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4 Axioms of ZFC

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The formalization of objectivity starts with the axioms of Zermelo Frankel Set Theory plus the axiom of choice ZFC, the most widely used formalization of mathematics. This is not the ideal starting point for formalizing objective mathematics but it is the best approach to clarify where in the existing mathematical hierarchy objective mathematics ends. To that end a restricted version of these axioms will be used to define an objective formalization of mathematics.
The following axioms are adapted from Set Theory and the Continuum Hypothesis[1]1.

4.1  Axiom of extensionality

Without the axiom that defines when two sets are identical (=) there would be little point in defining the integers or anything else. The axiom of extensionality says sets are uniquely defined by their members.

∀x ∀y    (∀z    z ∈ x ≡ z ∈ y) ≡ (x=y)
This axiom2 says a pair of sets x and y are equal if and only if they have exactly the same members.

4.2  Axiom of the empty set

The empty set must be defined before any other set can be defined.
The axiom of the empty set is as follows.

∃x ∀y    ¬(y ∈ x)
This axiom3 says there exists an object x that no other set belongs to. x contains nothing. The empty set is denoted by the symbol ∅.

4.3  Axiom of unordered pairs

From any two sets x and y one can construct a set that contains both x and y. The notation for that set is {x,y}.

∀x ∀y   ∃z  ∀w   w ∈ z ≡ ( w = x ∨w = y )
This says for every pair of sets x and y there exists a set w that contains x and y and no other members. This is written as w = x ∪y.

4.4  Axiom of union

A set is an arbitrary collection of objects. The axiom of union allows one to combine the objects in many different sets and make them members of a single new set. It says one can go down two levels taking not the members of a set, but the members of members of a set and combine them into a new set.

∀x ∃y   ∀z   z ∈ y ≡ (∃t   z ∈ t ∧t ∈ x)
This says for every set x there exists a set y that is the union of all the members of x. Specifically, for every z that belongs to the union set y there must be some set t such that t belongs to x and z belongs to t.

4.5  Axiom of infinity

The integers are defined by an axiom that asserts the existence of a set ω that contains all the integers. ω is defined as the set containing 0 and having the property that if n is in ω then n+1 is in ω. From any set x one can construct a set containing x by constructing the unordered pair of x and x. This set is written as {x}.

∃x  ∅ ∈ x ∧[∀y  (y ∈ x) →(y ∪{y} ∈ x)]
This says there exists a set x that contains the empty set ∅ and for every set y that belongs to x the set y+1 constructed as y ∪{y} also belongs to x.
The axiom of infinity implies the principle of induction on the integers.

4.6  Axiom scheme of replacement

The axiom scheme for building up complex sets like the ordinals is called replacement. They are an easily generated recursively enumerable infinite sequence of axioms.
The axiom of replacement scheme describes how new sets can be defined from existing sets using any relationship An(x,y) that defines y as a function of x. A function maps any element in its range (any input value) to a unique result or output value.
∃ ! y g(y) means there exists one and only one set y such that g(y) is true. The axiom of replacement scheme is as follows.
B(u,v) ≡ [ ∀y (y ∈ v ≡ ∃x [ x ∈ u ∧An(x,y)])]

[ ∀x ∃ ! y An(x,y) ] → ∀u ∃v (B(u,v))
That first line defines B(u,v) as equivalent to y ∈ v if and only if there exists an x ∈ u such that An(x,y) is true. One can think of An(x,y) as defining a function that may have multiple values for the same input. B(u,v) says v is the image of u under this function.
This second line says if An defines y uniquely as a function of x then the for all u there exists v such that B(u.v) is true.
This axioms says that, if An(x,y) defines y uniquely as a function of x, then one can take any set u and construct a new set v by applying this function to every element of u and taking the union of the resulting sets.
This axiom schema came about because previous attempts to formalize mathematics were too general and led to contradictions like the Barber Paradox4. By restricting new sets to those obtained by applying well defined functions to the elements of existing sets it was felt that one could avoid such contradictions. Sets are explicitly built up from sets defined in safe axioms. Sets cannot be defined as the universe of all objects satisfying some relationship. One cannot construct the set of all sets which inevitably leads to a paradox.

4.7  Power set axiom

The power set axiom says the set of all subsets of any set exists. This is not needed for finite sets, but it is essential to define the set of all subsets of the integers.

∀x ∃y ∀z [ z ∈ y ≡ z ⊆ x ]
This says for every set x there exists a set y that contains all the subsets of x. z is a subset of x (z ⊆ x) if every element of z is an element of x.
The axiom of the power set completes the axioms of ZF or Zermelo Frankel set theory. From the power set axiom one can conclude that the set of all subsets of the integers exists. From this set one can construct the real numbers.
This axioms is necessary for defining recursive ordinals which is part of objective mathematics. At the same time it allows for questions like the continuum hypothesis that are relative. Drawing the line between objective and relative properties is tricky.

4.8  Axiom of Choice

The Axiom of Choice is not part of ZF. It is however widely accepted and critical to some proofs. The combination of this axiom and the others in ZF is called ZFC.
The axiom states that for any collection of non empty sets C there exists a choice function f that can select an element from every member of C. In other words for every e ∈ C f(e) ∈ e.

∀C ∃f ∀e [ (e ∈ C ∧e ≠ ∅) →f(e) ∈ e]

4.9  The axioms of ZFC summary

  1. Axiom of extensionality (See Section 4.1).
    ∀x ∀y    (∀z    z ∈ x ≡ z ∈ y) ≡ (x=y)
  2. Axiom of the empty set (See Section 4.2).
    ∃x ∀y    ¬(y ∈ x)
  3. Axiom of unordered pairs (See Section 4.3).
    ∀x ∀y   ∃z  ∀w   w ∈ z ≡ ( w = x ∨w = y )
  4. Axiom of union (See Section 4.4).
    ∀x ∃y   ∀z   z ∈ y ≡ (∃t   z ∈ t ∧t ∈ x)
  5. Axiom of infinity (See Section 4.5).
    ∃x  ∅ ∈ x ∧[∀y  (y ∈ x) →(y ∪{y} ∈ x)]
  6. Axiom schema of replacement (See Section 4.6).
    B(u,v) ≡ [ ∀y (y ∈ v ≡ ∃x [ x ∈ u ∧An(x,y)])]

    [ ∀x ∃ ! y An(x,y) ] → ∀u ∃v (B(u,v))
  7. Axiom of the power set (See Section 4.7).
    ∀x ∃y ∀z [ z ∈ y ≡ z ⊆ x ]
  8. Axiom of choice (See Section 4.8).
    ∀C ∃f ∀e [ (e ∈ C ∧e ≠ ∅) →f(e) ∈ e]
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References

[1]
Paul J. Cohen. Set Theory and the Continuum Hypothesis. W. A. Benjamin Inc., New York, Amsterdam, 1966.

Footnotes:

1The axioms use the existential quantifier (∃) and the universal quantifier (∀). ∃x g(x) means there exists some set x for which g(x) is true. Here g(x) is any expression that includes x. ∀x g(x) means g(x) is true of every set x.
2a ≡ b means a and b have the same truth value or are equivalent. They are either both true or both false. It is the same as (a→ b)∧(b → a).
3The `¬' symbol says what follow is not true.
4 The barber paradox concerns a barber who shaves everyone in the town except those who shave themselves. If the barber shaves himself then he must be among the exceptions and cannot shave himself. Such a barber cannot exist.


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