Created by Browserling
This online utility generates the definitions of positive integers using von Neumann ordinals and prints the recursive sets in a vertical list. You can switch between the fully expanded recursive definition that uses subsets or the symbolic definition that references previous natural numbers. You can also specify the number of recursions and the starting integer, as well as adjust the format of the sets. Created by math nerds from team Browserling.
With this webapp, you can generate a sequence of von Neumann ordinals. These ordinals represent positive integers (natural numbers) as a list of recursively defined sets. Each next stage is obtained by forming a new set that combines the previous integer with the set of all previous integers. This way each ordinal is the well-ordered set of all smaller ordinals. The construction of von Neumann ordinals can be performed in two ways – referencing previous natural numbers symbolically or recursively using subsets. Symbolic ordinals start with the base case empty set {}, which represents 0. The next ordinal is constructed by adding the previous ordinal to the previous set. So, the next ordinal after 0 is created by adding 0 to {}. This creates {0}, which represents 1. The next one is 1 added to {0}, which creates {0, 1}, which represents 2. The next one is 2 added to {0, 1}, which creates {0, 1, 2}, which represents 3. Each subsequent iteration adds the previous integer number and this continues to infinity, which is called ω. Ordinals via subsets are constructed recursively by starting with the empty set and then taking all sets containing previously-defined sets as elements. The integer 0 is defined as ∅ (empty set). The next integer 1 is constructed by creating a new set with ∅ as its only element, so we get {∅}, which equals 1. The second recursive stage includes first and zeroth stages and creates the set {∅, {∅}}, which equals 2. The third stage is constructed from the 0th (∅), 1st ({∅}), and 2nd ({∅, {∅}}) subsets and we get {∅, {∅}, {∅, {∅}}} as integer 3. For both types of von Neumann ordinals, you can specify the open "{" and close "}" set symbols, and the empty set symbol, which is usually the ∅ Unicode symbol. You can also customize the set element separator (comma by default), the starting ordinal, and the recursive depth (how many integers to construct). Ordinabulous!
In this example, we generate ten stages of von Neumann ordinals symbolically, where each stage references the previous ones. The empty set {} is equal to 0 by definition. Then each next ordinal is obtained by referencing the previous ordinal and adding it to the currently constructed set. So, 1 is {0}, 2 is {0, 1}, 3 is {0, 1, 2}, and so on.
This example shows the first five ordinals written in the expanded von Neumann notation using sets and subsets. The empty set (∅) is the first element of this sequence and is equal to 0. The next element is 1 and it's equal to a set of all previous sets, which is a set of ∅, which equals {∅}. The next ordinal 2 is again a set of all previous sets and equals {∅, {∅}}, etc. To show what can be customized, we changed the set element separator from a comma to a semicolon.
In this example, we use custom set symbols and generate levels one to six (total of five) of von Neumann construction. We enter the glyph "◈" in the empty set field, glyph "<" in the open set symbol field, and glyph ">" in the close set symbol field, and leave the set element separator field empty that merges all set elements together. In the output, we get beautiful von Neumann patterns.
You can pass options to this tool using their codes as query arguments and it will automatically compute output. To get the code of an option, just hover over its icon. Here's how to type it in your browser's address bar. Click to try!
Quickly rewrite integer digits using pretty Unicode glyphs.
Quickly generate random two-dimensional integer points.
Quickly generate random three-dimensional integer points.
Quickly print a list of integers that are palindromes.
Quickly convert the given integer to a palindromic integer.
Quickly find all integers in the given text or string.
Quickly make integer digits go in a triangle wave.
Quickly unleash the destructive force of Zalgo on integers.
Quickly find two or more integers that add up to the given integer.
Quickly find several integers that multiply up to the given integer.
Quickly randomize the given sequence of integers.
Quickly mix the order of digits in an integer.
Quickly put integers in increasing or decreasing order.
Quickly sort the individual digits of an integer.
Quickly increase all integers by a certain amount.
Quickly increase individual digits by a certain amount.
Quickly decrease all integers by a certain amount.
Quickly decrease individual digits by a certain amount.
Quickly randomly select an integer from a list of integers.
Quickly create groups from the digits of an integer.
Quickly write integer digits backwards.
Quickly remove one or more digits from an integer.
Quickly replace one or more digits in an integer.
Quickly drop digits from an integer and make it shorter.
Quickly add the thousands separator to integers.
Quickly remove the thousands separator from integers.
Quickly find the largest integer in a list of integers.
Quickly find the smallest integer in a list of integers.
Quickly find the total sum of a bunch of integers.
Quickly find the sum of all the digits in an integer.
Quickly find the total product of a bunch of integers.
Quickly find the product of all the digits in an integer.
Quickly calculate the partial sum of a bunch of integers.
Quickly calculate the partial product of a bunch of integers.
Quickly flip the sign of one or more integers.
Quickly split an integer into chunks of a given length.
Quickly merge partial integers together to form a larger integer.
Quickly round integers up or down to the required precision.
Quickly convert integers to a downloadable picture.
Quickly generate a list of von Neumann ordinals.
Quickly generate a list of Zermelo ordinals.
Quickly generate an ordered sequence of integers.
Quickly generate an ordered list of even integers.
Quickly generate an ordered list of odd integers.
Quickly write an integer spelling in words.
Quickly convert spelled integers to regular integers with digits.
Quickly generate an array of integers for programming.
Quickly find the number of occurrences of digits in integers.
Create a drawing that visualizes von Neumann hierarchy of sets.
Create a sudoku puzzle.
Create a list of neat-looking integers (called magic integers).
Generate a list of tuples of integers with n elements.
Quickly convert integers to base one.
Quickly convert base one to integers.
Quickly convert integers to base two.
Quickly convert base two to integers.
Quickly convert integers to base eight.
Quickly convert base eight to integers.
Quickly convert integers to base sixteen.
Quickly convert base sixteen to integers.
Quickly encode integers to base-64.
Quickly decode base-64 to integers.
Quickly convert integers to a custom base.
Quickly encode integers to HTML encoding.
Quickly decode HTML entities to integers.
Quickly encode integers to URL (percent) encoding.
Quickly decode URL-encoded integers.
Quickly convert a signed integer to an unsigned integer.
Quickly convert an unsigned integer to a signed integer.
Generate a list of random integers.
Check if the given integers are palindromes.
Create a matrix whose entries are all integers.
Create a vector with integer coefficients.
Quickly calculate the average value of integers.
Quickly calculate the average value of integer digits.
Quickly randomly select a digit from an integer.
Find which of the given integers is the biggest or smallest.
Limit integer values to a range.
Limit integer digit values to a range.
Create multiple copies of the input integers.
Create multiple copies of digits of input integers.
Rotate the digits of an integer to the left or right.
Move the digits of an integer to the left or right.
Quickly find the difference of a bunch of integers.
Quickly apply the bitwise AND operation to integers.
Quickly apply the bitwise OR operation to integers.
Quickly apply the bitwise XOR operation to integers.
Quickly apply the bitwise NOT operation to integers.
Quickly apply the bitwise NAND operation to integers.
Quickly apply the bitwise NOR operation to integers.
Quickly apply the bitwise NXOR operation to integers.
Quickly divide two or more integers.
Quickly divide the digits of an integer.
Add -st, -nd, -rd, -th suffixes to integers to make them ordinals.
Remove -st, -nd, -rd, -th suffixes from ordinals to make them ints.
Find integers that match a filter (greater, less, equal).
Add padding to integers on the left side.
Add padding to integers on the right side.
Position all integers so that they align on the right.
Position all integers so that they align in the middle.
Turn all integers into positive integers.
Turn all integers into negative integers.
Rewrite an integer in fractional form.
Extract the numerator and denominator from a fraction.
Search for all occurrences of an integer and replace it.
Create a regex that matches the given integers.
Create integers that match the given regular expression.
Create relatively tiny integers.
Create relatively huge integers.
Create a sequence of oscillating integers, such as 123212321.
Create multiple integer sequences at once.
Slightly change an integer so it has an error.
Slightly change integer digits so there are errors.
Apply fuzzing to integers and add perturbations.
Apply fuzzing to integer digits and add digit perturbations.
Add highlighting to certain integers.
Add highlighting to certain integer digits.
Add color to integers based on a condition.
Add color to individual digits in the given integers.
Quickly assign colors to integers and draw them as pixels.
Quickly assign integer values to pixel colors and print them.
Make the digits of an integer go in a spiral shape.
Make the digits of an integer go in a circle.
Make the digits of an integer go in a diamond shape.
Fill a box with certain width and height with digits.
Use ASCII art to convert integers to 2-dimensional drawings.
Use ASCII art to convert integers to 3-dimensional drawings.
Decompose an integer into ones, tens, hundreds, etc.
Generate an ordered list of increasing integers.
Generate an ordered list of decreasing integers.
Quickly find various information about the given integers.
Find hidden patterns of numbers in integers.
Find the Shannon entropy of an integer.
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