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Single Idea 12321
[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 actually amounts to admitting an absolutely indeterminate infinite set whose existence is asserted albeit remaining linguistically indefinable. On the other hand, as a process, it is unconstructible.
Gist of Idea
The axiom of choice must accept an indeterminate, indefinable, unconstructible set
Source
Alain Badiou (Briefings on Existence [1998], 2)
Book Ref
Badiou,Alain: 'Briefings on Existence', ed/tr. Madarsz,Norman [SUNY 2006], p.52
A Reaction
If only constructible sets are admitted (see 'V = L') then there is a contradiction.
The
24 ideas
from 'Briefings on Existence'
12318
|
The female body, when taken in its entirety, is the Phallus itself
[Badiou]
|
12320
|
Ontology is (and always has been) Cantorian mathematics
[Badiou]
|
12331
|
Logic is definitional, but real mathematics is axiomatic
[Badiou]
|
12338
|
We must either assert or deny any single predicate of any single subject
[Badiou]
|
12332
|
The modern view of Being comes when we reject numbers as merely successions of One
[Badiou]
|
12337
|
There is 'transivity' iff membership ∈ also means inclusion ⊆
[Badiou]
|
12334
|
There is no single unified definition of number
[Badiou]
|
12335
|
Numbers are for measuring and for calculating (and the two must be consistent)
[Badiou]
|
12333
|
Each type of number has its own characteristic procedure of introduction
[Badiou]
|
12342
|
Topos theory explains the plurality of possible logics
[Badiou]
|
12341
|
Logic is a mathematical account of a universe of relations
[Badiou]
|
12340
|
There is no Being as a whole, because there is no set of all sets
[Badiou]
|
12323
|
Existence is Being itself, but only as our thought decides it
[Badiou]
|
12322
|
Must we accept numbers as existing when they no longer consist of units?
[Badiou]
|
12321
|
The axiom of choice must accept an indeterminate, indefinable, unconstructible set
[Badiou]
|
12324
|
Consensus is the enemy of thought
[Badiou]
|
12325
|
Philosophy has been relieved of physics, cosmology, politics, and now must give up ontology
[Badiou]
|
12327
|
The undecidability of the Continuum Hypothesis may have ruined or fragmented set theory
[Badiou]
|
12326
|
The primitive name of Being is the empty set; in a sense, only the empty set 'is'
[Badiou]
|
12329
|
If mathematics is a logic of the possible, then questions of existence are not intrinsic to it
[Badiou]
|
12328
|
Platonists like axioms and decisions, Aristotelians like definitions, possibilities and logic
[Badiou]
|
12330
|
In ontology, logic dominated language, until logic was mathematized
[Badiou]
|
12316
|
For Enlightenment philosophers, God was no longer involved in politics
[Badiou]
|
12317
|
The God of religion results from an encounter, not from a proof
[Badiou]
|