MAGIC Set theory lecture 1 - University of East Anglia

MAGIC Set theory

lecture 1

David Aspero

University of East Anglia

11 October 2018

Welcome

Welcome to this set theory course.

This will be an introduction to set theory. The only prerequisiteis some level of mathematical maturity. The course will bemostly self–contained but you are and will be invited to look inother sources.

Welcome

Welcome to this set theory course.

This will be an introduction to set theory. The only prerequisiteis some level of mathematical maturity. The course will bemostly self–contained but you are and will be invited to look inother sources.

Set theory plays a dual role. It provides a foundation formathematics and it is itself a branch of mathematics withapplications to other areas of mathematics.

Reducing everything to sets

Set theory was developed / discovered / instigated by GeorgCantor, in the second half of the 19th century, as a result of hisinvestigations of trigonometric series rather than out offoundational considerations.

However, set theory would soon become the prevalentfoundation of mathematics. In fact, it was born at a time whenmathematicians saw the need to define things carefully(specifically, to define the object of their study in a mathematicallanguage referring to reasonably ‘simple’ and well–understoodentities) and set theory provided the means to do exactly that.

Example: What is a differentiable function? What is acontinuous function? What is a function?









A case example: A relation is a set of ordered pairs (a, b). Anda function f is a functional relation (i.e., (a, b), (a, b0) 2 f impliesb = b

0).What is an ordered pair (a, b)? Well, given a, b, we can define

(a, b) = {{a}, {a, b}}

(this definition is due to Kuratowski).

FactGiven any ordered pairs (a, b), (a0, b0), (a, b) = (a0, b0) if and

only if a = a

0and b = b

0.

[Easy exercise: Check]

Similarly, for any given n, we can define the n–tuple(a0, . . . , an

, an+1) = ((a0, . . . , an

), an+1).



(a, b) = {{a}, {a, b}}



only if a = a

0and b = b

0.



, an+1) = ((a0, . . . , an

), an+1).



(a, b) = {{a}, {a, b}}



only if a = a

0and b = b

0.



, an+1) = ((a0, . . . , an

), an+1).

So we can successfully define the notion of function from thenotion of set (and the membership relation 2, of course). Andthe notion of set is presumably easier to grasp than the notionof function.

What about natural numbers, integers, rational, reals and so on?

We can define 0 = ; (the empty set, the unique set with noelements). The set ; has 0 members.

We can define 1 = {0} = {;}. The set {;} has 1 member.

We can define 2 = {0, 1} = {;, {;}}. The set {;, {;}} has 2members.

. . .In general, we can define n + 1 = n [ {n}. With this definitionn + 1 is a set with exactly n + 1 many members and thesemembers are all natural numbers m such that m n.
















With this definition, each n is an ordinal � which is either ; or ofthe form ↵ [ {↵} for some ordinal ↵ and all of whose membersare either the empty set or of the form ↵ [ {↵} for some ordinal↵, and every ordinal which is either ; or of the form ↵ [ {↵} andall of whose members are either the empty set or of the form↵ [ {↵} for some ordinal ↵ is a natural number (the notion ofordinal, which we will see later on, is defined only in terms ofsets).

What is nice about this is that it gives a definition of the set N ofnatural numbers involving only the notion of set:

N is the set of all those ordinals � such that every member of �is either the empty set or of the form ↵[ {↵} for some ordinal ↵.

In particular: We may want to say that a set x is finite iff there isa bijection between x and some member of N. By the abovedefinition of N we thus have a definition of finiteness purely interms of sets and the membership relation.

+ and · on N can be defined also in a satisfactory way using thenotion of set. Then we can define Z in the usual way as the setof equivalence classes of the equivalence relation ⇠ on N⇥ Ndefined by (a, b) ⇠ (a0, b0) if and only if a + b

0 = a

0 + b,

and we can define also Q from the natural arithmeticaloperations on Z in the usual way.

We can define R as the set of equivalence classes of theequivalence relation ⇠ on the set of Cauchy sequencesf : N �! Q where f ⇠ g if and only if lim

n!1 h = 0, whereh(n) = f (n)� g(n). Etc.

All these constructions involve only notions previously defined,together with the notion of set and the membership relation. Sothey ultimately involve only the notion of set and themembership relation.


0 = a

0 + b,






0 = a

0 + b,





If there is nothing fishy with the notion of set and the operationswe have used to build more complicated sets out of simplerones, then there cannot be anything fishy with these higherlevel objects.

Similarly: We feel confident with the existence of C (which, bythe way, contains “imaginary numbers” like i) once we becomeconfident with the existence of R and know how to build C fromR in a very simple set–theoretic way.

Also: We can derive everything we know about the higher levelobjects (like, say, the fact that ⇡ is transcendental) fromelementary facts about sets.

And, presumably, we would expect that the combination ofelementary facts about sets can ultimately answer everyquestion we’re interested in (is e + ⇡ transcendental?,Goldbach’s conjecture, ...). This would reduce mathematics toconsiderations of sets and their (elementary) properties.

Also: We can derive everything we know about the higher levelobjects (like, say, the fact that ⇡ is transcendental) fromelementary facts about sets.

And, presumably, we would expect that the combination ofelementary facts about sets can ultimately answer everyquestion we’re interested in (is e + ⇡ transcendental?,Goldbach’s conjecture, ...). This would reduce mathematics toconsiderations of sets and their (elementary) properties.

Some elementary facts about setsGiven sets A, B:

We say that A is of cardinality at most that of B, and write

|A| |B|,

if there is an injective (or one–to–one) function f : A �! B

(remember, a function is a special kind of set!).

We say that that A and B have the same cardinality, and write

|A| = |B|,

if and only if there is a bijection f : A �! B.

We say that A has cardiality strictly less than B, and write

|A| < |B|,

if and only if there is an injective function f : A �! B but thereis no bijection f : A �! B.

Clearly |A| |B| and |B| |C| together imply |A| |C|. Also, itis true, but not a trivial fact, that |A| = |B| holds if and only ifboth |A| |B| and |B| |A| hold (Cantor–Bernstein theorem,we will see this later on).

The notion of cardinality captures the notion of “size” of a set.(Example: |5| < |6|).

Notation: Given a set X , P(X ) is the set of all sets Y such thatY ✓ X . (P(X ) is the power set of X ).







The following theorem arguably marks the beginning of settheory.Theorem (Cantor, December 1873) Given any set X ,|X | < |P(X )|.

Proof: There is clearly an injection f : X �! P(X ): f sends x tothe singleton of x , i.e., to {x}.

Now suppose f : X �! P(X ) is a function. Let us see that f

cannot be a surjection: Let

Y = {a 2 X : a /2 f (a)}

Y 2 P(X ).But if a 2 X is such that f (a) = Y , then a 2 Y if and only ifa /2 f (a) = Y . This is a logical impossibility, so there is no sucha. ⇤

This theorem immediately yields that not all infinite sets are ofthe same size, and in fact there is a whole hierarchy of infinities!(which was not known prior to Cantor’s theorem):

|N| < |P(N)| < |P(P(N))| = |P2(N)| < . . .

. . . < |Pn(N)| < |Pn+1(N)| < . . .

. . . < |S

n2N Pn(N)| < |P(S

n2N Pn(N))| < . . .

More elementary facts

Let R be the collection of all those sets X such that

X /2 X

R is a collection of objects, and so it is therefore a set.

R contains many sets. For instance, ; 2 R, 1 2 R, everynatural number is in R, N 2 R, R 2 R, etc.

Does R belong to R?

Well, R 2 R if and only if R /2 R,

which is the same kind of contradiction that we obtained at theend of the proof of Cantor’s theorem! So R cannot be a set!!

(Russell’s paradox)

So, our naıve “theory” of sets is inconsistent and maybe it’s notso good a foundation of mathematics after all...

Is this the end of the story for set theory?

So, our naıve “theory” of sets is inconsistent and maybe it’s notso good a foundation of mathematics after all...

Is this the end of the story for set theory?

Well, we like to think in terms of objects built out of sets and likethe simplicity of the foundations set theory was intending toprovide.

Also, we find the multiplicities of infinities predicted by settheory an exciting possibility, and there was nothing obviouslycontradictory in Cantor’s theorem.

A retreat

A reasonable move at this point would be to retreat to a moremodest theory T such that

1 T should express true facts about sets (or should we sayplausible, desirable?),

2 T enables us to carry out enough constructions so as tobuild all usual mathematical objects (real numbers, spacesof functions, etc.),

3 T gives us an interesting theory of the infinite(|N| < |P(N)|, etc.), and such that

4 we can prove that T is consistent; or, if we cannot provethat, such that we have good reasons to believe T isconsistent.

A retreat






A retreat






A retreat






A retreat






First questions:

(1): What is a theory?

(2): Which should be our guiding principles for designing ourtheory T ?

We answer (1) first.

First questions:

(1): What is a theory?

(2): Which should be our guiding principles for designing ourtheory T ?

We answer (1) first.

The axiomatic method: A crash coursein first order logic.

For us a theory will be a first order theory or, more accurately, atheory in classical first order logic. A theory T will always be atheory in a given first order language L. It will be a set (!) ofL–sentences expressing facts about our intended domain ofdiscourse.

Talk of “sets” of L–sentences before we have even defined T

(which might end up being an intended theory of sets)? Well,those sets of sentences, as well as the sentences, thelanguage L, etc., are objects in our meta–theory. Presumablythey will obey laws expressible in some meta–meta–theory(perhaps the same laws the theory T is, in our understand / inits intended interpretation, trying to express!).

The axiomatic method: A crash coursein first order logic.

For us a theory will be a first order theory or, more accurately, atheory in classical first order logic. A theory T will always be atheory in a given first order language L. It will be a set (!) ofL–sentences expressing facts about our intended domain ofdiscourse.

Talk of “sets” of L–sentences before we have even defined T

(which might end up being an intended theory of sets)? Well,those sets of sentences, as well as the sentences, thelanguage L, etc., are objects in our meta–theory. Presumablythey will obey laws expressible in some meta–meta–theory(perhaps the same laws the theory T is, in our understand / inits intended interpretation, trying to express!).

A language L consists of

• a (possible empty) set of constant symbols c, d , ...• a (possibly empty) set of functional symbols f , g, ...,

together with their arities (this arity is a natural number; if f

is meant to represent a function f

M : M �! M it has arity1, if it is meant to represent a function f

M : M ⇥ M �! M,then it has arity 2, etc.)

• a (possibly empty) set of relational symbols R, S, ....together with their arities (this arity is a natural number; ifR is meant to represent a subset R

M ✓ M, then it has arity1, if it is meant to represent a binary relation R

M ✓ M ⇥ M,then it has arity 2, etc.)

These are the non-logical symbols and completely determine L.

We also have logical symbols, which are independent from L:

• ^, _, ¬, !, $ (connectives)• 8, 9 (quantifiers)• (, ), =

= is sometimes omitted. Also, many of these symbols are notnecessary; we could do with just ¬, _ and 9.

Finally, we have a sufficiently large supply of variables:Var = {v0, v1, . . . , vn

, . . .}. For most uses it is enough to takethe set of variables to have the same size as the naturalnumbers.

Language of set theory: Only non–logical symbol: A relationalsymbol 2 of arity 2.

Let’s focus on the language of set theory from now on:

1 every expression of the form (vi

2 v

j

) or (vi

= v

j

), with v

i

and v

j

variables, is a formula (an atomic formula).2 If ' and are formulas, then (¬'), (' _ ), (' ^ ),

('! ), ('$ ) are formulas. Also, if v is a variable,then (8v') and (9v') are formulas.

3 Something is a formula if and only if it is an atomic formulaor is obtained from formulas as in (2).

When referring to a formula, we often omit parentheses toimprove readability (these expressions are not actual officialformulas but refer to them in an unambiguous way).

A sentence is a formula ' without free variables, i.e., such thatfor every variable v and every atomic subformula '0 of ', if v

occurs in '0, then '0 is a subformula of some subformula of 'of the form 8v or of the form 9v .

Examples of formulas are the formulas abbreviated as:

8x8y(x = y $ 8z(z 2 x $ z 2 y))

(The axiom of Extensionality)

8x8y9z8w(w 2 z $ (w = x _ w = y))

or, even more abbreviated,

“for all x , y , {x , y} exists”

(Axiom of unordered pairs).

Another example:

9a9b8y(y 2 x $ ((8w(w 2 y $ (w = a _ w = b))) _ (8w(w 2y $ w = a)))))

(x is in ordered pair)

The first two formulas are sentences. The third one is not.

SatisfactionThis takes place of course in the meta–theory:

A pair M = (M,R), where M is a set and R ✓ M ⇥ M, is calledan L–structure.

Given an assignment

~a : Var �! M:

• M |= (vi

2 v

j

)[~a] if and only if (~a(vi

),~a(vj

)) 2 R.• M |= (v

i

= v

j

)[~a] if and only if ~a(vi

) = ~a(v

j

).• M |= (¬')[~a] if and only if M |= '[~a] does not hold.• M |= ('0 _ '1)[~a] if and only if M |= '0[~a] or |='1[~a]; and

similarly for the other connectives.• M |= (9v')[~a] if and only if there is some b 2 M such thatM |= '[~a(v/b)], where ~a(v/b) is the assignment ~b suchthat ~b(v

i

) = ~a(v

i

) if v

i

6= v and ~b(v) = b.• M |= (8v')[~a] if and only if for every b 2 M,M |= '[~a(v/b)].

We say that M satisfies ' with the assignment

~a if M |= '[~a].

Easy fact: If ' is a sentence, then M |= '[~a] for someassignment ~a if and only if M |= '[~a] for every assignment ~a. Inthat case we say that M is a model of '.

Given a set T of formulas and a formula ', we write

T |= '

if and only if for every L–structure M = (M,R) and everyassignment ~a : Var �! M,IF M |= �[~a] for every � 2 T ,THEN M |= '[~a].

The relation |= aims at capturing the notion of ‘logicalconsequence’: ' follows logically from T if and only if ' is truein every world in which T is true. |= is often called the relation

of logical consequence.

This framework is mostly due to the logician A. Tarski (1930’s).

‘First order’ in ‘first order logic’ refers to the fact that variablesrange in the above definition only over the individuals of theuniverse of the relevant L–structures M. In second order logicwe can have variables that range over (arbitrary) subsets of theuniverse of the relevant L–structures M. Etc.


T |= '







T |= '






Syntactical deductionLet T will be a set of formulas. We will view T as a set ofaxioms and deduce theorems from T : A theorem of T will bethe final member �

n

of a derivation

� = (�0,�1, . . . ,�n

)

from T , where we say that � = (�0,�1, . . . ,�n

) is a derivation

from T if it is a finite sequence of L–formulas and for every i ,

• �i

is either in T , or• �

i

is a logical axiom of first order logic, or• �

i

is obtained form �j

and �k

, for some j , k < i , by the rule

of Modus Ponens “If '! and ', then ” (for allL–formulas ', ). In other words, in this last case, thereare j , k < i and an L–formula ' such that �

j

= ' and �k

is'! �

i

.

Here, a logical axiom is a member of a certain infinite easilyspecifiable list of formulas that express logical / completelygeneral truths.

Typical members of this list are for example, '! ( ! ') forall formulas ', , or ' _ ¬' for all formulas '.

Indeed, we see it as a general truth that if ' is true, then it istrue that if is true then ' is true. And we seeit as a general truth that for every ' either ' is true or ¬' is true.1

This list of axioms is not unique: Many different lists of axiomsgive rise to the same system of logic.

1If we are classical logicians. There are weakening / versions of classicalfirst order logic in which ' _ ¬', also known as Law of Excluded Middle, isnot true for some choices of '.

If ' is a theorem from T , we write

T ` '

` is often called the relation of logical derivability.

A priori, |= and ` look like quite different relations, aimed atcapturing two apparently different notions: The notion of logical(semantical) consequence and the notion of deductibility in areasonable calculus.

However,

Theorem (Completeness theorem for first order logic) (K.Godel, 1930’s)

|==`

A theory T is consistent if no contradiction (say, 9x¬(x = x))can be derived from it:

T 0 9x¬(x = x)

A theory is inconsistent (i.e., not consistent) if and only if it istrivial, in the sense that it proves everything.

By the completeness theorem the following are equivalent:•

T is consistent.• There is an L–structure M such that M |= T (T is true

in some world).

We’ll be interested in whether or not T ` � for a given theory T

and a given sentence �.

The following are equivalent again by (the contrapositive of) thecompleteness theorem:

•T 0 �

• There is an L–structure M such that M |= T but M |= ¬�.

Axiomatic set theory: ZFC

Z is for Ernst Zermelo, F is for Abraham Fraenkel, C is for theAxiom of Choice.

The objects of set theory are sets. As in any axiomatic theory,they are not defined (they are feature–less objects; in thecontext of thetheory there is nothing to them apart from what the theory says).

ZFC expresses facts about sets expressible in the first orderlanguage of set theory. The same is true for any other first ordertheory in the language of set theory, like ZF, ZFC+“There is asupercompact cardinal”, ZFC+GCH, ZFC+V = L, ZFC+PFA, ...









Most ZFC axioms will be axioms saying that certain “classes”(built out of given sets) are actual sets (they are objects in theset–theoretic universe): Axiom 0, The Axiom of unorderedpairs, Union set Axiom, Power set Axiom, Axiom Scheme ofSeparation, Axiom Scheme of Replacement and Axiom ofInfinity will be of this kind.

Here, a class is any collection of objects, where this collectionis definable possibly with parameters. For example the class ofall sets. A proper class will be a class which is not a set.

ZFC will also have an axiom guaranteeing the existence of setswith a given property, even if these sets are not definable: TheAxiom of Choice

We will also have two “structural” axioms: Axiom ofExtensionality and Axiom of Foundation.

Most ZFC axioms will be axioms saying that certain “classes”(built out of given sets) are actual sets (they are objects in theset–theoretic universe): Axiom 0, The Axiom of unorderedpairs, Union set Axiom, Power set Axiom, Axiom Scheme ofSeparation, Axiom Scheme of Replacement and Axiom ofInfinity will be of this kind.

Here, a class is any collection of objects, where this collectionis definable possibly with parameters. For example the class ofall sets. A proper class will be a class which is not a set.

ZFC will also have an axiom guaranteeing the existence of setswith a given property, even if these sets are not definable: TheAxiom of Choice

We will also have two “structural” axioms: Axiom ofExtensionality and Axiom of Foundation.

A classification of the ZFC axioms

1 Structural axioms: Axioms of Extensionality, Axiom ofFoundation.

2 Constructive set–existence axioms: Axiom 0, TheAxiom of unordered pairs, Union set Axiom, Power setAxiom, Axiom Scheme of Separation, Axiom Scheme ofReplacement and Axiom of Infinity.

3 Non–constructive set–existence axiom: Axiom ofChoice.

The axioms

Axiom of Extensionality: Two sets are equal if and only if theyhave the same elements:

8x8y(x = y $ 8z(z 2 x $ z 2 y))

In other words, the identity of a set is completely determined byits members:

The sets• ;• {(a, b, c, n) : a

n+b

n = c

n, a, b, c, n 2 N, a, b, c � 2, n � 3}are the same set.

Axiom 0: ; exists.

9x8y(y 2 x $ y 6= y)

(of course y 6= y abbreviates ¬(y = y)).

Strictly speaking this axiom is not needed: It follows from theother axioms.

It is convenient to postulate it at this point, though.

In the theory given by the Axiom of Extensionality together withAxiom 0 we can only prove the existence of one set:

;

Not so interesting yet.

The theory T = { Axiom 0, Axiom of Extensionality } surely isconsistent: For any set a,

({a}, ;) |= T

But ({a, b}, ;) 6|= T if a 6= b.

Axiom of unordered pairs: For any sets x , y there is a setwhose members are exactly x and y ; in other words, {x , y}exists.

8x8y9z8w(w 2 z $ (w = x _ w = y))

Of course: If x = y , then {x , y} = {x}.

[Prove this using the Axiom of Extensionality.]

Recall:Definition: (x , y) = {{x}, {x , y}}

The theory laid down so far gives us already the existence ofinfinitely many sets! :

;, {;}, {{;}}, {{{;}}}, {{{{;}}}}, {;, {;}}, {;, {;, {;}}},{{;}, {;, {;}}}, {;, {;, {;, {;}}}}, ...

With the definition of the natural numbers given in lecture 1,these sets are: 0, 1, {1} = (0, 0), {{1}} = {(0, 0)},((0, 0), (0, 0)), 2 = (0, 1), {0, 2}, {1, 2} = (0, 1), {0, {0, 2}}, ...

All sets whose existence is proved by the theory given so farhave at most two elements (!).

This theory proves the existence of (a, b) for all a, b.

MAGIC Set theory lecture 1 - University of East Anglia

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