Overview

The experiment demonstrates Cantor's diagonalization technique using strings over customizable alphabets. The simulation works with the following principles:

1. It displays a "Purported Bijection Table" showing a mapping from natural numbers to infinite strings
2. You can customize the alphabet size (2-4 symbols), window size (5-15), and window start position (0-50)
3. Each string is infinite in length, but only the first n characters are visible (where n is the window size)
4. Your goal is to construct the diagonalized string that proves no such bijection can exist

Steps to Perform the Experiment

1. Access the Controls Panel

   • Click the settings button (floating button with gear icon) to open the Parameters panel
   • Adjust the following parameters as needed:
     • Alphabet Size: Choose between 2-4 symbols (default: 2 for binary {0,1})
     • Window Size: Select 5-15 rows to display (default: 10)
     • Window Start: Set starting natural number 0-50 (default: 0)

2. Study the Purported Bijection Table

   • The table shows natural numbers (n ∈ ℕ) mapped to strings (f(n) ∈ Σ^ω)
   • Each row represents f(n) for natural numbers in your chosen window
   • Position columns (0,1,2,...) show individual characters of each string

3. Understand the Negation Reference

   • Below the table, you'll see the alphabet negation mapping
   • For binary: 0 → 1, 1 → 0
   • For larger alphabets: each symbol maps to the next one cyclically

4. Construct the Diagonalized String

   • Identify the diagonal elements: position i in row i
   • For each diagonal element, apply the negation mapping
   • The diagonalized string differs from every row at its diagonal position

5. Enter Your Answer

   • Type the complete diagonalized string in the input field
   • The string must be exactly the same length as the window size
   • Use only valid alphabet characters

6. Verify Your Solution

   • Click the "Check Answer" button
   • The system will validate your answer and provide feedback
   • Upon correct answer, a new table will be automatically generated after 2 seconds

Example Walkthrough

1. Parameter Setup:

   • Set Alphabet Size to 2 (binary: {0, 1})
   • Set Window Size to 6
   • Set Window Start to 0

2. Generated Table Example:

   n ∈ ℕ | f(n) ∈ Σ^ω | Position | 0 | 1 | 2 | 3 | 4 | 5 |
             ------|------------|----------|---|---|---|---|---|---|
             0     | 101010     | 0        | 1 | 0 | 1 | 0 | 1 | 0 |
             1     | 010101     | 1        | 0 | 1 | 0 | 1 | 0 | 1 |
             2     | 110011     | 2        | 1 | 1 | 0 | 0 | 1 | 1 |
             3     | 001100     | 3        | 0 | 0 | 1 | 1 | 0 | 0 |
             4     | 111000     | 4        | 1 | 1 | 1 | 0 | 0 | 0 |
             5     | 000111     | 5        | 0 | 0 | 0 | 1 | 1 | 1 |
             

3. Diagonal Construction Process:

   • Position 0: Take diagonal element f(0)[0] = 1, negate to get 0
   • Position 1: Take diagonal element f(1)[1] = 1, negate to get 0
   • Position 2: Take diagonal element f(2)[2] = 0, negate to get 1
   • Position 3: Take diagonal element f(3)[3] = 1, negate to get 0
   • Position 4: Take diagonal element f(4)[4] = 0, negate to get 1
   • Position 5: Take diagonal element f(5)[5] = 1, negate to get 0
   • Final diagonalized string: 001010

4. Result Verification:

   • Enter the string "001010" in the input field
   • Click "Check Answer"
   • You will receive feedback: "Correct! '001010' is the diagonalized string..."
   • A new table will be generated automatically for the next challenge