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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]
14445 Choice shows that if any two cardinals are not equal, one must be the greater [Russell]
14444 Choice is equivalent to the proposition that every class is well-ordered [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]
13462 With the Axiom of Choice every set can be well-ordered [Hart,WD]
13461 We can choose from finite and evident sets, but not from infinite opaque ones [Hart,WD]
13024 Efforts to prove the Axiom of Choice have failed [Maddy]
13025 Modern views say the Choice set exists, even if it can't be constructed [Maddy]
13026 A large array of theorems depend on the Axiom of Choice [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]