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Single Idea 14444
[filed under theme 4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / j. Axiom of Choice IX
]
Full Idea
Zermelo has shown that [the Axiom of Choice] is equivalent to the proposition that every class is well-ordered, i.e. can be arranged in a series in which every sub-class has a first term (except, of course, the null class).
Gist of Idea
Choice is equivalent to the proposition that every class is well-ordered
Source
Bertrand Russell (Introduction to Mathematical Philosophy [1919], XII)
Book Ref
Russell,Bertrand: 'Introduction to Mathematical Philosophy' [George Allen and Unwin 1975], p.123
A Reaction
Russell calls Choice the 'Multiplicative' Axiom.
The
34 ideas
with the same theme
[axiom for a new set by sampling]:
14443
|
The British parliament has one representative selected from each constituency
[Russell]
|
14444
|
Choice is equivalent to the proposition that every class is well-ordered
[Russell]
|
14445
|
Choice shows that if any two cardinals are not equal, one must be the greater
[Russell]
|
14446
|
We can pick all the right or left boots, but socks need Choice to insure the representative class
[Russell]
|
10196
|
The Axiom of Choice needs a criterion of choice
[Black]
|
10542
|
To associate a cardinal with each set, we need the Axiom of Choice to find a representative
[Dummett]
|
13205
|
We can only define functions if Choice tells us which items are involved
[Enderton]
|
10775
|
The axiom of choice now seems acceptable and obvious (if it is meaningful)
[Tharp]
|
12321
|
The axiom of choice must accept an indeterminate, indefinable, unconstructible set
[Badiou]
|
18139
|
The Axiom of Choice relies on reference to sets that we are unable to describe
[Bostock]
|
13418
|
The old problems with the axiom of choice are probably better ascribed to the law of excluded middle
[Parsons,C]
|
13036
|
Choice: ∀A ∃R (R well-orders A)
[Kunen]
|
18270
|
Choice suggests that intensions are not needed to ensure classes
[Coffa]
|
13461
|
We can choose from finite and evident sets, but not from infinite opaque ones
[Hart,WD]
|
13462
|
With the Axiom of Choice every set can be well-ordered
[Hart,WD]
|
13024
|
Efforts to prove the Axiom of Choice have failed
[Maddy]
|
13026
|
A large array of theorems depend on the Axiom of Choice
[Maddy]
|
13025
|
Modern views say the Choice set exists, even if it can't be constructed
[Maddy]
|
17610
|
The Axiom of Choice paradoxically allows decomposing a sphere into two identical spheres
[Maddy]
|
13647
|
Choice is essential for proving downward Löwenheim-Skolem
[Shapiro]
|
10301
|
The axiom of choice is controversial, but it could be replaced
[Shapiro]
|
10208
|
Axiom of Choice: some function has a value for every set in a given set
[Shapiro]
|
10252
|
The Axiom of Choice seems to license an infinite amount of choosing
[Shapiro]
|
15717
|
Using Choice, you can cut up a small ball and make an enormous one from the pieces
[Kaplan/Kaplan]
|
10879
|
Choice: For every set a mechanism will choose one member of any non-empty subset
[Clegg]
|
10146
|
Cantor's theories needed the Axiom of Choice, but it has led to great controversy
[Feferman/Feferman]
|
10147
|
The Axiom of Choice is consistent with the other axioms of set theory
[Feferman/Feferman]
|
10148
|
Axiom of Choice: a set exists which chooses just one element each of any set of sets
[Feferman/Feferman]
|
10149
|
Platonist will accept the Axiom of Choice, but others want criteria of selection or definition
[Feferman/Feferman]
|
10150
|
The Trichotomy Principle is equivalent to the Axiom of Choice
[Feferman/Feferman]
|
15898
|
The controversy was not about the Axiom of Choice, but about functions as arbitrary, or given by rules
[Lavine]
|
15920
|
Pure collections of things obey Choice, but collections defined by a rule may not
[Lavine]
|
10676
|
The Axiom of Choice is a non-logical principle of set-theory
[Hossack]
|
10686
|
The Axiom of Choice guarantees a one-one correspondence from sets to ordinals
[Hossack]
|