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Some notes for class

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Blackboard Stu
ﬀ
: 1
/
30 and 2
/
1
M
/
PHL 344K
1. Two principles of naive set theory:(a)
Extensionality
: Twosetsareidenticalifandonlyiftheyhaveexactlythe same members.
ã ∀
x
∀
y
(
x
=
y
↔ ∀
z
(
z
∈
x
↔
z
∈
y
)(b)
Comprehension
: Given any property P, or any open sentence S inone free variable, there is a set of all and only things having P, orsatisfying S.
ã ∃
x
∀
y
(
y
∈
x
↔
S
(
y
))(For the cognoscenti: this is an axiom schema, and hence speci-ﬁesaninﬁnitesetofaxioms(oneforeachopensentenceSinourpreferred logical language).)2. A contradiction in naive set theory:
ã
Let S be the open sentence
x
x
. Then by Comprehension,
∃
x
∀
y
(
y
∈
x
↔
y
y
. By instantiating
y
to
x
, we get the contradiction
∃
x
(
x
∈
x
↔
x
x
).3. The standard revised axiomatization of set theory (we didn’t go over thisin class, and won’t really need these details, but here they are, in caseyou’re interested):(a)
Extensionality
: twosetsareidenticalifandonlyiftheyhaveexactlythe same elements. (As before.)(b)
Pairing
: Given any two sets
x
and
y
, there is a set
{
x
,
y
}
.(c)
Union
: Given any set
x
, there is a set of all members of members of
x
.(d)
Restricted Comprehension
: Given any set
x
and any open sentence
S
, there is a set of all members of
x
that satisfy
S
.(e)
Inﬁnity
: There is an inﬁnite set.(f)
Replacement
: Given any set
x
and function
f
, there is a set of theimages of the members of
x
under
f
.(g)
Foundation
: Every (non-empty) set contains a member with whichit has an empty intersection.1
(h)
Power Set
: Given any set
x
, there is a set of all subsets of
x
.(i)
Choice
: Given any set
x
, there is a set that contains exactly onemember of every non-empty member of
x
.4. Some properties of binary relations (in class, these were given treating
R
asasetoforderedpairs. HereIgivethemtreatingRasabinarypredicate):(a)
Reﬂexive
:
∀
x Rxx
(b)
Irreﬂexive
:
∀
x
¬
Rxx
(c)
Symmetric
:
∀
x
∀
y
(
Rxy
→
Ryx
)(d)
Asymmetric
:
∀
x
∀
y
(
Rxy
→ ¬
Ryx
)(e)
Antisymmetric
:
∀
x
∀
y
((
Rxy
∧
Ryx
)
→
x
=
y
)(f)
Transitive
:
∀
x
∀
y
∀
z
((
Rxy
∧
Ryz
)
→
Rxz
)(g)
Intransitive
:
∀
x
∀
y
∀
z
((
Rxy
∧
Ryz
)
→ ¬
Rxz
)5. Equivalence relations:(a) AnequivalencerelationonAisabinaryrelationonAthatisreﬂexive,symmetric, and transitive.(b) Given an equivalence relation R on A, the
equivalence classes
gen-erated by R are sets of the form
{
b
:
b
∈
A
∧
Rab
}
, for each
a
∈
A
.(c) A
partition
on A is a collection of non-empty subsets of A which arepairwise disjoint and collectively exhaustive of A.(d)
Fact1
: If
R
isanequivalencerelationonA,thenthesetofequivalenceclasses generated by
R
form a partition of A.(e)
Fact 2
: If
P
is a partition of
A
, the relation
being in the same partitionelement as
is an equivalence relation on A.(f)
Fact 3
: If
R
is an equivalence relation, then the equivalence relationgenerated by the partition generated by
R
is
R
. And if
P
is a parti-tion on
A
, then the partition generated by the equivalence relationgenerated by
P
is
P
.2

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