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.

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!

Click to try

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 {{{{{{{{∅}}}}}}}}.

∅
{∅}
{{∅}}
{{{∅}}}
{{{{∅}}}}
{{{{{∅}}}}}
{{{{{{∅}}}}}}
{{{{{{{∅}}}}}}}
{{{{{{{{∅}}}}}}}}

Create ordinals by wrapping

the previous set in a set.

the previous set in a set.

Start Zermelo sequence from

this number.

this number.

How many Zermelo sets

to generate?

to generate?

Enter left set character.

Enter right set character.

Enter character for the

set with no elements.

set with no elements.

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}

Create ordinals by wrapping

the previous number in a set.

the previous number in a set.

Start Zermelo sequence from

this number.

this number.

How many Zermelo sets

to generate?

to generate?

Enter left set character.

Enter right set character.

Enter character for the

set with no elements.

set with no elements.

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.

[[[*]]]
[[[[*]]]]
[[[[[*]]]]]
[[[[[[*]]]]]]
[[[[[[[*]]]]]]]
[[[[[[[[*]]]]]]]]

Create ordinals by wrapping

the previous set in a set.

the previous set in a set.

Start Zermelo sequence from

this number.

this number.

How many Zermelo sets

to generate?

to generate?

Enter left set character.

Enter right set character.

Enter character for the

set with no elements.

set with no elements.

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 add the thousands separator to integers.

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Quickly remove the thousands separator from integers.

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Quickly find the largest integer in a list of integers.

Find the Min Integer

Quickly find the smallest integer in a list of integers.

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Quickly find the total sum of a bunch of integers.

Add Integer Digits

Quickly find the sum of all the digits in an integer.

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Multiply Integer Digits

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Quickly calculate the partial sum of a bunch of integers.

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Quickly flip the sign of one or more integers.

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Quickly split an integer into chunks of a given length.

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Quickly merge partial integers together to form a larger integer.

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Quickly round integers up or down to the required precision.

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Quickly generate a list of Zermelo ordinals.

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Quickly generate an ordered sequence of integers.

Print Even Integers

Quickly generate an ordered list of even integers.

Print Odd Integers

Quickly generate an ordered list of odd integers.

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Created with love by

51K

@browserling

51K

@browserling

We're Browserling — a friendly and fun cross-browser testing company powered by alien technology. At Browserling our mission is to make people's lives easier, so we created this collection of integer tools. Our tools have the simplest user interface that doesn't require advanced computer skills and they are used by millions of people every month. Our integer tools are actually powered by our programming tools that we created over the last couple of years. Check them out!

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Quickly randomly select a digit from an integer.

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Limit integer values to a range.

Clamp Integer Digits

Limit integer digit values to a range.

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Duplicate Integer Digits

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Rotate Integers

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AND Integers

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OR Integers

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XOR Integers

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NOT Integers

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NAND Integers

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NOR Integers

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XNOR Integers

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Divide Integer Digits

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Center Integers

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Make All Integers Negative

Turn all integers into negative integers.

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Rewrite an integer in fractional form.

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Generate Big Integers

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Create multiple integer sequences at once.

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Make Integers Digits Fuzzy

Apply fuzzing to integer digits and add digit perturbations.

Highlight Integers

Add highlighting to certain integers.

Highlight Integer Digits

Add highlighting to certain integer digits.

Color Integers

Add color to integers based on a condition.

Color Integer Digits

Add color to individual digits in the given integers.

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Convert Pixels to Integers

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Draw an Integer Circle

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Draw an Integer Diamond

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Create 3D Integers

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

Print Increasing Integers

Generate an ordered list of increasing integers.

Print Decreasing Integers

Generate an ordered list of decreasing integers.

Print Integer Information

Quickly find various information about the given integers.

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Find hidden patterns of numbers in integers.

Calculate Integer Entropy

Find the Shannon entropy of an integer.

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