15: Terminal Objects and Initial Objects
Topics
1. Terminal Objects
2. Points
3. Map-Separating Objects
4. Initial Objects
1. Terminal Objects
Definition:
An object S of a category C is a terminal object of C if for each object X of C, there is exactly on
14: Generalized Elements in S
Structure-preserving maps from a cycle to another endomap
Let X and Y be the S -objects (i.e., dynamical systems):
0
y
X =
3
Y =
1
f
X
f
Y
X
such that f(0) = y.
f=f
or
f(x) = (f(x), for all x in X
Y
f
w
z
2
Find an S -map X
14: More Categories
Topics
1. The Category of
Endomaps of Sets
2. The Category of Graphs
Recall: So far we've been talking about the category of sets, call it S.
I.
S-objects.
Sets
II. S-maps.
Maps between sets. Each is defined by specifying:
(1)
a domain
12: Article II - Isomorphisms
Topics
1. Isomorphisms
2. Retractions and Sections
1. Isomorphisms
Idea: Isomorphisms are maps that preserve "similarity".
Suppose we have 4 collections:
A
B
mother
C
feather
father
bone
rock
stone
child
D
sling
arrow
11: Category Theory
Theory of objects and maps (relations between objects).
A category consists of a type of object and an associated map
Topics
I.
Category of Finite Sets
II. Endomap, Identity Map, Composite Map
III. Definition of a Category
IV. Points o
10. Problems with ZF Set Theory: Gdel's Incompleteness Theorems
Gdels 1st
Incompleteness
Theorem
Gdels 2nd
Incompleteness
Theorem
implies
There are claims about sets that cannot
be proven or refuted within ZF.
To refute a claim A is
to prove its negation.
9: Problems with ZF Set Theory: The Skolem Paradox
Recall: ZF Set Theory - attempt to mathematically codify the concept of infinity
I. Advantages of ZF Set Theory:
(A) Precise notion of set: avoids paradoxes of the One and the Many (Russells paradox, Set
08. Zermelo-Fraenkel (ZF) Formal Set Theory
Motivation: To translate Naive Set Theory into a formal system (see lecture on Rigorization and Proof)
Primitives of ZF:
Individuals (infinite):
sets (pure iterative sets)
One Property:
set-membership (denoted b
07. Cantors Theory of Ordinal and Cardinal Numbers
Ordinals - measure the "length" or "shape" of a set
Cardinals - measure the "size" of a set (in correlation sense)
Topics
I.
Ordinals
II. Iterative Conception of a Set
III. Ordinals as Sets
IV. Cardinal N
06. Naive Set Theory
Topics
I. Sets and Paradoxes of the
Infinitely Big
II. Naive Set Theory
III. Cantor and Diagonal Arguments
I. Sets and Paradoxes of the Infinitely Big
Recall: Paradox of the Even Numbers
Claim: There are just as many even natural numb
05. Rigorization and Proof
(Reference : Brown, J. (1999) Philosophy of Mathematics , Chap. 3.)
I. The Notion of a Proof in a Formal System
Topics
I. Formal System and Proof
II. Analytical Proofs
III. Picture Proofs
One way to view mathematics: mathematics
04. The Calculus
Topics
I. The Calculus and Infinitesimals
II. The Status of Infinitesimals
III. The Concept of a Limit
IV. Limits and Derivatives
V. Limits and Infinite Sums.
VI. Infinite Sums and Integrals.
I. The Calculus and Infinitesimals
2 Sets of P
03. Early Greeks & Aristotle
Topics
I. Early Greeks
II. The Method of Exhaustion
III. Aristotle
I. Early Greeks
1. Anaximander (b. 610 B.C.)
to apeiron -
the unlimited, unbounded
fundamental substance of reality
underlying substratum for change
neutral su
Topics
I. Paradoxes of the Infinitely Small
II. Paradoxes of the Infinitely Big
III. Paradoxes of the One and the Many
IV. Paradoxes of Thought about the Infinite
02. Paradoxes
The Infinite - Two clusters of concepts:
boundlessness
endlessness
unlimitedne
Topics
I. The Branches of Mathematics
II. A Beastiary of Number Systems
Course Intro
I. The Branches of Mathematics
According to the Greeks.
Mathematics
The discrete
The continuous
The absolute
The relative
The static
The moving
Arithmetic
Music
Geometry