Building Natural numbers from Sets with python
Jupyter Note book python implementation using default set data structure
We can axiomatically build the Set-interpretation of natural numbers by recursively nesting sets within each other.
\[0 := \{ \} = \emptyset \] \[1:= 0 \cup \{ 0 \} = \{ 0 \} = \{ \{ \} \} \] \[2 := 1 \cup \{1 \} = \{ 0, 1 \} = \{ 0, \{ 0 \} \} = \{ \{ \} ,\{ \{ \} \} \}\] \[3:= 2 \cup \{2 \} = \{ 0, 1, 2 \} = \{ 0, 1 \} \cup \{ \{ 0, 1 \} \} = \{ \{ \},\{ \{ \} \} , \{ \{ \} ,\{ \{ \} \} \} \} \]
0.0.0.1 General Form
\[ N := {\color{blue}{N-1}} \cup {\color{red} \{ {N-1} \}} = \{ {\color{blue}0,1,2...} {\color{red}N-1} \} \]
Given a goal to define the Set-interpretation of the k natural
We take the Set-interpretation of the (k-1) natural, which we have previously defined, and union it with itself but nested inside a set.
Another way to look at it.
To define the k natural, we form a set that contains all the Set-interpretation of numbers from 0 to k-1.
Given the above knowledge we can define a successor function: \[ Succ(K) = K \cup \{ K \} \]
Notice that since the k natural contains all the numbers,(0,1,2…k-1) before it,
we can define a Predecessor function by simply fetching the Maximum element inside the Set-interpretation of K.
\[ Pred(K) = max(\{ x | x \in K \}) \]
