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.

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!

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!

Click to try!

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### Symbolic References

**Required options**

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.

{}
{0}
{0, 1}
{0, 1, 2}
{0, 1, 2, 3}
{0, 1, 2, 3, 4}
{0, 1, 2, 3, 4, 5}
{0, 1, 2, 3, 4, 5, 6}
{0, 1, 2, 3, 4, 5, 6, 7}
{0, 1, 2, 3, 4, 5, 6, 7, 8}

Generate ordinals symbolically
by referencing previous ordinals.

Starting ordinal.

Number of ordinals.

Set element separator symbol.

Open set symbol.

Close set symbol.

Empty set symbol.

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### Expanded Sets

**Required options**

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.

∅
{∅}
{∅; {∅}}
{∅; {∅}; {∅; {∅}}}
{∅; {∅}; {∅; {∅}}; {∅; {∅}; {∅; {∅}}}}

Generate ordinals recursively
by combining sets of subsets.

Starting ordinal.

Number of ordinals.

Set element separator symbol.

Open set symbol.

Close set symbol.

Empty set symbol.

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### Von Neumann Ornaments

**Required options**

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.

<◈>
<◈<◈>>
<◈<◈><◈<◈>>>
<◈<◈><◈<◈>><◈<◈><◈<◈>>>>
<◈<◈><◈<◈>><◈<◈><◈<◈>>><◈<◈><◈<◈>><◈<◈><◈<◈>>>>>

Generate ordinals recursively
by combining sets of subsets.

Starting ordinal.

Number of ordinals.

Set element separator symbol.

Open set symbol.

Close set symbol.

Empty set symbol.

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-von-neumann-ordinals?natural-ordinals=true&start=0&count=10&separator=%252C%20&open-symbol=%257B&close-symbol=%257D&empty-symbol=

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