 # Zermelo ordinals generator

## World's simplest integer tool

This online utility generates integers through Zermelo ordinals and prints a sequence of sets in the output. You can construct the full expansion of Zermelo ordinals using sets or abbreviated expansion using references to previous natural numbers. You can also specify the starting ordinal and the number of recursions. The ordinal output format can be customized as well by changing the set symbols. Created by math nerds from team Browserling.
a new math project! announcement
Love math as much as we do? Then you'll also love our new project – MATHURLS – a super neat math news aggregator. Check it out!
fullscreen fullscreen_exit
Type of Zermelo Ordinals
Create ordinals by wrapping the previous set in a set.
Create ordinals by wrapping the previous number in a set.
Start and Cardinality
Start Zermelo sequence from this number.
How many Zermelo sets to generate?
Customize Set Format
Enter left set character.
Enter right set character.
Enter character for the set with no elements.
Zermelo ordinals generator tool What is a zermelo ordinals generator?
With this browser-based application, you can create a list of Zermelo ordinals, which are similar to the Von Neumann ordinals as you can find a bijection between both of them. The Zermelo ordinals are created by recursively adding the set of the preceding integer to a new empty set. There are two ways to write Zermelo ordinals. The first follows the recursive definition. It defines 0 to be the empty set ∅. Then, each next number n is constructed by wrapping the previous number n-1 in a set n = {n-1}. The first few levels of Ernst Zermelo's construction goes as follows: 0 = ∅, 1 = {∅}, 2 = {{∅}}, 3 = {{{∅}}}, …. The other way to write Zermelo ordinals is to use natural numbers as set elements. Zero is defined to be {} and each next number n is simply a set of the previous number {n-1}. The first few levels of this notation are as follows: 0 = {}, 1 = {0}, 2 = {1}, 3 = {2}, …. You can generate as many ordinals as you want by specifying the required number of output items in the options. It is not necessary to generate the sequence from the zeroth element and you can specify any starting integer. You can also customize the notation of Zermelo ordinals. In particular, you can specify the left set character (by default, it's an open curly bracket), the right set character (by default, it's a close curly bracket), and the empty set element (by default, it's a crossed zero ∅). Zermelabulous!
Zermelo ordinals generator examples Click to use
Wrapped Sets
In this example, we construct Zermelo ordinals by wrapping the empty set in more sets. Each ordinal stage is generated by creating a set from the set at the previous stage. We output nine ordinals from zero to eight. Zero is ∅ and eight is ∅ in eight sets {{{{{{{{∅}}}}}}}}.
∅ {∅} {{∅}} {{{∅}}} {{{{∅}}}} {{{{{∅}}}}} {{{{{{∅}}}}}} {{{{{{{∅}}}}}}} {{{{{{{{∅}}}}}}}}
Required options
These options will be used automatically if you select this example.
Create ordinals by wrapping the previous set in a set.
Start Zermelo sequence from this number.
How many Zermelo sets to generate?
Enter left set character.
Enter right set character.
Enter character for the set with no elements.
Wrapped Natural Numbers
This example creates a sequence of Zermelo numbers via positive integers. Zero is defined to be {} and each next integer is the previous integer wrapped in a single set. So, zero is {}, one is {0}, two is {1}, and so on.
{} {0} {1} {2} {3} {4} {5} {6}
Required options
These options will be used automatically if you select this example.
Create ordinals by wrapping the previous number in a set.
Start Zermelo sequence from this number.
How many Zermelo sets to generate?
Enter left set character.
Enter right set character.
Enter character for the set with no elements.
Custom Sets
In this example, we change the set notation. We use square brackets instead of curly brackets to open and close a set. We also use the star symbol for the set with no elements. The generated ordinals start from the 3rd ordinal and six more are printed.
[[[*]]] [[[[*]]]] [[[[[*]]]]] [[[[[[*]]]]]] [[[[[[[*]]]]]]] [[[[[[[[*]]]]]]]]
Required options
These options will be used automatically if you select this example.
Create ordinals by wrapping the previous set in a set.
Start Zermelo sequence from this number.
How many Zermelo sets to generate?
Enter left set character.
Enter right set character.
Enter character for the set with no elements.
Pro tips Master online integer tools
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!
https://onlineintegertools.com/generate-zermelo-ordinals?&subset-ordinals=true&start=0&count=9&open-symbol=%7B&close-symbol=%7D&empty-symbol=%E2%88%85
All integer tools
Didn't find the tool you were looking for? Let us know what tool we are missing and we'll build it!
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 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.
Coming soon These integer tools are on the way
Draw von Neumann Universe
Create a drawing that visualizes von Neumann hierarchy of sets.
Create Integer n-Tuples
Generate a list of tuples of integers with n elements.
Convert Integers to Unary Base
Quickly convert integers to base one.
Convert Unary Base to Integers
Quickly convert base one to integers.
Convert Integers to Binary Base
Quickly convert integers to base two.
Convert Binary Base to Integers
Quickly convert base two to integers.
Convert Integers to Octal Base
Quickly convert integers to base eight.
Convert Octal Base to Integers
Quickly convert base eight to integers.
Convert Integers to Hex Base
Quickly convert integers to base sixteen.
Convert Hex Base to Integers
Quickly convert base sixteen to integers.
Convert Integers to Any Base
Quickly convert integers to a custom base.
Convert Integers to Base64
Quickly encode integers to base-64.
Convert Base64 to Integers
Quickly decode base-64 to integers.
Convert Integers to HTML Entities
Quickly encode integers to HTML encoding.
Convert HTML Entities to Integers
Quickly decode HTML entities to integers.
Create Random Integers
Generate a list of random integers.
Check if an Integer is Symmetric
Check if the given integers are palindromes.
Generate an Integer Matrix
Create a matrix whose entries are all integers.
Generate an Integer Vector
Create a vector with integer coefficients.
Rotate Integers
Rotate the digits of an integer to the left or right.
Shift Integer Digits
Move the digits of an integer to the left or right.
Subtract Integers
Quickly find the difference of a bunch of integers.
AND Integers
Quickly apply the bitwise AND operation to integers.
OR Integers
Quickly apply the bitwise OR operation to integers.
XOR Integers
Quickly apply the bitwise XOR operation to integers.
NOT Integers
Quickly apply the bitwise NOT operation to integers.
NAND Integers
Quickly apply the bitwise NAND operation to integers.
NOR Integers
Quickly apply the bitwise NOR operation to integers.
XNOR Integers
Quickly apply the bitwise NXOR operation to integers.
Convert an Integer to Ordinal
Add -st, -nd, -rd, -th suffixes to integers to make them ordinals.
Convert an Oridinal to Integer
Remove -st, -nd, -rd, -th suffixes from ordinals to make them ints.
Filter Integers
Find integers that match a filter (greater, less, equal).
Right-align Integers
Position all integers so that they align on the right.
Center Integers
Position all integers so that they align in the middle.
Take Absolute Value of Integers
Turn all integers into positive integers.
Make All Integers Negative
Turn all integers into negative integers.
Convert Integers to Fractions
Rewrite an integer in fractional form.
Convert Fractions to Integers
Extract the numerator and denominator from a fraction.
Find and Replace Integers
Search for all occurrences of an integer and replace it.
Generate Integers from Regex
Create integers that match the given regular expression.