Basics of Set Theory: Natural and Real Numbers

"Set theory started around 1870’s → late development in mathematics but now taught early in one’s maths education due to the Bourbaki school."
1. A set is a collection of objects. x ∈ A means the element x is in the set A (i.e. belongs to A).
  1. e.g. all students in a class
  2. N the set of natural numbers starting at 0
    1. 0 ∈ N
    2. if x ∈ N, then x + 1 ∈ N (x ∈ N → x + 1 ∈ N)
R is the set of real numbers. It is governed by the following axioms:
1. CLOSURE
  1. Additive
    1. ∀x, y ∃z(x + y = z)
  2. Multiplicative
    1. ∀x, y, ∃z(x × y = z)
2. Associativity
  1. Additive
    1. ∀x, y, z: x + (y + z) = (x + y) + z
  2. Multiplicative
    1. ∀x, y, z: x × (y × z) = (x × y) × z
3. Commutativity
  1. Additive
    1. ∀x, y: x + y = y + x
  2. Multiplicative
    1. ∀x, y: x × y = y × x
4. Distributivity
  1. ∀x, y, z: x × (y + z) = (x × y) + (x × z) and (y + z) × x = (y × x) + (z × x)
5. Identity
  1. Additive
    1. There is a number, denoted 0, such that for all x, x + 0 = x.
  2. Multiplicative
    1. There is a number, denoted 1, such that for all x, x * 1 = 1 * x = x.
6. Inverses
  1. Additive
    1. For every x there is a number, denoted -x, such that x + (-x) = 0
  2. Multiplicative
    1. For every nonzero x there is a number, denoted x^−1, such that (x * x^-1) = (x^-1 * x) = 1.
7. 0 != 1
8. Irreflexivity of <
  1. ~(x < x)
9. Transitivity of <
  1. If x < y and y < z, then x < z
10. Trichotomy
  1. Either x < y, y < x, or x = y
11. Completeness
  1. If a nonempty set of real numbers has an upper bound, then it has a least upper bound.
12. If x < y, then x + y < y + z.
  1. If x < y and 0 < z, then x * z < y * z and z * x < z * y.

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Basics of Set Theory: Natural and Real Numbers

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