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Single Idea 12337

[filed under theme 4. Formal Logic / F. Set Theory ST / 2. Mechanics of Set Theory / b. Terminology of ST ]

Full Idea

'Transitivity' signifies that all of the elements of the set are also parts of the set. If you have α∈Β, you also have α⊆Β. This correlation of membership and inclusion gives a stability which is the sets' natural being.

Gist of Idea

There is 'transivity' iff membership ∈ also means inclusion ⊆

Source

Alain Badiou (Briefings on Existence [1998], 11)

Book Ref

Badiou,Alain: 'Briefings on Existence', ed/tr. Madarsz,Norman [SUNY 2006], p.128

Related Ideas

Idea 13443 ∈ relates across layers, while ⊆ relates within layers [Hart,WD]

Idea 13201 ∈ says the whole set is in the other; ⊆ says the members of the subset are in the other [Enderton]

Idea 15500 Classes divide into subclasses in many ways, but into members in only one way [Lewis]

Idea 15499 A subclass of a subclass is itself a subclass; a member of a member is not in general a member [Lewis]

The 30 ideas from Alain Badiou

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]
12335 Numbers are for measuring and for calculating (and the two must be consistent) [Badiou]
12334 There is no single unified definition of number [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]
12321 The axiom of choice must accept an indeterminate, indefinable, unconstructible set [Badiou]
12324 Consensus is the enemy of thought [Badiou]
12322 Must we accept numbers as existing when they no longer consist of units? [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]
9808 Philosophy aims to reveal the grandeur of mathematics [Badiou]
9809 Mathematics inscribes being as such [Badiou]
9811 It is of the essence of being to appear [Badiou]
9812 In mathematics, if a problem can be formulated, it will eventually be solved [Badiou]
9813 Mathematics shows that thinking is not confined to the finite [Badiou]
9814 All great poetry is engaged in rivalry with mathematics [Badiou]