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Single Idea 10542

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

Full Idea

We may suppose that with each set is associated an object as its cardinal number, but we have no systematic way, without appeal to the Axiom of Choice, of selecting a representative set of each cardinality.

Gist of Idea

To associate a cardinal with each set, we need the Axiom of Choice to find a representative

Source

Michael Dummett (Frege Philosophy of Language (2nd ed) [1973], Ch.14)

Book Ref

Dummett,Michael: 'Frege Philosophy of Language' [Duckworth 1981], p.484

The 23 ideas from 'Frege Philosophy of Language (2nd ed)'

10320 If a genuine singular term needs a criterion of identity, we must exclude abstract nouns [Dummett, by Hale]
19168 Concepts only have a 'functional character', because they map to truth values, not objects [Dummett, by Davidson]
10549 Since abstract objects cannot be picked out, we must rely on identity statements [Dummett]
10516 A realistic view of reference is possible for concrete objects, but not for abstract objects [Dummett, by Hale]
10547 Abstract objects can never be confronted, and need verbal phrases for reference [Dummett]
10531 There is a modern philosophical notion of 'object', first introduced by Frege [Dummett]
17621 What matters in mathematics is its objectivity, not the existence of the objects [Dummett]
10537 The ordered pairs <x,y> can be reduced to the class of sets of the form {{x},{x,y}} [Dummett]
10542 To associate a cardinal with each set, we need the Axiom of Choice to find a representative [Dummett]
10554 Intuitionists find the Incompleteness Theorem unsurprising, since proof is intuitive, not formal [Dummett]
10540 We can't say that light is concrete but radio waves abstract [Dummett]
10552 Intuitionism says that totality of numbers is only potential, but is still determinate [Dummett]
10515 Ostension is possible for concreta; abstracta can only be referred to via other objects [Dummett, by Hale]
10544 The concrete/abstract distinction seems crude: in which category is the Mistral? [Dummett]
10546 We don't need a sharp concrete/abstract distinction [Dummett]
10548 The context principle for names rules out a special philosophical sense for 'existence' [Dummett]
10281 The objects we recognise the world as containing depends on the structure of our language [Dummett]
10532 We can understand universals by studying predication [Dummett]
10534 'Nominalism' used to mean denial of universals, but now means denial of abstract objects [Dummett]
10545 Abstract objects may not cause changes, but they can be the subject of change [Dummett]
10541 Concrete objects such as sounds and smells may not be possible objects of ostension [Dummett]
10555 If we can intuitively apprehend abstract objects, this makes them observable and causally active [Dummett]
10543 Abstract objects must have names that fall within the range of some functional expression [Dummett]