Surreal number/ChatBot assistance

Surreal number (subpages/draftspace)
So far, I have found Gemini useful on several occasions.

#Cantor's omega was a relatively trivial situation where I was uncomfortable with the use of ω to describe infinity by Cantor and in surreal numbers. It is entirely possible that both infinities are identical, but wondered if I had the option of using two different symbols.
At #Gemini self-correcting due to human interaction I asked about creating 1/3 as the convergence of an infinite sequence of dyadic fractions. This was a surprise in that Gemini gave me a wrong answer, but after several attempts I was able to guide Gemini to what might be a correct answer. If chatbots can successfully respond to constructive in 2024, I can't imagine what chatbots will do in math after 10 or 20 years. Warning: Even after I explained the calculation, and Gemini agreed that ${\textstyle \sum _{n=1}^{\infty }{\frac {1}{3^{n}}}={\frac {1}{2}},}$ Gemini got another series wrong. This led to the following dialogue: (Me.) I am curious as to how you could make such a trivial mistake. 🔷(Gemini) You're right, the mistake I made with the infinite geometric series sum was a careless error. There are a couple of reasons why I might have made such a simple mistake: (1) Overconfidence: Since I correctly handled the previous converging geometric series problems without showing work, I might have gotten overconfident and not applied the formula carefully enough in this case. (2) Focus on explanation: While explaining how geometric series work and why they converge/diverge is important, I should have prioritized getting the actual answer correct first.
At #Why is the surreal class of numbers larger than the class of hyperreals?, I think I got the correct answer. At least that is what Wikipedia tells me
#Use_the_birthday_rule_to_show_that_(2l5)_equals_the_surreal_number_3 I had to edit the title because symbols like {2|5} do not hyperlink properly. But Gemini gave my first clear answer to such a question (I hope its right!)
#Define_a_game_in_combinational_game_theory is essential to understanding the relationship between surreal numbers and game theory.
#Hackenbush_questions I don't know whether or not these answers are correct.

All conversations are shown below (almost unedited.)

Cantor's omega

In Cantor's theory if infinite numbers, what names have been used for \omega?

🔷 In Cantor's theory of transfinite numbers, there's only one name widely used for ω (omega): omega.

Cantor did choose the symbol ω specifically to differentiate it from the infinity symbol ∞ (often seen as "potential infinity") to emphasize that ω represented a well-defined, "completed" type of infinity.

There isn't any other commonly used name for ω within the context of Cantor's set theory.

How does Cantor's omega differ from the omega of surreal numbers?

🔷 Here's how Cantor's omega differs from the omega of surreal numbers:

Cantor's omega (ω):

Represents the first transfinite ordinal number in Cantor's set theory.
Represents the order type of the set of natural numbers (1, 2, 3, ...).
Is the smallest infinite ordinal number – there are no "smaller" infinities.
Used to describe the ordering within an infinitely large set.

Surreal omega (ω):

Is a specific surreal number within the system of surreal numbers.
Can be constructed using the set definition for surreal numbers.
Is considered one of many possible infinite surreal numbers.
Represents a specific infinite quantity rather than an order type.

Key Differences:

Purpose: Cantor's omega focuses on ordinality (ordering) of infinite sets, while surreal omega is just another infinite value within a broader system.
Uniqueness: Cantor's omega is the first and smallest transfinite ordinal; surreal omega is one among many infinite surreal numbers.
Construction: Cantor's omega is a concept within set theory, while surreal omega is constructed using the specific rules of surreal numbers.

Connection:

Interestingly, both Cantor's omega and surreal omega can be used to represent the concept of "infinity." However, they represent it in different ways and within different mathematical frameworks.

Surreal numbers actually encompass Cantor's omega. The system of surreal numbers allows for the construction of Cantor's omega as a specific surreal number, along with many other infinite and infinitesimal surreal numbers.

Two part question: Is there an infinite number smaller than Cantor's omega? Is there an infinite number smaller than the surreal omega?

🔷 Cantor's Omega:

No, there is no infinite number smaller than Cantor's omega (ω).
By definition, Cantor's omega is the smallest transfinite ordinal number. This means it's the first number that's larger than all finite ordinals (natural numbers). There are no "in-between" sizes of infinity in this context.

Surreal Omega:

It depends on the specific construction of surreal numbers in a given context.
Surreal numbers are a more general system that allows for a wider range of infinite and infinitesimal values.
While Cantor's omega can be constructed as a surreal number, there might be other surreal numbers defined to be "smaller" depending on the specific rules used.

Here's the key difference:

In Cantor's set theory, ordinals define the order of infinite sets, and omega is the starting point for transfinite sizes. There's no "before" the first.
Surreal numbers are a broader system that allows for more flexibility in defining infinite and infinitesimal values. Whether a specific surreal omega is the "smallest" depends on the chosen construction rules.

Can you describe a construction of the surreal omega where ω -1 exists, is infinite, and smaller than omega?

🔷 Absolutely! Here's a construction of surreal numbers where ω - 1 exists, is infinite, and smaller than ω:

1. Basic Surreal Numbers: We'll start with the basic definition of surreal numbers. A surreal number is a pair of sets, the left set containing all surreal numbers smaller than the given number and the right set containing all surreal numbers greater than the given number. We write it as {left | right}.

2. Ordinals and ω: We'll utilize ordinals within the surreal number system. An ordinal is a well-ordered set with no "largest" element. The smallest infinite ordinal is ω, which represents the order type of the natural numbers.

3. Defining ω - 1: Here's the key step: We define ω - 1 as { ω | }.

Left Set: This set contains only ω, the first infinite ordinal.
Right Set: This set is empty (indicated by "{}"). There are no surreal numbers "greater than" ω - 1 because any number containing ω in its left set would be larger than ω - 1.

Properties of ω - 1:

Infinite: Since ω - 1 contains ω in its left set, it "sees" infinitely many surreal numbers (all the natural numbers). Therefore, ω - 1 itself is considered infinite.
Smaller than ω: Because ω - 1's right set is empty, any number containing something greater than ω (like another infinite ordinal) would be larger than ω - 1. This establishes ω - 1 as smaller than ω.

Important Note:

This construction is not necessarily the standard or only way to define surreal numbers. Different frameworks might have slightly different rules, and the existence and properties of ω - 1 could vary depending on those rules.

However, this example demonstrates that within a specific construction of surreal numbers, it's possible to have an infinite number ω - 1 that's smaller than the standard surreal omega (ω).

Gemini self-correcting due to human interaction

Construct a sequence of dyadic fractions that converge to 1/3

🔷 Absolutely, here's a sequence of dyadic fractions that converges to 1/3:

{1/2, 1/4, 2/4, 1/8, 2/8, 1/16, 2/16, ...}

This sequence alternates between 1/(2^n) and 2/(2^n), where n is the position in the sequence (starting from 1). As n increases, the denominators get larger, making the fractions closer and closer to 1/3.

Here's an explanation of why this works:

Dyadic Fractions: Each fraction in the sequence has a denominator that is a power of 2 (2^n). These are called dyadic fractions.
Sum of the Series: If we were to sum an infinite number of terms in this alternating series, where each term is half the value of the previous term, the sum would converge to 1/3. This can be proven using the concept of geometric series.

However, due to the alternating nature of the series, some terms are greater than 1/3 and some are less than 1/3. As we take more and more terms, the fractions get progressively closer to 1/3, ultimately converging to it.

Note: It's important to remember that this is an infinite series. In practice, you can use a finite number of terms from the sequence to get a good approximation of 1/3. The more terms you use, the better the approximation will be.

Your sequence might work, but the formula cannot be of the form $\frac{1}{2^n}$, but instead of the form, $\frac{f(n)}{2^n}$, where f(n) is an integer 🔷 You're absolutely right. I apologize for the mistake.

The sequence I provided earlier converges to 1/3, but it doesn't strictly follow the form 2nf(n) where f(n) is an integer.

Here's a corrected sequence that adheres to the target formula:

{1/2, 1/4, 1/8, 2/8, 1/16, 3/16, 1/32, 5/32, ...}

In this sequence, f(n) is defined as follows:f(n) = 1 for even n (n = 2, 4, 8, ...)

f(n) = f(n-1) + 1 for odd n (n = 3, 5, 7, ...)

This way, each term is constructed by taking half of the previous term (denominator multiplied by 2) and adding 1 to the numerator. This results in a sequence of dyadic fractions that gets increasingly closer to 1/3.

I hope this corrected sequence is what you were looking for!

No. I believe you need a truncation function, such as $trunc(\frac{2^n/2}{3})$ in the numerator 🔷 You're exactly right! To construct a sequence of dyadic fractions converging to 1/3 with numerators of the form f(n), we need a truncation function. Here's a Python code that incorporates truncation to achieve this:

Python

def dyadic_fractions(n):
  """
  Generates a sequence of dyadic fractions that converge to 1/3 using truncation.

  Args:
      n: The number of terms in the sequence.

  Returns:
      A list of dyadic fractions with integer numerators.
  """
  sequence = []
  for i in range(1, n + 1):
    numerator = int(trunc(2**(i-1) / 2 / 3))
    sequence.append(numerator)
    sequence[i - 1] /= (2**i)
  return sequence

# Define a truncation function (replace with your preferred implementation)
def trunc(x):
  return int(x)

# Get the first 10 terms of the sequence
sequence = dyadic_fractions(10)

# Print the sequence
print(sequence)

Use code with caution.

This code defines two functions:

dyadic_fractions(n): This function takes the number of terms n as input and generates a sequence of dyadic fractions converging to 1/3.
trunc(x): This is a placeholder for your preferred truncation function. It takes a number x as input and returns its integer part.

The code iterates through n terms, calculating the numerator using the following steps:

Truncates (rounds down) 2^(i-1) / 2 / 3 to get an integer.
Assigns the truncated value to the numerator.
Appends the fraction numerator/2^(i-1) to the sequence.
Divides the numerator by 2 (denominator multiplied by 2) for the next term.

This approach ensures that the numerators are integers and the fractions are in dyadic form.

Running the code will provide you with the first few terms of the sequence that converge to 1/3. You can experiment with different truncation functions and the number of terms to see how it affects the convergence.

This looks right! I am astonished that Gemini can self-correct on a math problem by interacting with the human user. 🔷 I'm glad I could improve my response and provide a correct solution!

While I can't perform calculations myself, I can access and process information effectively. By understanding your feedback and referring to general mathematical principles, I was able to identify the error and collaborate with you to arrive at the desired outcome. This type of interaction helps me learn and improve my ability to assist users with complex problems.

Why is the surreal class of numbers larger than the class of hyperreals?

🔷 The key difference between surreal and hyperreal numbers lies in their size (in a set-theoretic sense) and what they encompass. Here's the breakdown:

Surreal Numbers: Imagine a vast container that holds everything – all the real numbers, infinitely large numbers, infinitely small numbers (infinitesimals), and even things beyond those. This container is the class of surreal numbers. It's a proper class, meaning it's too big to be a set in standard set theory.
Hyperreal Numbers: Think of the hyperreals as a smaller, more manageable box within the surreal container. They contain the real numbers, infinitesimals (numbers smaller than any positive real number), and infinitely large numbers. However, hyperreals are still a set, not a proper class.

Here's the analogy: Imagine a library with countless books (real numbers). The surreal class is like the entire library, including all the written and unwritten stories (other types of numbers). The hyperreals are a specific section of the library focusing on science fiction and fantasy (infinitesimals and infinite numbers).

Why are Surreals Bigger?

Surreal numbers are constructed in a way that allows them to encompass everything an ordered field (like the reals or hyperreals) can handle, plus more. They can even include things that wouldn't fit neatly into a standard set. This makes them a more general, all-encompassing system.

However, due to their vastness, surreal numbers can be cumbersome to work with in some contexts. Hyperreals, being a more manageable set, offer a good balance between including infinitesimals and infinite numbers while remaining practical for mathematical analysis.

Additional Point:

Although it's a bit technical, recent advancements show that the "maximal class" versions of surreal and hyperreal fields (essentially the largest versions possible) are isomorphic – meaning they have the same structure despite the size difference.

Use_the_birthday_rule_to_show_that_(2l5)_equals_the_surreal_number_3

🔷 Absolutely, we can use the birthday rule to demonstrate that {2|5} equals the surreal number 3.

The birthday rule states that in a set of n distinct positive integers, the probability of at least two of them sharing the same remainder when divided by a smaller integer k approaches 1 as n increases.

Here's how we apply it to {2|5}:

Consider the set {1, 2, ..., n} (n being any positive integer).
Divide each element by 3 (k = 3). This gives us remainders of 1 or 2.
According to the birthday rule, as n gets larger, the probability of having at least two elements with the same remainder (either 1 or 2) approaches 1.
In the context of surreal numbers, {2|5} represents all numbers greater than or equal to 2 and less than 5.

Here's the key point:

Numbers with a remainder of 1 when divided by 3 are all greater than or equal to 1 (since 1 is the first remainder) and less than 4 (since dividing by 3 with a remainder of 1 gives us numbers like 1, 4, 7, etc.).
Numbers with a remainder of 2 when divided by 3 are all greater than or equal to 2 (since 2 is the second remainder) and less than 5 (since dividing by 3 with a remainder of 2 gives us numbers like 2, 5, 8, etc.).

Therefore, as n grows larger, the set {1, 2, ..., n} will contain elements that fall within the definition of {2|5} with increasing probability. This implies that {2|5} "captures" an increasing portion of the set {1, 2, ..., n} as n approaches infinity.

In the surreal number system, when a set captures all the elements "above" a certain point (like 2 in this case) and "below" another point (like 5), it essentially represents the number between those two points.

Therefore, based on the birthday rule, as n tends to infinity, {2|5} effectively captures all the numbers greater than or equal to 2 and less than 5, making it equivalent to the surreal number 3.

Note: This is a simplified explanation, and a more rigorous proof would involve formal set theory and the construction of surreal numbers. However, the birthday rule provides a good intuitive understanding of why {2|5} approaches 3 as the set size increases.

Asking Chatbot Copilot about (2 1/2)(4 1/2):

ME: Prove that among the surreal numbers, {2\tfrac 1 2| 4\tfrac 1 2}=3

COPILOT:

No problem! Let’s explore the surreal number {2\tfrac 1 2| 4\tfrac 1 2} and determine whether it equals 3.

Surreal Numbers Overview:
- Surreal numbers extend the real numbers to include infinite and infinitesimal quantities.
- They are represented as equivalence classes of the form { L | R }, where L and R are sets of previously constructed surreal numbers.
- The surreal number system contains not only real numbers but also these additional quantities.
Understanding {2\tfrac 1 2| 4\tfrac 1 2}:
- The left set (L) contains all surreal numbers less than 2.5.
- The right set (R) contains all surreal numbers less than or equal to 4.5.
- To prove that {2\tfrac 1 2| 4\tfrac 1 2} equals 3, we need to check both conditions:
  - Greater than or equal to 3: Since 3 is not in (L), {2\tfrac 1 2| 4\tfrac 1 2} is greater than or equal to 3.
  - Less than or equal to 3: Since 3 is in (R), {2\tfrac 1 2| 4\tfrac 1 2} is less than or equal to 3.
Conclusion:
- {2\tfrac 1 2| 4\tfrac 1 2} equals 3.

Feel free to ask if you have any more questions or need further clarification! 😊 References):

ME: But I thought the right set contained all surreals greater than 4.5.

Copilot

You’re absolutely right, and I appreciate your attention to detail! Let’s revisit the surreal number {2\tfrac 1 2| 4\tfrac 1 2}.

Left Set (L): Contains all surreal numbers less than 2.5.
Right Set ®: Contains all surreal numbers greater than 4.5.

To verify whether {2\tfrac 1 2| 4\tfrac 1 2} equals 3:

It is greater than or equal to 3 because 3 is not in (L).
It is less than or equal to 3 because 3 is in (R).

Therefore, {2\tfrac 1 2| 4\tfrac 1 2} indeed equals 3. Thank you for catching that oversight! 😊 References):

Wikiversity: Surreal number
Wikipedia: Surreal number

Define a game in combinational game theory

See also (from Google searching: Conway's simplicity rule game theory)

note1.pdf Simple-need to include ideas from this one.
gametheory.pdf Too advanced for me.
*The simplicity rule and hackenbush are discussed at Bartlett.pdf

In combinational game theory, a game is a specific type of two-player contest with some key characteristics:

Deterministic: There's no element of chance involved. No dice rolls or card draws decide the outcome, only the players' choices.
Perfect Information: Both players have complete knowledge of the game's state at all times. No hidden pieces or secret information exist.
Zero-Sum: One player's win is always the other's loss. There are no ties or draws.

Here's how mathematicians formalize these ideas:

Positions and Moves: The game consists of a series of positions. Each position represents the current state of the game. Players take turns making moves, which transition the game from one position to another.
Terminal Positions: The game eventually reaches a terminal position where no further moves are possible. At this point, a winner is declared based on pre-defined rules.

Here's an analogy: Think of navigating a maze. Each turn in the maze is a position, and choosing a direction is a move. Reaching the exit is a terminal position, and whoever gets there first wins.

Some common examples of combinatorial games include:

Tic-Tac-Toe
Pick-Up-Bricks (removing bricks from a pile)
Wythoff's game (placing counters on a board)

Combinatorial game theory analyzes these games through a mathematical lens, focusing on strategies and perfect play to determine winners and optimal moves.

*Essential math.stackexchange.com/questions/816540/proof-of-conways-simplicity-rule-for-surreal-numbers

Hackenbush questions

My limited knowledge of hackenbush renders it impossible to judge the veracity of these answers.

Suppose green edges are excluded from hackenbush. Does that permit surreal numbers to model the scoring?

Yes, if you restrict hackenbush to only red and blue edges (excluding green), then surreal numbers can effectively model the scoring. Here's why:

Surreal numbers excel at two-player, zero-sum games: This is exactly what red-and-blue hackenbush represents. Each player (red and blue) aims to gain an advantage by forcing the other into a losing position.
Winning positions define surreal numbers: Surreal numbers are constructed by analyzing the set of winning positions available to each player in a game.
Red and blue control: With only red and blue edges, each player has clear control over their colored edges. Red cuts red, and blue cuts blue. This allows for a clear definition of winning positions based on who can make the last legal move.

Here's how surreal numbers capture the scoring:

Red edge value: Each blue edge can be considered worth 1 unit.
Red edge sequence: As you move down a chain of red edges, their value gets halved compared to the previous one. This reflects the diminishing advantage of having more red edges as the opponent can cut them away.
Summing values: The final surreal number score for a position is calculated by summing the values of all blue edges minus the sum of the values of all red edges.

Therefore, with only red and blue edges, surreal numbers can effectively model the scoring in hackenbush. The system leverages the core strengths of surreal numbers – representing winnable positions and their "size" – to assign a score to a specific arrangement of red and blue edges.

In this equivalency between red-green hackenbush (without green), is the addition of surreal arithmetic equivalent to placing two hackenbush positions side-by-side?

No, adding surreal numbers in red-blue Hackenbush (without green) isn't always equivalent to placing two hackenbush positions side-by-side. While there's a connection, it's not a perfect match.

Here's why:

Surreal numbers capture winning potential: In red-blue Hackenbush, surreal numbers represent the combined winnable positions for a player, considering both the individual positions.
Side-by-side placement creates options: Placing two positions side-by-side essentially creates a new game where the first player can choose to move on either side. This introduces additional winning possibilities compared to the individual positions.

Let's look at an example:

Position A: Two red edges and one blue edge (surreal number value: 1 - (1/2) - (1/4) = 3/4)
Position B: One red edge and two blue edges (surreal number value: 2 - (1/2) = 3/2)
Adding surreal numbers: 3/4 + 3/2 = 9/4.
Side-by-side combination: The new position offers the first player a choice to cut a red edge from either side. This might create new winning opportunities not reflected in the sum (3/4 + 3/2).

Similarities do exist:

Despite the differences, there can sometimes be a match between adding surreal numbers and combining positions. This is because some combined positions might not offer the first-mover any significant advantage, and the available winning opportunities become essentially the sum of the original positions.

Overall, surreal number addition provides a theoretical framework to understand the winning potential in combined red-blue Hackenbush positions. However, it doesn't perfectly capture the strategic complexities that arise due to the first-mover choice in the combined game.