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Single Idea 10149

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

Full Idea

The Axiom of Choice seems clearly true from the Platonistic point of view, independently of how sets may be defined, but is rejected by those who think such existential claims must show how to pick out or define the object claimed to exist.

Gist of Idea

Platonist will accept the Axiom of Choice, but others want criteria of selection or definition

Source

Feferman / Feferman (Alfred Tarski: life and logic [2004], Int I)

Book Ref

Feferman,S/Feferman,A.B.: 'Alfred Tarski: life and logic' [CUP 2008], p.47

A Reaction

The typical critics are likely to be intuitionists or formalists, who seek for both rigour and a plausible epistemology in our theory.

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]