Download Models of Set Theory

Models of Set Theory
Benjamin Cohen Wallace
December 31, 2012
1
Introduction
The early 20th century saw a great surge of activity amongst mathematicians
and philosophers who were trying to ground mathematics on a solid foundation. This situation, known today as the foundational crisis, arose due to disagreements about the nature of mathematics and mathematical truth, along
with the discovery of several disturbing paradoxes in mathematics. Perhaps
the most famous of these is Russell’s paradox, named after Bertrand Russell,
which addresses the issue of the set that contains all sets which do not contain
themselves.
One of the leading mathematicians of the time, David Hilbert, brought these
issues to the forefront by calling out for a complete and consistent axiomatization
of mathematics. The developments that followed had great impact on the way
that mathematics is presently understood.
Modern mathematicians commonly see mathematical objects as being defined in terms of sets. The definition of “set” itself has been reduced to a
collection of commonly accepted “self-evident truths”, or axioms, in the same
way that Euclid axiomatized geometry. The theory built up from these axioms
is known as Zermelo-Fraenkel set theory plus the axiom of choice, or ZFC. Other
set theories have been proposed, but this is by far the most commonly accepted
one, and the one we shall be working with.
Mathematicians pursue understanding in the form of theorems, which are
proven from axioms via formal rules of deduction. The underlying belief is that
if the axioms are true, and if the rules make sense, then theorems obtained in
this way, too, should be true. We will discuss these topics in some more detail
later on.
Several techniques and methodologies for studying the aforementioned concepts were developed in the first half of the 20th century. Today, many of these
have developed into fields of their own. Model theory is amongst these, and
is used principally to determine the consistency of axiomatic systems. In the
following paper, we explain how model theory can be used to study the consistency of ZFC set theory. But first, we must introduce the fundamentals of
mathematical logic.
1
2
Logic
Logic is one of the main pillars of the foundations of mathematics. By formal
logic is meant the strict codification of rules that are generally accepted as
true. While there are many different variants of formal logic, the one most
used by mathematicians, deliberately or not, is known as first-order logic or
the first-order predicate calculus. There is a very good reason for this that will
be discussed in the next section. For now, let us give a brief, non-rigorous
treatment of this logic.
A formal system consists of a language, some axioms, and rules of inference or rules of deduction. The language consists of symbols, finitely many of
which can be combined into strings or expressions. These strings are considered
meaningful if the combinations respect a formal grammar. Meaningful strings
are called formulas, but we will also refer to them as statements or propositions.
All axioms must be formulas in the language of the system. We will discuss rules
of inference in the specific case of the first-order logic. First, let us introduce
its language.
The language of first-order logic is a type of first-order language. First-order
languages consist of the following symbols:
∀, ∃;
∧, ∨, →, ↔, ¬;
(, );
=.
(quantifiers)
(logical connectives)
(parentheses)
(equality)
In addition, we may use any number of variables, n-ary function symbols, and
n-ary predicate symbols, represented by letters.
For brevity, we will not give a detailed description of the grammar of firstorder logic. An example of one of the rules of this grammar is that any quantifier
must immediately be followed by a variable. In fact, when this occurs, we say
that this variable is bound. Any variables appearing within a formula that
are not bound are free. Formulas with no free variables will be referred to as
sentences.
The purpose of all these formalities is the eventual reduction of mathematics
to formal logic. There are, however, many mathematical symbols that are not
given in the above list. One can convince oneself that most of these symbols
may be defined in terms of the above logical alphabet. We will see later that
there is another crucial mathematical symbol that cannot be defined logically.
Of course, we wish to apply the predicate calculus to the discovery of new
truths from the old. We do this by proving the new truths via the rules of
inference. These rules are highly specific instructions as to how the symbols in
a string may be manipulated. For example, suppose p is a formula, and suppose
p is true (we usually say instead, “suppose p”). Then given any formula q,
we can safely say that p ∨ q (p or q). While most of the time, such a simple
fact is taken for granted, in formal logic, this deduction follows from the direct
application of a specific rule.
2
It is important to understand these rules as being grammatical in nature.
They dictate “thoughtless” symbolic manipulation. It is in this sense that formal
logic is driven by the study of syntax.
We can now define a proof of the formula φ from the formula set Φ as a
finite sequence of formulas ending in φ, each of which is either logically deduced
from a previous one or is an element of Φ. All formulas of Φ may be introduced
at will into the sequence (they behave as axioms). We remind the reader that
these proofs study syntactical truth. Methods for investigating the semantics
underlying these symbols are to be discussed in the following section.
3
Model Theory
Model theory is concerned with the rigorous application of mathematics and
logic to the study of mathematical theories in and of themselves. It may be
quite puzzling, at first, how one should go about doing this. Things begin to
fall into perspective once an adequate definition of the subject’s main objects
of study has been given.
Definition. A theory is a set of sentences in a formal language. This formal
language is usually built off of a formal logic, but may extend the grammar
and symbol set. Roughly speaking, a first-order theory is a theory based in
first-order logic.
More formally, a first-order theory consists foremost of a first-order language,
along with the logical axioms in the language. Such a theory uses the predicate
calculus’s rules of inference. Finally, nonlogical axioms and nonlogical symbols
may be added to the theory. We will only deal with first-order theories.
In order to give meaning to the abstract symbols in a theory, we construct
a model for the theory. We will not define models here with complete rigour.
The study of models goes well beyond their use in set theory, and we will
only discuss the tools that we will borrow from the subject. A model for a
theory is a structure (a set along with some additional machinery) that satisfies
each sentence of the theory. Of course, we haven’t said yet what we mean by
satisfaction. First, we give the following definition.
Definition. The formula φ relativised to the model M, denoted by φM , is
obtained by restricting the domain of all bound variables in φ to the domain
M.
For example, relativising
∃x∀y(x ≤ y)
to M gives
∃x ∈ M ∀y ∈ M(x ≤ y).
We then say that a formula φ is satisfied by the model M if φM is true.
This is our notion of semantic truth. A set of formulas Φ is said to semantically
entail a formula φ if φ is true in every structure that models Φ. First-order logic
3
is said to be sound in that syntactic entailment (proof) always implies semantic
entailment.
In discussing a formal theory, we will really be referring to more than just a
simple list of axioms. All statements derivable from the theory are essentially
considered a part of it. An essential feature that we desire of these statements
is that they be consistent, which is to say that no contradictions exist within
the theory. A contradiction occurs when both φ and ¬φ, for some formula φ. A
theory which is inconsistent is of absolutely no use to us. Not only is it obvious
that some statements of this theory can’t be trusted, but by the principle of
explosion, which states that in an inconsistent theory, φ is true for all φ, all
statements derived from the theory are worthless. By the following theorem of
Gödel’s, model theory steps into play as our main tool for showing that a theory
is consistent.
Theorem 1 (Gödel’s Completeness Theorem). A first-order theory is consistent
if and only if it has a model. Equivalently, φ is true in a theory T if and only
if there is a proof of φ in T .
The fact that the above result has only been proven in the case of firstorder theories is an excellent reason for why most of mathematical theories are
first-order. The proof of this statement is beyond the scope of this paper, but
one may be able to see why this theorem is quite reasonable. If a theory is
inconsistent, then it is full of contradictions, and it would be impossible for a
structure to satisfy all of its axioms. On the other hand, it is nice to see that
one should always be able to model any consistent theory.
While Gödel has provided us with a wonderful tool for showing that certain
theories are consistent, this same great mathematician discovered two somewhat
discouraging counterparts to it. Before going any further, let us clarify that the
completeness theorem deals with a specific meaning of the word complete (ie.,
semantic truth and syntactic truth are equivalent in a first-order theory). This
same word has an entirely different meaning which is used in the incompleteness
theorems below. Here, we say that a theory is complete if all statements within
the theory have a truth value; that is, all statements are either true or false
within the theory.
Theorem 2 (Gödel’s First Incompleteness Theorem). Any effectively generated theory powerful enough to express elementary arithmetic cannot be both
consistent and complete.
Theorem 3 (Gödel’s Second Incompleteness Theorem). Any effectively generated theory powerful enough to express elementary arithmetic is able to prove
its own consistency if and only if it is inconsistent.
We shall not give a precise definition of the phrase “effectively generated.”
The reader may think of a theory as being effectively generated if there is a computer program that can generate the axioms and their implications. Going into
this any further would require a whole section on the theory of recursion, computability, and Turing machines. The main point is that Gödel’s incompleteness
4
theorems apply to a very broad class of “reasonable” theories, including ones
with infinitely many axioms.
An important consequence of the first incompleteness theorem is the inevitable existence of undecidable statements within consistent theories. A statement is said to be undecidable if it can neither be proved nor disproved within
the theory. Such statements may also be called independent. Moreover, the
theorem tells us that there are infinitely many such statements. In building an
axiomatic system, we wish to ensure that our axioms are all independent of each
other. Otherwise, the presence of some axioms would be redundant.
Turning to Gödel’s second incompleteness theorem, let us begin by assuring
the reader that the statement of his theorem does indeed make sense. It may at
first seem contradictory that there should be a proof that an inconsistent theory
is consistent. But let us remind the reader that, by the principle of explosion,
all statements are true in an inconsistent theory, and this includes the statement
of the theory’s own consistency1 .
Gödel’s second incompleteness theorem is perhaps even more important to
the particular subject of this paper: the consistency of set theory. Set theory,
being one of the primary foundations of all of mathematics is obviously capable
of expressing arithmetical truths. Therefore, this theorem tells us that if, as we
hope, set theory is in fact consistent, then it is impossible to prove it. Finding
a model of set theory would only prove that it is inconsistent. But then what
should we hope to possibly accomplish? We answer this question after the
following important theorem.
Theorem 4. Let T be a consistent theory and let S be some other theory.
Suppose, working in T , that we can prove that S has a model. Then S is
consistent.
Proof. If S were inconsistent, we could derive a contradiction from S. But then
we would really be deriving a contradiction from T , and so T would need to be
inconsistent.
Thus, our standard approach when trying to prove the consistency of powerful enough formal systems takes the following form. We begin with a theory
which, by Gödel, cannot be proved consistent, such as ZFC. We assume that
this theory is consistent. Then, working from these axioms, we derive results
about their consistency after adding some additional statements, such as the
continuum hypothesis (CH). This is known as proving relative consistency or
inconsistency. In this example, we would conclude by writing
Con(ZFC) =⇒ Con(ZFC + CH).
4
Some Facts about Set Theory
We now turn to the language of mathematics: set theory. We already mentioned that mathematics operates according to the rules of first-order logic.
1 We
are taking for granted the fact that this statement can even be formed.
5
But in order to actually talk about real mathematical objects, which have more
substance than logic alone can provide, we must enrich logic with the language
of set theory. We are adding the following (nonlogical) symbol to our alphabet:
∈.
We think of this symbol as inclusion, and the rules for its manipulation are
given by the (nonlogical) axioms of ZFC set theory. The full list of axioms
may be found in the appendix2 . Just as set theory is grounded in logic, it can
be seen that mathematics may be reduced to set theory. Whenever we use a
symbol that we have not yet introduced, we are working under the (verifiable)
belief that it can be defined in terms of the language of logic and set theory.
For example, the subset inclusion symbol may be defined as
x ⊂ y iff z ∈ x → z ∈ y.
Let us now recall some basic notions of set theory that we will require in the
following sections.
A set is said to be transitive if the ∈ relation is transitive with respect to
its elements. Equivalently, a set is transitive when all of its elements are also
subsets.
An extremely useful concept in building up models of set theory is the cumulative hierarchy, a collection of transitive sets that are strictly increasing with
respect to subset inclusion. These sets are denoted Vα and defined recursively
as follows.
V0 = 0
Vα+1 = P(Vα )
[
Vβ =
Vγ , Lim(β).
γ<β
We also let denote the union of all the Vα by V. However, V is not a set,
but a proper class.
Statements about the cumulative hierarchy and their proofs are greatly simplified by introducing the rank of a set within the hierarchy. We define the
rank of a set x, denoted rk(x), as the least ordinal α for which x ∈ Vα+1 , or
equivalently for which x ⊂ Vα . It is easy to see that every element y of x has
rank strictly less than rk(x). Less obvious are the following facts.
Lemma 5. For any set x,
(a) rk(x) = sup{rk(y) + 1 : y ∈ x},
(b) if x is an ordinal, then rk(x) = x, and
2 We will often refer to these axioms by a single word with the first letter capitlized (so
Infinity means the axiom of infinity)
6
(c) rk(P(x)) = rk(x) + 1.
Proof. We only prove (c). If rk(x) = α, then x ⊂ Vα , and so for each y ⊂ x,
y ∈ Vα+1 . But then P(x) = {y : y ⊂ x} ⊂ Vα+1 . Furthermore, if P(x) were a
subset of Vα , then x would be an element of Vα , and we would have rk(x) < α,
and so we are done.
In order to build our models in Section 6, the following is of absolute necessity.
Definition. The transitive closure of a set x is defined by
[
tr cl(x) = {∪n x : n ∈ ω},
where we use ∪n to denote the n-fold iterate of the union operation.
We now list some useful properties of transitive closure.
Lemma 6.
(a) If x ⊂ T and T is transitive, then tr cl(x) ⊂ T .
(b) If x ∈ y, then tr cl(x) ⊂ tr cl(y).
S
(c) tr cl(x) = x ∪ {tr cl(y) : y ∈ x}.
(d) tr cl(x ∪ y) = tr cl(x) ∪ tr cl(y)
Proof. We will prove (c) S
and (d).
For (c), let T = x ∪ {tr cl(y) : y ∈ x}. For any z ∈ T , either z ∈ x or
z ∈ tr cl(y), for some y ∈ x. In the latter case, z ⊂ tr cl(y) ⊂ T by transitivity.
If, on the other hand, z ∈ x, then we have z ⊂ tr cl(z) ⊂ T . Hence, T is
transitive, and so by (a), x ⊂ T ⇒ tr cl(x) ⊂ T . But by (b), T ⊂ tr cl(x), and
we are done.
Part (d) follows from (c):
[
tr cl(x ∪ y) = x ∪ y ∪ {tr cl(z) : z ∈ x ∪ y}
[
[
= x ∪ y ∪ {tr cl(z) : z ∈ x} ∪ {tr cl(z) : z ∈ y}
= tr cl(x) ∪ tr cl(y).
In order to construct our models of set theory later on, we will require
the notions of regular cardinals and of strongly inaccessible cardinals. We first
introduce the following definitions leading up to these concepts.
Definition.
(a) A subset A of an ordinal α is said to be unbounded in α if for every β < α,
there is a γ ∈ A such that β ≤ γ.
7
(b) A function f : β → α is cofinal in α if the range of f is unbounded in α.
(c) The cofinality of α, denoted cf(α), is the least β such that there is a cofinal
function from β to α.
We are now equipped to talk about the following:
Definition. A cardinal α is regular if cf(α) = α.
It is easily seen that the set of natural numbers ω is regular. We also provide
the following useful fact about regular cardinals.
Lemma 7. Let κ be a regular cardinal and let λ < κ be a cardinal. Given any
collection
{ρα : α < λ} of cardinals such that for each α < λ, ρα < κ. Then
S
ρ
<
κ.
α
α<λ
S
Proof. It is a simple fact that for any cardinal κ, α<λ ρα ≤ κ. Suppose we
have equality. Let f : λ → κ be defined by f (α) = ρα for each α < λ. We
show that this map is cofinal, contradicting the regularity of κ. Suppose to the
contrary that this map is not cofinal. That is, let β < κ and suppose ρα < β
for each α. Then, depending on whether β ≤ λ or β > λ,
[
ρα ≤ max{|β|, λ} < κ.
α<λ
But this is a contradiction, so we must have ρα ≥ β for some α.
Definition. A cardinal κ is said to be strongly inaccessible if it is uncountable,
regular, and for every λ < κ, 2λ < κ.
An important fact about these cardinals is that their existence is independent
of ZFC. That is, one cannot prove or disprove within ZFC set theory that such
cardinals exist. As we shall soon see, strongly inaccessible cardinals are deeply
entwined with the consistency of set theory.
5
Useful Lemmas on Models of Set Theory
This section develops results that simplify the verification of several axioms of
ZFC. These will help us in the next chapter, where we shall build some models
of set theory. We shall refer to a generic model as M.
Lemma 8. If M is transitive, the axiom of extensionality holds in M.
Proof. Extensionality is equivalent to the statement
∀x, y ∈ M({z ∈ M : z ∈ x} = {z ∈ M : z ∈ y} → x = y).
But by transitivity of M,
{z ∈ M : z ∈ x} = {z ∈ x} = x, and similarly, {z ∈ M : z ∈ y} = y,
8
so extensionality reduces to
∀x, y ∈ M(x = y → x = y),
which is true.
Lemma 9. Suppose that for each formula φ with only x, z, w1 , . . . , wn free,
∀x, z, w1 , . . . , wn ({x ∈ z : φM (x, z, w1 , . . . , wn )} ∈ M).
Then comprehension holds in M.
Proof. Given any z, w1 , . . . , wn , let y = {x ∈ z : φM (x, z, w1 , . . . , wn )}. By
hypothesis, y ∈ M. But then we have found the required y in the axiom of
comprehension relativized to M.
Corollary 10. If ∀z ∈ M(P(z) ⊂ M), then comprehension holds in M.
Proof. If P(z) ⊂ M, then ∀y ⊂ z(y ∈ M), so for any formula φ(x, z, w1 , . . . , wn )
with only x, z, w1 , . . . , wn free, {x ∈ z : φM (x, z, w1 , . . . , wn )} ∈ M.
Lemma 11. If ∀F ∈ M ∃A ∈ M(∪F ⊂ A), then union holds in M.
Proof. Since ∪F ⊂ A if and only if ∀Y ∀x(x ∈ Y ∧ Y ∈ F → x ∈ A), the
statement in the hypothesis is simply a rephrasing of the relativization of Union
to M.
Lemma 12. Suppose we can show, for each φ(x, y, A, w1 , . . . , wn ), that for each
A, w1 , . . . , wn ∈ M,
if
then
∀x ∈ A ∃!y ∈ M φM (x, y, A, w1 , . . . , wn )
∃Y ∈ M({y : ∃x ∈ A φM (x, y, A, w1 , . . . , wn )} ⊂ Y ).
Then the replacement scheme is true in M.
Proof. The only difference between our hypothesis and the replacement scheme
itself can be found in their respective consequent phrases. But the former implies
the latter. That is,
{y : ∃A φM (x, y, A, w1 , . . . , wn )} ⊂ Y
implies
∀x ∈ A ∃y ∈ Y (φM (x, y, A, w1 , . . . , wn )).
Lemma 13. For any M ⊂ V, foundation holds.
9
Proof. For any nonempty x ∈ M, we shall find an -minimal y ∈ x in M. Simply
choose a y ∈ x of minimal rank; ie., a y such that ∀z ∈ x(rk(y) ≤ rk(z)). We
can see, then, that if
∃z ∈ M(z ∈ x ∧ z ∈ y),
then
∃z ∈ M(z ∈ x ∧ rk(z) < rk(y)),
a contradiction. We conclude that
∀x ∈ M(∃y ∈ M(y ∈ x) → ∃y ∈ M(y ∈ x ∧ @z ∈ M(z ∈ x ∧ z ∈ y))).
Remark. The requirement that M ⊂ V in the result above is needed in order
to make use of the notion of rank.
6
The H(κ) Models and Consistency Results
We now introduce a special class of sets, indexed by infinite cardinals, and show
that they can serve as models of most ZFC, or in a special case, all of ZFC set
theory.
Definition. For any infinite cardinal κ, H(κ) = {x : | tr cl(x)| < κ}.
The following lemma shows that H(κ) is indeed a set.
Lemma 14. For any infinite κ, H(κ) ⊂ Vκ .
Proof. Let x ∈ H(κ). We will show that rk(x) < κ.
Let t = tr cl(x), so that |t| < κ, and let S = {rk(y) : y ∈ t} ⊂ ON. To
show that S is an ordinal, let α be the least ordinal not in S, let β be the least
ordinal in S that is greater than α, and choose a y ∈ t of rank β. If z ∈ y,
then rk(z) < rk(y) = β and z ∈ t, by transitivity of t, so rk(z) < α. But
then, rk(y) = sup{rk(z) + 1 : z ∈ y} ≤ α, a contradiction, so we must have
∀β > α(β ∈
/ S). Therefore, S consists precisely of all ordinals below α, which
means that S = α.
We then get α = |S| ≤ |t| < κ, and so x ⊂ t ⊂ Vα ⇒ rk(x) ≤ α < κ.
Lemma 15. If κ is regular, H(κ) = Vκ iff κ = ω or κ is strongly inaccessible.
Proof. Suppose that κ = ω or that κ is strongly inaccessible. Assuming that
|Vα | < κ for all α < κ, |Vα+1 | = | P(Vα )| = 2|Vα | < κ. It follows then by
induction that ∀α < κ(|Vα | < κ). Now
rk(x) = α
⇒ x ⊂ Vα
⇒ tr cl(x) ⊂ Vα
10
by Lemma 6(a) and the transitivity of Vα . So letting α = rk(x),
x ∈ Vκ
⇒
rk(x) < κ
⇒ | tr cl(x)| ≤ |Vα | < κ
⇒ x ∈ H(κ).
Hence, Vκ ⊂ H(κ), and so Vκ = H(κ) by the previous lemma.
For the other direction, suppose κ is uncountable and not strongly inaccessible, so that there is some λ < κ with κ ≤ 2λ = | tr cl(P(x))|. Then P(λ) ∈
/ H(κ).
But, as we now show, P(λ) ∈ Vκ .
λ = rk(λ) ⇒ λ ⊂ Vλ
⇒
P(λ) ⊂ P(Vλ ) = Vλ+1 .
⇒
P(λ) ∈ Vλ+2
⇒ rk(P(λ)) < λ + 2 < κ.
We conclude that H(κ) 6= Vκ .
Theorem 16. For any infinite κ :
(a) H(κ) is transitive;
(b) H(κ) ∩ ON = κ;
(c) x ∈ H(κ) ⇒ ∪x ∈ H(κ);
(d) x, y ∈ H(κ) ⇒ {x, y} ∈ H(κ);
(e) x ∈ H(κ) and y ⊂ x ⇒ y ∈ H(κ);
(f ) κ regular ⇒ ∀x(x ∈ H(κ) ↔ x ⊂ H(κ) ∧ |x| < κ).
Proof.
(a) By Lemma 6(b), y ∈ x ⇒ tr cl(y) ⊂ tr cl(x), so
y ∈ x ∈ H(κ) ⇒
| tr cl(y)| ≤ | tr cl(x)| < κ
⇒ y ∈ H(κ).
(b) By definition, H(κ) = {x : | tr cl(x)| < κ} ⊃ {α ∈ ON : | tr cl(α)| = α < κ},
and for any α ≥ κ,
| tr cl(α)| = α ≥ κ ⇒ α ∈
/ H(κ).
(c)
tr cl(∪x)
⊂
[
{∪n (∪x) : n ∈ ω}
[
{∪n+1 x : n ∈ ω}
[
{∪n x : n ∈ ω}
=
tr cl(x).
=
=
So | tr cl(∪x)| ≤ | tr cl(x)| < κ.
11
(d) We have | tr cl(x)|, | tr cl(y)| < κ, and κ is infinite, so since
[
tr cl({x, y}) =
{∪n {x, y} : n ∈ ω}
[
=
{∪n (x ∪ y) : n ∈ ω}
=
tr cl(x ∪ y)
=
tr cl(x) ∪ tr cl(y), by Lemma 6 (d),
we get that | tr cl({x, y})| < κ.
(e) It is easy to see that tr cl(y) ⊂ tr cl(x), and so | tr cl(y)| ≤ | tr cl(x)| < κ.
(f) If x ∈ H(κ), then by transitivity, x ⊂ H(κ), and |x| ≤ | tr cl(x)| < κ.
Conversely, assume that x ⊂ H(κ) and |x| < κ. Then
S
tr cl(x) = x ∪ {tr cl(y) : y ∈ x}
is a union of fewer than κ sets of size less than κ each, so by regularity,
| tr cl(x)| < κ.
Lemma 17. If κ is an infinite, regular cardinal, then H(κ) is a model of
ZFC – Power – Infinity3 .
Proof. Most of the axioms hold as a direct consequence of the properties of
H(κ) proved above. Extensionality holds since H(κ) is transitive. Foundation
holds since H(κ) ⊂ Vκ . By Theorem 16(e), for every x ∈ H(κ), P(x) ⊂ H(κ),
so comprehension holds by Corollary 10. Theorem 16(c) and (d) give us union
and pairing, respectively.
The next two axioms, replacement and choice, are the only ones that require
the regularity of κ.
We will consider a formula φ with free variables x, y, A, and w1 , . . . , wn .
Take any set A ∈ H(κ) and for each x ∈ A, assume there is a unique y ∈ H(κ)
such that φ(x, y, A, w1 , . . . , wn ) holds. Let Y be the collection of all such y.
Then Y is a subet of H(κ). Moreover, by the uniqueness of the y ∈ Y ,
|Y | ≤ |A| ≤ | tr cl(A)| < κ. So by Theorem 16(f), Y ∈ H(κ). In other words,
the replacement scheme holds in H(κ).
Finally, for any A in H(κ), let R ⊂ A × A well-order A. We must show
that R ∈ H(κ). Given any hx, yi ∈ R, x, y ∈ A, so by transitivity, x, y ∈ H(κ).
Applying Theorem 16(d) twice, {x}, {x, y} ∈ H(κ) and hx, yi ∈ H(κ). Now
that we have shown that R ⊂ H(κ), we use the fact that |R| < κκ = κ and
Theorem 16(f) to see that R ∈ H(κ). So the axiom of choice holds in H(κ).
Theorem 18.
(a) H(ω) = Vω models ZFC – Infinity + ¬Infinity.
3 ZFC
with the axioms of power set and infinity removed.
12
(b) For κ regular and uncountable, H(κ) models ZFC – Power.
Proof.
(a) Since ω is regular, the result of the previous lemma applies. Now any x ∈ Vω
has finite rank, so by Lemma 5(c), P(x) does too, and so Power set holds.
Finally, since any element of H(ω) has cardinality less than ω, Infinity does
not hold here.
(b) Similarly, we need only show that Infinity holds, here. But by Theorem
16 (b), κ ∈ H(κ). But if κ is uncountable, ω ∈ κ, and so by transitivity,
ω ∈ H(κ).
Notice that the first result above shows that the axiom of infinity is not
provable from the other axioms of ZFC, since its negation is consistent with
them. In other words,
Con(ZFC) =⇒ Con(ZFC − Power + ¬Power).
Working under the assumption that Infinity is consistent with the rest of ZFC,
this means that Infinity is in fact independent of the other axioms.
Theorem 19. For κ regular and uncountable, the following are equivalent:
(a) H(κ) models ZFC;
(b) H(κ) = Vκ ;
(c) κ is strongly inaccessible.
Proof. We have already shown the equivalence between (b) and (c) (Lemma
15).
Now assuming (b) and (c), we have that for any x ∈ H(κ), rk(x) < κ, and
so by Lemma 5(c), Power set holds.
On the other hand, if (c) is false, then there is some λ < κ with 2λ ≥ κ.
But it follows directly that λ, being its own transitive closure, is in H(κ), while
P(λ) is not, and so power set fails in H(κ). Thus (a) implies (c).
Remark. The above does not show that ZFC is consistent from within ZFC
itself. For if that were the case, the second incompleteness theorem would tell
us that that was not the case at all! The model H(κ) only models ZFC if κ is
strongly inaccessible. But the existence of such a cardinal must be postulated as
an axiom of its own. Therefore, the most we have shown is that the consistency
of ZFC can be proven from ZFC + “there exists a strongly inaccessible cardinal.”
But the consistency of this larger axiomatic system is itself in doubt.
13
We now see that
Con(ZFC) =⇒ Con(ZFC + ¬Power)
and conclude that the axiom of power set is independent from the rest of ZFC.
The previous two theorems partially justify having the axiom of power set
and infinity as axioms of set theory. Their consistency with the other axioms
show that they are acceptable, but on top of that, their independence gives their
presence a purpose in the formal theory. That they are necessary for doing mathematics is a more nebulous, philosophical issue. However, these two axioms are
so firmly entrenched and accepted by the mathematical community that there is
really no question as to whether or not they should be used. A better question
would be whether there exists some more fundamental, yet undiscovered, axiom
which could replace the two. It is fortunate that any “candidate” axioms that
may end up being discovered can be studied by the methods and techniques of
model theory and mathematical logic.
7
Conclusion
We have introduced the elements of mathematical logic, including the model
theoretic formulation of semantic truth, in order to look at the way models may
be built and used to test the consistency of an axiomatic system. In particular,
we have built two special types of models, the H(κ) and the Vκ , and used
them to show the consistency and independence of Power Set and Infinity. By
experimenting with particular values of κ, many other results could be derived.
But this just gives us a glimpse of the possibilities. Other models, such as
Gödel’s L, may be used to prove results such as the consistency of the generalized
continuum hypothesis (GCH) in ZFC. Other, more advanced techniques have
been developed, such as Paul Cohen’s method of forcing, which he used to prove
the independence of the axiom of choice and of CH from ZFC. Model theory
itself is a much larger subject than we have made it appear here, with its own
collection of important results. Many mathematical objects come equipped with
their own additional axioms and their own theories for which models may be
built and studied.
There are presently several important axioms which, though not in use by the
majority of mathematicians, are subjects of study for set theorists and logicians.
These include GCH, the axiom of determinacy, and the large cardinal axioms,
to name a few. It is believed by some that an important discovery may one day
bring a new axiom into the main light as a fundamental mathematical truth.
Such a discovery would likely have an enormous impact, and the consequenes
could include the resolution of a variety of mathematical problems and the
illumination of important problems in the philosophy of mathematics.
14
A
The ZFC Axioms
Extensionality
∀x∀y(∀z(z ∈ x ↔ z ∈ y) → x = y).
Foundation
∀x(∃y(y ∈ x) → ∃y(y ∈ x ∧ @(z ∈ x ∧ z ∈ y))).
Comprehension Scheme For any formula φ with free variables x, z, w1 , . . . , wn ,
∀z, w1 , . . . , wn ∃y∀x(x ∈ y ↔ z ∈ x ∧ φ).
Pairing
∀x, y∃z(x ∈ z ∧ y ∈ z).
Union
∀F∃A∀Y, x(x ∈ Y ∧ Y ∈ F → x ∈ A).
Replacement Scheme For any formula φ with free variables x, y, A, w1 , . . . , wn ,
∀A, w1 , . . . , wn (∀x ∈ A∃!yφ → ∃Y ∀x ∈ A∃y ∈ Y φ).
Infinity
∃x(0 ∈ x ∧ ∀y ∈ x(y ∪ {y} ∈ x)).
Power
∀x∃y∀z(z ⊂ x → z ∈ y).
Choice
∀A∃R(R well-orders A).
References
[1] H.-D. Ebbinghaus, J. Flum, and W. Thomas. Mathematical logic. Undergraduate Texts in Mathematics. Springer-Verlag, New York, 1984. Translated from the German by Ann S. Ferebee.
[2] Wilfrid Hodges. Model theory, volume 42 of Encyclopedia of Mathematics
and its Applications. Cambridge University Press, Cambridge, 1993.
[3] Stephen Cole Kleene. Mathematical logic. John Wiley & Sons Inc., New
York, 1967.
[4] Joseph R. Shoenfield. Mathematical logic. Association for Symbolic Logic,
Urbana, IL, 2001. Reprint of the 1973 second printing.
15