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Single Idea 14446

[filed under theme 4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / j. Axiom of Choice IX ]

Full Idea

Among boots we distinguish left and right, so we can choose all the right or left boots; with socks no such principle suggests itself, and we cannot be sure, without the [Axiom of Choice], that there is a class consisting of one sock from each pair.

Gist of Idea

We can pick all the right or left boots, but socks need Choice to insure the representative class

Source

Bertrand Russell (Introduction to Mathematical Philosophy [1919], XII)

Book Ref

Russell,Bertrand: 'Introduction to Mathematical Philosophy' [George Allen and Unwin 1975], p.126


A Reaction

A deservedly famous illustration of a rather tricky part of set theory.


The 34 ideas with the same theme [axiom for a new set by sampling]:

The British parliament has one representative selected from each constituency [Russell]
Choice shows that if any two cardinals are not equal, one must be the greater [Russell]
Choice is equivalent to the proposition that every class is well-ordered [Russell]
We can pick all the right or left boots, but socks need Choice to insure the representative class [Russell]
The Axiom of Choice needs a criterion of choice [Black]
To associate a cardinal with each set, we need the Axiom of Choice to find a representative [Dummett]
We can only define functions if Choice tells us which items are involved [Enderton]
The axiom of choice now seems acceptable and obvious (if it is meaningful) [Tharp]
The axiom of choice must accept an indeterminate, indefinable, unconstructible set [Badiou]
The Axiom of Choice relies on reference to sets that we are unable to describe [Bostock]
The old problems with the axiom of choice are probably better ascribed to the law of excluded middle [Parsons,C]
Choice: ∀A ∃R (R well-orders A) [Kunen]
Choice suggests that intensions are not needed to ensure classes [Coffa]
With the Axiom of Choice every set can be well-ordered [Hart,WD]
We can choose from finite and evident sets, but not from infinite opaque ones [Hart,WD]
Efforts to prove the Axiom of Choice have failed [Maddy]
Modern views say the Choice set exists, even if it can't be constructed [Maddy]
A large array of theorems depend on the Axiom of Choice [Maddy]
The Axiom of Choice paradoxically allows decomposing a sphere into two identical spheres [Maddy]
Choice is essential for proving downward Löwenheim-Skolem [Shapiro]
The axiom of choice is controversial, but it could be replaced [Shapiro]
Axiom of Choice: some function has a value for every set in a given set [Shapiro]
The Axiom of Choice seems to license an infinite amount of choosing [Shapiro]
Using Choice, you can cut up a small ball and make an enormous one from the pieces [Kaplan/Kaplan]
Choice: For every set a mechanism will choose one member of any non-empty subset [Clegg]
Cantor's theories needed the Axiom of Choice, but it has led to great controversy [Feferman/Feferman]
The Axiom of Choice is consistent with the other axioms of set theory [Feferman/Feferman]
Axiom of Choice: a set exists which chooses just one element each of any set of sets [Feferman/Feferman]
Platonist will accept the Axiom of Choice, but others want criteria of selection or definition [Feferman/Feferman]
The Trichotomy Principle is equivalent to the Axiom of Choice [Feferman/Feferman]
The controversy was not about the Axiom of Choice, but about functions as arbitrary, or given by rules [Lavine]
Pure collections of things obey Choice, but collections defined by a rule may not [Lavine]
The Axiom of Choice is a non-logical principle of set-theory [Hossack]
The Axiom of Choice guarantees a one-one correspondence from sets to ordinals [Hossack]