PDFnextuppreviousThe 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}.
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 axiom^{2} says a pair of sets x and y are equal
if and only if they have exactly the same members.
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.
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.
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
A_{n}(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
∧A_{n}(x,y)])]
[ ∀x
∃ ! y A_{n}(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 A_{n}(x,y) is
true. One can think of A_{n}(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 A_{n} 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 A_{n}(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 Paradox^{4}. 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.
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.
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.
Paul J. Cohen. Set Theory and the Continuum
Hypothesis. W. A. Benjamin Inc., New York, Amsterdam,
1966.
Footnotes:
^{1}The 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.
^{2}a ≡ 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).
^{3}The `¬' 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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