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Zermelo Ordinal 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.

᠎᠎᠎          Tool Options

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.

What Is a Zermelo Ordinal 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 Ordinal Generator Examples

Click to try!

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://onlinetools.com/integer/generate-zermelo-ordinals?subset-ordinals=true&start=0&count=9&open-symbol=%257B&close-symbol=%257D&empty-symbol=%25E2%2588%2585

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Quickly convert integers to base one.

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Quickly convert base one to integers.

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Quickly convert integers to base two.

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Quickly convert integers to base eight.

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