Transcript Cardinals - Purdue University

Cardinals
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Georg Cantor (1845-1918) thought of a cardinal as a special represenative.
Bertrand Russell (1872-1970) and Gottlob Frege (1848-1925) used the following
definition:
M  {N | N ~ M}
This type of formulation is subject to Russell’s paradox given by the “set”
X  {M | M  M }
X cannot be a set since if it is either
a)
X  X or
b)
X X
1. If a) holds then by the definition of X: X  X which contradicts the
assumption a)
2. On the other hand if b) holds the by the definition of X: X  X
which contradicts the assumption b).
To avoid this one has to be careful which formulas of the type {x| p(x)} are allowed
to build sets.
This leads to the concept of class. There is a class of all sets, but the “set of all sets”
is not well defined.
In fact the cardinals would not be a set, but a class.
One can go back to Cantor and choose a representative, but this involves the axiom
of choice.
Nowadays one uses John von Neumann’s (1903-1957) definition in terms of
ordinals. For this one needs well ordering and thus the axiom of choice.
Orders of Sets
• A set S is called partially ordered if there exists a relation
r (usually denoted by the symbol ≤) between S and itself
such that the following conditions are satisfied:
• reflexive: a ≤ a for any element a in S
• transitive: if a ≤ b and b ≤ c then a ≤ c
• antisymmetric: if a ≤ b and b ≤ a then a = b
• A set S is called ordered if it is partially ordered and every
pair of elements x and y from the set S can be compared
with each other via the partial ordering relation.
• A set S is called well-ordered if it is an ordered set for
which every non-empty subset contains a smallest element.
Ordinals
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Consider pair (S,<S) of a set S and
a well-ordering <S on S.
We call (S,<S) and (T,<T)
equivalent
(S,<S)~(T,<T)
if there is a 1-1 correspondence,
which preserves the orders, i.e. a
bijection: f:S T, s.t.
a<b <=> f(a)<f(b)
According to Cantor, the
equivalence classes of this
equivalence relation are ordinals,
abstract from the nature of the
elements to obtain the order type.
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Ordinals can be constructed, from the ZF
system.
Von Neumann (1903-1957) constructs
ordinals as special types of sets, i.e.
representatives
A set S is an ordinal if and only if S is
totally ordered with respect to set
containment and every element of S is
also a subset of S.
Call (S,<S)<(T,<T) if and only if S is
order isomorphic to an initial segment T1
of T, i.e. there is a k in T such that
S~T1={a|a<k}
Ordinals themselves are well-ordered with respect to the order induced by <
Orders on Ordinals and Cardinals
• Questions:
– Can every set be well-ordered?
• Yes (Zermelo), if one assumes the axiom of choice.
– Is there an order for ordinals and cardinals?
• This is the case for the ordinals and cardinals of finite sets.
• Ordinals can be ordered.
• Cardinals can be ordered if all sets can be well-ordered which
is equivalent (Zermelo) to the axiom of choice.
Arithmetic of Ordinals and the sequence 
• Let w the ordinal of N in its natural
order.
• To add two ordinals A=(A,<A) and
B=(B,<B) in the following way:
A+B:=(AUB,<AUB)
where x <AUBy if either
1. x,y  A and x<Ay
2. x,y  B and x<By
3. x  A and y  B
• Caveat: + is not commutative
–
3+w=w
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w+3>w
Ordinals form a ascending sequence
1,2,…,w, w+1, w+2,…, w+w,w+w+1,…
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Now to each ordinal, we can associate
its cardinal. E.g. the cardinal of w is 0.
This is also the cardinal for w+1, w+w
Going along the sequence of ordinals,
we will however discover more
cardinals, one after the other. In this
way let 1 be the first new cardinal
after 0 and denote the sequence of
cardinals obtained in this way by 
Zermelo showed that assuming the
axiom of choice every set can be well
ordered.
Thus assuming the axiom of choice all
cardinals are among the  and the
cardinals are well ordered.