Naive Set Theory - École normale supérieure de...

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Naive Set Theory

Jacques Jayez, ENS de Lyon and L2C2

Cours Introduction à la logique, version du 8 octobre 2016

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Basics I

I Set = finite or infinite unordered collection without repetition.

I Notation : {a,b, c, . . .}, where a, b, c, etc. are the objects in the collection(the elements).

I Writing {a,a, . . .}, {a,b,a, . . .}, etc. does not make sense.I Identity criterion : two sets are the same iff they have the same elements.I Ex. : {a,b} = {b,a} (order is not relevant)

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Basics I

I Set = finite or infinite unordered collection without repetition.I Notation : {a,b, c, . . .}, where a, b, c, etc. are the objects in the collection

(the elements).

I Writing {a,a, . . .}, {a,b,a, . . .}, etc. does not make sense.I Identity criterion : two sets are the same iff they have the same elements.I Ex. : {a,b} = {b,a} (order is not relevant)

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Basics I


(the elements).I Writing {a,a, . . .}, {a,b,a, . . .}, etc. does not make sense.

I Identity criterion : two sets are the same iff they have the same elements.I Ex. : {a,b} = {b,a} (order is not relevant)

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Basics I


(the elements).I Writing {a,a, . . .}, {a,b,a, . . .}, etc. does not make sense.I Identity criterion : two sets are the same iff they have the same elements.

I Ex. : {a,b} = {b,a} (order is not relevant)

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Basics I


(the elements).I Writing {a,a, . . .}, {a,b,a, . . .}, etc. does not make sense.I Identity criterion : two sets are the same iff they have the same elements.I Ex. : {a,b} = {b,a} (order is not relevant)

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Basics II

I Definition of ∈ : x ∈ X means that x is an element of the set X .

I 6∈ notes the contrary property : x 6∈ X means that x is not an element of X .I Ex. : a ∈ {a,b}, b ∈ {a,b}, c 6∈ {a,b}.I Is that clear ? Not enough !I Can we write a ∈ {b, {a,b}, c} ?I NO, NEVER ! Why ?

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Basics II

I Definition of ∈ : x ∈ X means that x is an element of the set X .I 6∈ notes the contrary property : x 6∈ X means that x is not an element of X .

I Ex. : a ∈ {a,b}, b ∈ {a,b}, c 6∈ {a,b}.I Is that clear ? Not enough !I Can we write a ∈ {b, {a,b}, c} ?I NO, NEVER ! Why ?

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Basics II

I Definition of ∈ : x ∈ X means that x is an element of the set X .I 6∈ notes the contrary property : x 6∈ X means that x is not an element of X .I Ex. : a ∈ {a,b}, b ∈ {a,b}, c 6∈ {a,b}.

I Is that clear ? Not enough !I Can we write a ∈ {b, {a,b}, c} ?I NO, NEVER ! Why ?

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Basics II

I Definition of ∈ : x ∈ X means that x is an element of the set X .I 6∈ notes the contrary property : x 6∈ X means that x is not an element of X .I Ex. : a ∈ {a,b}, b ∈ {a,b}, c 6∈ {a,b}.I Is that clear ? Not enough !

I Can we write a ∈ {b, {a,b}, c} ?I NO, NEVER ! Why ?

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Basics II

I Definition of ∈ : x ∈ X means that x is an element of the set X .I 6∈ notes the contrary property : x 6∈ X means that x is not an element of X .I Ex. : a ∈ {a,b}, b ∈ {a,b}, c 6∈ {a,b}.I Is that clear ? Not enough !I Can we write a ∈ {b, {a,b}, c} ?

I NO, NEVER ! Why ?

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Basics II

I Definition of ∈ : x ∈ X means that x is an element of the set X .I 6∈ notes the contrary property : x 6∈ X means that x is not an element of X .I Ex. : a ∈ {a,b}, b ∈ {a,b}, c 6∈ {a,b}.I Is that clear ? Not enough !I Can we write a ∈ {b, {a,b}, c} ?I NO, NEVER ! Why ?

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Basic III

I Because of the Matriochka Principle.

(1) The Matriochka Principle If a set is seen as a Matriochka, only thedaughters are elements.

Good

Bad

Masha1 (the biggest) . . .Masha8 (thesmallest)Masha8 ∈ Masha7 ∈ Masha6 ∈ . . .∈Masha1Masha8 6∈ Masha6, Masha4 6∈ Ma-sha1, etc.if j 6= i + 1, Mashai 6∈ Mashaj.

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Basics IV

I Within the limits of the Matriochka Principle, the elements of a set canbe :

1. atomic elements or2. sets or3. sets of sets, etc.

I Ex. : a ∈ {a,b},I {a,b} ∈ {c, {a,b},a},I {a, {a,b}} ∈ {b, c, {a,b}, {a, {a,b}}, e, {a}}I * : a 6∈{b, {a, c}}

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Basics IV


1. atomic elements or

2. sets or3. sets of sets, etc.


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Basics IV


1. atomic elements or2. sets or

3. sets of sets, etc.


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Basics IV




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Basics IV



I Ex. : a ∈ {a,b},

I {a,b} ∈ {c, {a,b},a},I {a, {a,b}} ∈ {b, c, {a,b}, {a, {a,b}}, e, {a}}I * : a 6∈{b, {a, c}}

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Basics IV



I Ex. : a ∈ {a,b},I {a,b} ∈ {c, {a,b},a},

I {a, {a,b}} ∈ {b, c, {a,b}, {a, {a,b}}, e, {a}}I * : a 6∈{b, {a, c}}

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Basics IV



I Ex. : a ∈ {a,b},I {a,b} ∈ {c, {a,b},a},I {a, {a,b}} ∈ {b, c, {a,b}, {a, {a,b}}, e, {a}}

I * : a 6∈{b, {a, c}}

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Basics IV




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Basics V

I Correspondence between properties and sets.

I {x : P(x)} denotes the set of objects that satisfy the property P.I P can be any formula of a formal language :{x : integer(x) ∧ even(x)} = {0,2,4, . . .}.

I * Only in naive set theory.I The extensional view of a set : {0,2,4, . . .}.

The intensional view of the same set : {x : integer(x) ∧ even(x)}.I When P is contradictory, we have the null or empty set, noted ∅.I * There is one and only one empty set (∅).

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Basics V

I Correspondence between properties and sets.I {x : P(x)} denotes the set of objects that satisfy the property P.

I P can be any formula of a formal language :{x : integer(x) ∧ even(x)} = {0,2,4, . . .}.



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Basics V

I Correspondence between properties and sets.I {x : P(x)} denotes the set of objects that satisfy the property P.I P can be any formula of a formal language :{x : integer(x) ∧ even(x)} = {0,2,4, . . .}.



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Basics V


I * Only in naive set theory.

I The extensional view of a set : {0,2,4, . . .}.The intensional view of the same set : {x : integer(x) ∧ even(x)}.

I When P is contradictory, we have the null or empty set, noted ∅.I * There is one and only one empty set (∅).

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Basics V



The intensional view of the same set : {x : integer(x) ∧ even(x)}.

I When P is contradictory, we have the null or empty set, noted ∅.I * There is one and only one empty set (∅).

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Basics V



The intensional view of the same set : {x : integer(x) ∧ even(x)}.I When P is contradictory, we have the null or empty set, noted ∅.

I * There is one and only one empty set (∅).

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Basics V




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Operations on sets

Operations I

I Take one set or several sets and return a set.

I IntersectionX ∩ Y =df {x : x ∈ X ∧ x ∈ Y}.

I UnionX ∪ Y =df {x : x ∈ X ∨ x ∈ Y}.

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Operations on sets

Operations I

I Take one set or several sets and return a set.I Intersection

X ∩ Y =df {x : x ∈ X ∧ x ∈ Y}.

I UnionX ∪ Y =df {x : x ∈ X ∨ x ∈ Y}.

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Operations on sets

Operations I

I Take one set or several sets and return a set.I Intersection

X ∩ Y =df {x : x ∈ X ∧ x ∈ Y}.I Union

X ∪ Y =df {x : x ∈ X ∨ x ∈ Y}.

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Operations on sets

Operations II

I How to represent negation ?

I The complement of a set X with respect to another set Y = the set ofelements of Y which are not in X .

I {Y X = {z : z ∈ Y ∧ z 6∈ X}, other notations Y − X ,Y \ X .

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Operations on sets

Operations II

I How to represent negation ?I The complement of a set X with respect to another set Y = the set of

elements of Y which are not in X .

I {Y X = {z : z ∈ Y ∧ z 6∈ X}, other notations Y − X ,Y \ X .

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Operations on sets

Operations II

I How to represent negation ?I The complement of a set X with respect to another set Y = the set of

elements of Y which are not in X .I {Y X = {z : z ∈ Y ∧ z 6∈ X}, other notations Y − X ,Y \ X .

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Operations on sets

Operations III

I Introducing order : sets are unordered, but, sometimes, we need order !

I Ordered pairs/couples : 〈x,y〉 =df {{x}, {x,y}}.I Triples : 〈x,y, z〉 abbreviates 〈x, 〈y, z〉〉.I * 〈x,y〉 6= 〈y, x〉 (order does matter).I In general : 〈x1, x2, . . . , xn〉 =df 〈x1, 〈x2, x3, . . . , xn〉〉.I Hideously complex when you look into triples, pairs with pairs, etc.〈x, 〈y, z〉〉 = {{x}, {x, 〈y, z〉}} = {{x}, {x, {y, {y, z}}}}.

I The nice thing is : you can forget the { } structure and just use 〈〉.I * Repetitions are allowed : 〈a,a〉 is actually {{a}, {a,a}}, which collapses

to {{a}, {a}}, which collapses to {{a}}.I Can we have infinite lists, like 〈1,2,3, . . .〉 ? More on this in section 3.

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Operations on sets

Operations III

I Introducing order : sets are unordered, but, sometimes, we need order !I Ordered pairs/couples : 〈x,y〉 =df {{x}, {x,y}}.

I Triples : 〈x,y, z〉 abbreviates 〈x, 〈y, z〉〉.I * 〈x,y〉 6= 〈y, x〉 (order does matter).I In general : 〈x1, x2, . . . , xn〉 =df 〈x1, 〈x2, x3, . . . , xn〉〉.I Hideously complex when you look into triples, pairs with pairs, etc.〈x, 〈y, z〉〉 = {{x}, {x, 〈y, z〉}} = {{x}, {x, {y, {y, z}}}}.



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Operations on sets

Operations III

I Introducing order : sets are unordered, but, sometimes, we need order !I Ordered pairs/couples : 〈x,y〉 =df {{x}, {x,y}}.I Triples : 〈x,y, z〉 abbreviates 〈x, 〈y, z〉〉.

I * 〈x,y〉 6= 〈y, x〉 (order does matter).I In general : 〈x1, x2, . . . , xn〉 =df 〈x1, 〈x2, x3, . . . , xn〉〉.I Hideously complex when you look into triples, pairs with pairs, etc.〈x, 〈y, z〉〉 = {{x}, {x, 〈y, z〉}} = {{x}, {x, {y, {y, z}}}}.



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Operations on sets

Operations III

I Introducing order : sets are unordered, but, sometimes, we need order !I Ordered pairs/couples : 〈x,y〉 =df {{x}, {x,y}}.I Triples : 〈x,y, z〉 abbreviates 〈x, 〈y, z〉〉.I * 〈x,y〉 6= 〈y, x〉 (order does matter).

I In general : 〈x1, x2, . . . , xn〉 =df 〈x1, 〈x2, x3, . . . , xn〉〉.I Hideously complex when you look into triples, pairs with pairs, etc.〈x, 〈y, z〉〉 = {{x}, {x, 〈y, z〉}} = {{x}, {x, {y, {y, z}}}}.



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Operations on sets

Operations III

I Introducing order : sets are unordered, but, sometimes, we need order !I Ordered pairs/couples : 〈x,y〉 =df {{x}, {x,y}}.I Triples : 〈x,y, z〉 abbreviates 〈x, 〈y, z〉〉.I * 〈x,y〉 6= 〈y, x〉 (order does matter).I In general : 〈x1, x2, . . . , xn〉 =df 〈x1, 〈x2, x3, . . . , xn〉〉.

I Hideously complex when you look into triples, pairs with pairs, etc.〈x, 〈y, z〉〉 = {{x}, {x, 〈y, z〉}} = {{x}, {x, {y, {y, z}}}}.



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Operations on sets

Operations III

I Introducing order : sets are unordered, but, sometimes, we need order !I Ordered pairs/couples : 〈x,y〉 =df {{x}, {x,y}}.I Triples : 〈x,y, z〉 abbreviates 〈x, 〈y, z〉〉.I * 〈x,y〉 6= 〈y, x〉 (order does matter).I In general : 〈x1, x2, . . . , xn〉 =df 〈x1, 〈x2, x3, . . . , xn〉〉.I Hideously complex when you look into triples, pairs with pairs, etc.〈x, 〈y, z〉〉 = {{x}, {x, 〈y, z〉}} = {{x}, {x, {y, {y, z}}}}.



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Operations on sets

Operations III


I The nice thing is : you can forget the { } structure and just use 〈〉.

I * Repetitions are allowed : 〈a,a〉 is actually {{a}, {a,a}}, which collapsesto {{a}, {a}}, which collapses to {{a}}.

I Can we have infinite lists, like 〈1,2,3, . . .〉 ? More on this in section 3.

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Operations on sets

Operations III



to {{a}, {a}}, which collapses to {{a}}.

I Can we have infinite lists, like 〈1,2,3, . . .〉 ? More on this in section 3.

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Operations on sets

Operations III




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Operations on sets

Operations IV

I Cartesian product.X × Y =df {〈x,y〉 : x ∈ X ∧ y ∈ Y}.

I The Cartesian product is not commutative. It can happen that X × Y 6=Y × X .

I {a,b} × {a, c} = {〈a,a〉, 〈a, c〉, 〈b,a〉, 〈b, c〉} and . . .

{a, c} × {a,b} = {〈a,a〉, 〈a,b〉, 〈c,a〉, 〈c,b〉}.I X1 × X2 × . . .× Xn =df X1 × (X2 × . . .× Xn).I It is identical to : {〈x1 . . . xn〉 : x1 ∈ X1 ∧ . . . ∧ xn ∈ Xn}. Why ?

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Operations on sets

Operations IV




{a, c} × {a,b} = {〈a,a〉, 〈a,b〉, 〈c,a〉, 〈c,b〉}.

I X1 × X2 × . . .× Xn =df X1 × (X2 × . . .× Xn).I It is identical to : {〈x1 . . . xn〉 : x1 ∈ X1 ∧ . . . ∧ xn ∈ Xn}. Why ?

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Operations on sets

Operations IV




{a, c} × {a,b} = {〈a,a〉, 〈a,b〉, 〈c,a〉, 〈c,b〉}.I X1 × X2 × . . .× Xn =df X1 × (X2 × . . .× Xn).

I It is identical to : {〈x1 . . . xn〉 : x1 ∈ X1 ∧ . . . ∧ xn ∈ Xn}. Why ?

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Operations on sets

Operations IV




{a, c} × {a,b} = {〈a,a〉, 〈a,b〉, 〈c,a〉, 〈c,b〉}.I X1 × X2 × . . .× Xn =df X1 × (X2 × . . .× Xn).I It is identical to : {〈x1 . . . xn〉 : x1 ∈ X1 ∧ . . . ∧ xn ∈ Xn}. Why ?

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Operations on sets

Operations V

I X is a subset of Y , in symbols X ⊆ Y =df for every x, if x ∈ X , then x ∈ Y .

I E.g. {a,b} ⊆ {a,b, c}.I For every set X , X ⊆ X and ∅ ⊆ X *

I The ℘ operator (powerset) returns the set of subsets of a given set (itspowerset).

I ℘(X) =df {Y : Y ⊆ X}I E.g. ℘({a,b, c}) = {∅, {a,b, c}, {a,b}, {a, c}, {b, c}, {a}, {b}, {c}}.I Test your understanding : what is ℘(∅) ?

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Operations on sets

Operations V

I X is a subset of Y , in symbols X ⊆ Y =df for every x, if x ∈ X , then x ∈ Y .I E.g. {a,b} ⊆ {a,b, c}.

I For every set X , X ⊆ X and ∅ ⊆ X *



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Operations on sets

Operations V

I X is a subset of Y , in symbols X ⊆ Y =df for every x, if x ∈ X , then x ∈ Y .I E.g. {a,b} ⊆ {a,b, c}.I For every set X , X ⊆ X and ∅ ⊆ X *



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Operations on sets

Operations V



I ℘(X) =df {Y : Y ⊆ X}

I E.g. ℘({a,b, c}) = {∅, {a,b, c}, {a,b}, {a, c}, {b, c}, {a}, {b}, {c}}.I Test your understanding : what is ℘(∅) ?

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Operations on sets

Operations V



I ℘(X) =df {Y : Y ⊆ X}I E.g. ℘({a,b, c}) = {∅, {a,b, c}, {a,b}, {a, c}, {b, c}, {a}, {b}, {c}}.

I Test your understanding : what is ℘(∅) ?

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Operations on sets

Operations V




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Relations and functions

Relations and functions I

I Intensional view of a relation : the concept, e.g. taller than.

I Extensional view : the set of lists from a given set that satisfy the relation(concept).

I taller than : concept.Domain : {Paul, Marie, Lucien} ;Relation :[[taller than]]{Paul,Marie,Lucien} = {〈x,y〉 : x,y ∈ {Paul,Marie,Lucien} ∧taller than(x,y)}.

Paul

Marie

Lucien

Extensional view :{〈Marie,Paul〉, 〈Marie,Lucien〉, 〈Paul,Lucien〉}

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I Intensional view of a relation : the concept, e.g. taller than.I Extensional view : the set of lists from a given set that satisfy the relation

(concept).

I taller than : concept.Domain : {Paul, Marie, Lucien} ;Relation :[[taller than]]{Paul,Marie,Lucien} = {〈x,y〉 : x,y ∈ {Paul,Marie,Lucien} ∧taller than(x,y)}.

Paul

Marie

Lucien


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I Intensional view of a relation : the concept, e.g. taller than.I Extensional view : the set of lists from a given set that satisfy the relation

(concept).I taller than : concept.

Domain : {Paul, Marie, Lucien} ;Relation :[[taller than]]{Paul,Marie,Lucien} = {〈x,y〉 : x,y ∈ {Paul,Marie,Lucien} ∧taller than(x,y)}.

Paul

Marie

Lucien


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Relations and functions II

I Relations are everywhere in maths.

I [[Between]]N =df {〈x,y, z〉} : x,y, z ∈ N ∧ x < y < z.I An infinite set of triples : {〈1,2,3〉, 〈78,96,22223〉}, . . ..I Functions are just special relations.I Arguments and result are distinguished and the same arguments always

give the same result.I Ex. : the sum function on integers.

[[Sum]]=df {〈x,y, z〉} : z = x + y.x and y are the arguments, z is the result.

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I Relations are everywhere in maths.I [[Between]]N =df {〈x,y, z〉} : x,y, z ∈ N ∧ x < y < z.

I An infinite set of triples : {〈1,2,3〉, 〈78,96,22223〉}, . . ..I Functions are just special relations.I Arguments and result are distinguished and the same arguments always



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I Relations are everywhere in maths.I [[Between]]N =df {〈x,y, z〉} : x,y, z ∈ N ∧ x < y < z.I An infinite set of triples : {〈1,2,3〉, 〈78,96,22223〉}, . . ..

I Functions are just special relations.I Arguments and result are distinguished and the same arguments always



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I Relations are everywhere in maths.I [[Between]]N =df {〈x,y, z〉} : x,y, z ∈ N ∧ x < y < z.I An infinite set of triples : {〈1,2,3〉, 〈78,96,22223〉}, . . ..I Functions are just special relations.

I Arguments and result are distinguished and the same arguments alwaysgive the same result.

I Ex. : the sum function on integers.[[Sum]]=df {〈x,y, z〉} : z = x + y.x and y are the arguments, z is the result.

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I Relations are everywhere in maths.I [[Between]]N =df {〈x,y, z〉} : x,y, z ∈ N ∧ x < y < z.I An infinite set of triples : {〈1,2,3〉, 〈78,96,22223〉}, . . ..I Functions are just special relations.I Arguments and result are distinguished and the same arguments always

give the same result.

I Ex. : the sum function on integers.[[Sum]]=df {〈x,y, z〉} : z = x + y.x and y are the arguments, z is the result.

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I Relations are everywhere in maths.I [[Between]]N =df {〈x,y, z〉} : x,y, z ∈ N ∧ x < y < z.I An infinite set of triples : {〈1,2,3〉, 〈78,96,22223〉}, . . ..I Functions are just special relations.I Arguments and result are distinguished and the same arguments always



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Relations and functions III

I Relations can be represented as graphs.

Height comparison : Jean > Marie, Marie > Paul, Paul = Louis, Louis < Sophie, Sophie > Jean.

Jean

Marie Paul

Louis Sophie

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Complements

The comprehension axiom I

I Problem with the correspondence between sets and properties.

I Russell’s (1902, letter to Frege) paradox.I Let Y = {X : X 6∈ X}.I Either Y ∈ Y or Y 6∈ Y .I Suppose that Y ∈ Y , then Y 6∈ Y .I Suppose that Y 6∈ Y , then Y ∈ Y .

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Complements


I Problem with the correspondence between sets and properties.I Russell’s (1902, letter to Frege) paradox.

I Let Y = {X : X 6∈ X}.I Either Y ∈ Y or Y 6∈ Y .I Suppose that Y ∈ Y , then Y 6∈ Y .I Suppose that Y 6∈ Y , then Y ∈ Y .

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Complements


I Problem with the correspondence between sets and properties.I Russell’s (1902, letter to Frege) paradox.I Let Y = {X : X 6∈ X}.

I Either Y ∈ Y or Y 6∈ Y .I Suppose that Y ∈ Y , then Y 6∈ Y .I Suppose that Y 6∈ Y , then Y ∈ Y .

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Complements


I Problem with the correspondence between sets and properties.I Russell’s (1902, letter to Frege) paradox.I Let Y = {X : X 6∈ X}.I Either Y ∈ Y or Y 6∈ Y .

I Suppose that Y ∈ Y , then Y 6∈ Y .I Suppose that Y 6∈ Y , then Y ∈ Y .

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Complements


I Problem with the correspondence between sets and properties.I Russell’s (1902, letter to Frege) paradox.I Let Y = {X : X 6∈ X}.I Either Y ∈ Y or Y 6∈ Y .I Suppose that Y ∈ Y , then Y 6∈ Y .

I Suppose that Y 6∈ Y , then Y ∈ Y .

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Complements


I Problem with the correspondence between sets and properties.I Russell’s (1902, letter to Frege) paradox.I Let Y = {X : X 6∈ X}.I Either Y ∈ Y or Y 6∈ Y .I Suppose that Y ∈ Y , then Y 6∈ Y .I Suppose that Y 6∈ Y , then Y ∈ Y .

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Complements

The comprehension axiom II

I To eliminate this paradox, axiomatic set theory (non-naive) posits a specialaxiom, the Comprehension axiom (2).

I Intuitive interpretation : choose an arbitrary set (z) and a constraint φ,any subset of z whose elements satisfy φ is a set.

I To construct a set from a property, you must apply the property to analready existing set.

I And if I have no set in the first place ? An Existence axiom takes care ofthis (3)

(2) Comprehension For each formula φ where y is not free and each setdenoted by z, there is a set denoted by y such that x ∈ y iff x ∈ z and φ istrue.

(3) Existence There exists (at least) one set.

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Complements

The comprehension axiom III

I What happens with Russell’s paradox ?

I {X : X 6∈ X} must be replaced by {X : X ∈ Y ∧ X 6∈ X}.I Y must be the set of all sets, so Y ∈ Y .I This is impossible, as we are going to see shortly.

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Complements


I What happens with Russell’s paradox ?I {X : X 6∈ X} must be replaced by {X : X ∈ Y ∧ X 6∈ X}.

I Y must be the set of all sets, so Y ∈ Y .I This is impossible, as we are going to see shortly.

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Complements


I What happens with Russell’s paradox ?I {X : X 6∈ X} must be replaced by {X : X ∈ Y ∧ X 6∈ X}.I Y must be the set of all sets, so Y ∈ Y .

I This is impossible, as we are going to see shortly.

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Complements


I What happens with Russell’s paradox ?I {X : X 6∈ X} must be replaced by {X : X ∈ Y ∧ X 6∈ X}.I Y must be the set of all sets, so Y ∈ Y .I This is impossible, as we are going to see shortly.

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Complements

Hypersets I

I A set can be infinite.

I Infinite is vague.

1. Horizontal infinity : a set can have infinitely many elements but a finite numberof ‘levels’.

2. Vertical infinity (depth) : a set can have an infinite number of ‘levels’

*NOT INSTANDARD SET THEORY

Forbiden configuration

{. . .a . . .}

{. . .b . . .}

{. . . c . . .}

∞

. . . ∈ c ∈ b ∈ a

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Complements

Hypersets I

I A set can be infinite.I Infinite is vague.


2. Vertical infinity (depth) : a set can have an infinite number of ‘levels’

*NOT INSTANDARD SET THEORY


{. . .a . . .}

{. . .b . . .}

{. . . c . . .}

∞

. . . ∈ c ∈ b ∈ a

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Complements

Hypersets I

I A set can be infinite.I Infinite is vague.


2. Vertical infinity (depth) : a set can have an infinite number of ‘levels’ *NOT INSTANDARD SET THEORY


{. . .a . . .}

{. . .b . . .}

{. . . c . . .}

∞

. . . ∈ c ∈ b ∈ a

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Complements

Hypersets II

I In standard set theory the Foundation axiom precludes the existence ofsets of infinite depth.

I In particular X ∈ X is not possible (as a result, no Russell’s paradox). Lookcarefully into ([Foundation-axiom]).

I Every subset must respect Foundation. So (4) is equivalent to saying thatevery non-empty subset of a set must have a ∈-minimal element.

I Test your understanding : does an infinite list respect the Foundationaxiom ? Solution

(4) Every non-empty set has at least one element x such that nothing is anelement of x (a ∈ –minimal element in mathematical terminology).

∀x((∃y ∈ x)⇒ ∃z ∈ x(¬∃u(u ∈ x ∧ u ∈ z)))

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Complements

Hypersets III

I In the theory of non-well-founded sets (Aczel 1988), sets can have infinitedepth.

I There, sets are actually finite or infinite graphs. Loops and infinitebranches are possible.

I Hypersets defined by equational systems (finite or infinite).I Examples of hypersets (the arrows are the ∈–relation).

X

X = {X}

X

Y

Z

3

3

X = {Y}

Y = {Z}

Z = {. . .}

etc.

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Complements

Hypersets III



I Hypersets defined by equational systems (finite or infinite).

I Examples of hypersets (the arrows are the ∈–relation).

X

X = {X}

X

Y

Z

3

3

X = {Y}

Y = {Z}

Z = {. . .}

etc.

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Complements

Hypersets III



I Hypersets defined by equational systems (finite or infinite).I Examples of hypersets (the arrows are the ∈–relation).

X

X = {X}

X

Y

Z

3

3

X = {Y}

Y = {Z}

Z = {. . .}

etc.

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Complements

{THE, END}

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Complements

I Does an infinite list respect the Foundation axiom ?

I If you have an infinite consecutive repetition, you clearly violate the axiom,

because you will end up with a set of the form

∞︷︸︸︷{. . . {{a

∞︷︸︸︷}} . . .} (see slide 9).

I What happens in other cases, ? Start from an infinite list Λ =〈x1, x2, . . . , xα〉, where α is some non-finite ordinal.

I We rewrite Λ.Λ ; {{x1}, {x1, 〈x2, . . . , xα〉}};

{{x1}, {x1, {{x2}, {x2, 〈x3, . . . , xα〉}}}};{{x1}, {x1, {{x2}, {x2, {{x3}, {x3, 〈x4, . . . , xα〉}}}}}; etc.,

i.e. {{x1}, {x1, {{x2}, {x2, {{x3}, {x3, . . . {{xk}, {xk , 〈xk+1, . . . , xα〉}2k−1

} . . .}

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Complements

I Does an infinite list respect the Foundation axiom ?I If you have an infinite consecutive repetition, you clearly violate the axiom,

because you will end up with a set of the form

∞︷︸︸︷{. . . {{a

∞︷︸︸︷}} . . .} (see slide 9).

I What happens in other cases, ? Start from an infinite list Λ =〈x1, x2, . . . , xα〉, where α is some non-finite ordinal.

I We rewrite Λ.Λ ; {{x1}, {x1, 〈x2, . . . , xα〉}};

{{x1}, {x1, {{x2}, {x2, 〈x3, . . . , xα〉}}}};{{x1}, {x1, {{x2}, {x2, {{x3}, {x3, 〈x4, . . . , xα〉}}}}}; etc.,

i.e. {{x1}, {x1, {{x2}, {x2, {{x3}, {x3, . . . {{xk}, {xk , 〈xk+1, . . . , xα〉}2k−1

} . . .}

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Complements

I If we write this list as a tree, we ‘see’ that its rightmost branch is infinite,which is not compatible with Foundation.

I Conclusion : no infinite list is well-founded.

Λ

{x1} Λ1

{x2} Λ2

x3 Λ3

Λ4

The branches correspond to ∈. Therightmost branch is then an infinitedescending ∈-chain.

Naive Set Theory - École normale supérieure de...

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