# 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.

We created a cloud browser!
Browserling

Check out our project Browserling – anonymous cloud browser.

A link to this tool, including input, options and all chained tools.

Can't convert.

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**

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**

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**

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

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.

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 Base64

Quickly encode integers to base-64.

Convert Base64 to Integers

Quickly decode base-64 to integers.

Change Integer Base

Quickly convert integers to a custom base.

Convert Integers to HTML Entities

Quickly encode integers to HTML encoding.

Convert HTML Entities to Integers

Quickly decode HTML entities to integers.

Convert Integers to URL Encoding

Quickly encode integers to URL (percent) encoding.

Convert URL Encoding to Integers

Quickly decode URL-encoded integers.

Convert Signed Integer to Unsigned

Quickly convert a signed integer to an unsigned integer.

Convert Unsigned Integer to Signed

Quickly convert an unsigned integer to a signed integer.

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.

Find the Average

Quickly calculate the average value of integers.

Find the Digit Average

Quickly calculate the average value of integer digits.

Clamp Integers

Limit integer values to a range.

Clamp Integer Digits

Limit integer digit values to a range.

Duplicate Integers

Create multiple copies of the input integers.

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).

Left-pad Integers

Add padding to integers on the left side.

Right-pad Integers

Add padding to integers on the right side.

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 a Regex

Create integers that match the given regular expression.

Generate Small Integers

Create relatively tiny integers.

Generate Big Integers

Create relatively huge integers.

Generate Oscillating Integers

Create a sequence of oscillating integers, such as 123212321.

Color Integers

Add color to integers based on a condition.

Color Integer Digits

Add color to individual digits in the given integers.

Draw an Integer Spiral

Make the digits of an integer go in a spiral shape.

Draw an Integer Circle

Make the digits of an integer go in a circle.

Draw an Integer Box

Fill a box with certain width and height with digits.

Create 2D Integers

Use ASCII art to convert integers to 2-dimensional drawings.

Create 3D Integers

Use ASCII art to convert integers to 3-dimensional drawings.

Decompose an Integer

Decompose an integer into ones, tens, hundreds, etc.

Find Patterns in Integers

Find hidden patterns of numbers in integers.

Subscribe!
Never miss an update

Cool!

Notifications
We'll let you know when we add this tool

Cool!