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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]
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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]
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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]
|
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]
|
10150
|
The Trichotomy Principle is equivalent to the Axiom of Choice
[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]
|
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]
|