On day 1, two numbers are born: $-1$, and $1$) are born on day 1. Note that $2^{0}=1$ and $2^{1}=2$, so that on day 2 we have $2^{2}=$ four new numbers: $-2$, $-{\tfrac {1}{2}}$, ${\tfrac {1}{2}}$, and $2.$) Each day the the count doubles, so that on day 3, we obtain eight numbers: $-3$, $-1{\tfrac {1}{2}}$, $-2{\tfrac {1}{4}}$, $-2{\tfrac {1}{2}}$, $2{\tfrac {1}{4}}$, $2{\tfrac {1}{2}}$, $2{\tfrac {3}{4}}$, $3$.

The rules for creating new positive numbers are as follows:

1. Each day a new positive integer is created by adding 1 to the previous day's new positive integer.
2. All other new numbers are dyadic fractions created as the midpoint between the fractions created on the previous day.
3. The negative numbers are created in a symmetric fashion. On the same day that p/q is created, −p/q is also created.

It is impossible to create all of the $2^{n}$ surreal numbers created on day $n$ until all $2^{n-1}$ surreal numbers had been created on on day $n-1.$ But after the previous day's numbers were created, the next day's numbers can be created in any order.

What does it mean to count numbers?

Fig. 3: Counting rational numbers

By definition, "counting" doesn't mean you finish, but that you can create a list that includes every item if you count long enough. Stating a formal definition of "countability" is not a trivial task. But figure 3 illustrate a well-known countable infinity.

The rational numbers are countable

0	.	9	3	3	2	4	7	0	2	0	...
0	.	6	3	7	4	9	3	3	6	1	...
0	.	2	7	7	5	5	2	4	6	6	...
0	.	4	7	5	5	6	8	2	9	7	...
0	.	8	3	8	1	4	2	2	4	9	...
0	.	2	5	2	3	5	2	5	7	6	...
0	.	8	4	4	2	2	1	1	4	1	...
0	.	6	8	6	2	1	5	1	8	3	...
0	.	1	7	8	6	6	2	8	1	5	...
⋮	.	⋮	⋮	⋮	⋮	⋮	⋮	⋮	⋮	⋮	⋱
Fig. 4: Counting real numbers

Figure 3 shows you can count all fractions of the form $p/q$ where $p$ and $q$ are integers. Simply follow the arrows. You should be able to convince yourself that if $X$ is larger than both $p$ and $q$, then you will reach $p/q$ in no more than $X^{2}$ steps. This fact is essential to the formal proof that the rational numbers are countable.

Real numbers are not countable

Figure 4 establishes that the real numbers are uncountable. To prove this, imagine that a list of all numbers between 0 and 1 has been created, as shown in the table to the right. It is possible to find an irrational number that is not on this list as follows:

• The first number on the list is 0.9332... . Change the first digit after the decimal point from 9 to 0.
• On the second number (0.6374...), change the second digit from 3 to 4
• On the third number, change the third digit (7) to 8.

You have to do this to infinity, but when you are "done", you will have a number not on your original list. Therefore it is impossible to make an infinite list that contains all the irrational numbers in the interval (0,1). This argument can be found on a website found on Carnegie Mellon University, and it seems to contradict two ideas essential to the construction of surreal numbers:

1. Fractions are countable, while the real numbers are not.
2. The real numbers can be represented by a sequence of (dyadic) fractions.

This inspires the following:

(unsolved) Problem

This section is posed as an unsolved problem, with the moral to the story being that pure mathematics cannot be based on intuition. Careful logic and rigorous language are required.

It seems reasonable to suppose that the set of dyadic fractions is "smaller" than the set of all fractions, since the set of fractions contains elements that are not dyadic. Yet it is claimed that the set of dyadic fractions contains an element that comes arbitrarily close to any given real number. The paradox likely cannot be resolved without setting these ideas into a formal mathematical language. But one can identify what makes the collection of dyadic fractions illustrated in figure 1 somehow "larger" than the list of infinite decimal elements of figure 3: Changing one digit in a decimal expression removes all hope of expressing a decimal representation of any number. Without the 4 in 3.14159..., it is impossible to write $\pi$ in decimal form. In contrast, if $22/7$ is removed from the list of approximations to $\pi$, we still have $355/113$. And an infinite number of approximations better than $355/113$ also exist.

Another distinction between the two representations of "all numbers" is that for the decimal representation, each digit is one of of only ten choices (0-9). In contrast, each fraction in the construction of dyadic ratios represents one of 2ⁿ options. This exponential growth in the creation of "numbers" may explain how the set of dyadic fractions is "larger" than the set of fractions as depicted in figure 3.

About the "sets" and "numbers"

Figure 1 equates what look like sets with what look like numbers. This is an attempt to translate the set-theory language into conversational english.

"0"={Φ|Φ} ...(Day zero)
   "0" is created. "Nothing" is to the left and "nothing" to the right.
   Here, "nothing" means the null set Φ, and to the left (right) means "smaller" ("larger")

"−1"={Φ|0}  &  "1"={0|Φ} ...(Day one)
   "−1" is created with "nothing" to the left and "0" to the right.
   "1" is created with "0" to the left and "nothing" to the right.

"−2"={Φ|−1}  ,  "−½"={−1|−1}  ,  "½"={1|2}  &  "2"={1|Φ} ...(Day two)

   "−2" is created with "nothing" to the left and "−1" to the right.
   "−½" is created with "−1" to the left and "0" to the right.
   "½" is created with "0" to the left and "1" to the right.
   "2" is created with "−1" to the left and "nothing" to the right.

The instructions might have been more clear if the (positive) integer N was declared one unit to the right of N−1. Similarly, it could have been mentioned that "½" was midway between "0" and "1". Perhaps the author wanted to keep the reader in suspense.

The rest of this resource is under construction

Infinitesimals (ε) and infinities (ω) on the hyperreal number line (1/ε = ω/1). This image appears in the Wikipedia articles w:Hyperreal number, w:Infinitesimal, and w:Infinity.

This section is under construction. It will summarize some (but not all) of the following Wikipedia articles:

Reading list:

• w:Infinity#Real_analysis
• w:Transfinite number#Examples
• w:Hyperreal number
• w:Von Neumann universe
• w:
• w:
• w:
• w:

References

• Introduction to Category Theory/Products and Coproducts of Sets
• Introduction to Category Theory/Sets and Functions
• Set theory
• Review on amazon.com]
• staxeschange definition of all dyadics
• Scientific American badiou-and-science part-1 I might copy graph

• Steven Charlton - An Very Brief Introduction to Surreal Numbers (alternate link)

• what is a game? (mit)
• .9999
• wikipedia:Surreal number
• whitman.edu (Grimm.pdf) Includes games
• scietificamerican.com
• Roughan Directed Acyclic Graphs (DAG); 2*3 & 3/4 + 3/4/ (too complicated for me)

Staxexchange

proof of conways simplicity rule
https://math.stackexchange.com/questions/816540/proof-of-conways-simplicity-rule-for-surreal-numbers $0=\{|\}$ $n+1=\{n|\}$ $-n-1=\{|-n\}$ ${\tfrac {2p+1}{2^{q+1}}}=\left\{{\tfrac {p}{2q}}|{\tfrac {p+1}{2q}}\right\}$

Limits and analysis

Infinity is easy to imagine, but difficult to incorporate

Infinity is easy to imagine, but difficult to incorporate into rigorous mathematics. The following calculation certainly violates the rules of mathematics: $\lim _{\epsilon \to 0}{\frac {1}{\epsilon }}=\infty \;$ and $\;\lim _{\epsilon \to 0}{\frac {2}{\epsilon }}=\infty \;$ implies $\;{\frac {\infty }{\infty }}=2.$ This is why expressions like $\infty /\infty$ and $0/0$ of often called indeterminant. Most of the time, mistakes like this can be avoided by utilizing concepts taught in a course on mathematical analysis. Both the Wikipedia article and a query sent to an online chatbot suggest that no widely known problems in this field have been solved using surreal numbers. For that reason, any forrey into surreal numbers should probably be viewed as recreational.