Selected items from the Introduction to set theory and to methodology and philosophy of mathematics and computer programming.

1. Introduction to set theory and to methodology and philosophy of
mathematics and computer programming
Von Neumann natural numbers
An overview
by Jan Plaza
c 2017 Jan Plaza
Use under the Creative Commons Attribution 4.0 International License
Version of February 5, 2017

2. Definition
The successor operation is defined as successor(n) = n ∪ {n}
The set of von Neumann natural numbers , denoted ω , is defined as follows.
1. ∅ ∈ ω
2. If n ∈ ω then successor(n) ∈ ω.
3. Nothing belongs to ω unless it can be constructed using the preceding rules.

3. Definition of numerals
0 = ∅,
1 = successor(0) = ∅ ∪ {∅} = {∅} = {0},
2 = successor(1) = {∅} ∪ {{∅}} = {∅, {∅}} = {0, 1},
3 = successor(2) = {∅, {∅}} ∪ {{∅, {∅}}} = {∅, {∅}, {∅, {∅}}} = {0, 1, 2},
...
So, ω = {0, 1, 2, ...} )
Exercise
List von Neumann natural numbers 4, 5 and 6.

4. Programming Exercise (optional, extra credit)
Write a program that takes as input an integer and output the corresponding von
Neumann integer.
Programming Exercise (optional, extra credit)
Write a program that takes as input a string of left and right curly braces and outputs
the corresponding integer in the decimal system or a message “not an integer”.