more on this theme | more from this thinker
Full Idea
Given any x we have the singleton {x}, which is defined by the pairing axiom to be {x,x}.
Gist of Idea
The singleton is defined using the pairing axiom (as {x,x})
Source
Herbert B. Enderton (Elements of Set Theory [1977], 2:19)
Book Ref
Enderton,Herbert B.: 'Elements of Set Theory' [Posts + Telecoms 2006], p.19
A Reaction
An interesting contrivance which is obviously aimed at keeping the axioms to a minimum. If you can do it intuitively with a new axiom, or unintuitively with an existing axiom - prefer the latter!
| 13200 | Note that {Φ} =/= Φ, because Φ ∈ {Φ} but Φ ∉ Φ [Enderton] |
| 13199 | The empty set may look pointless, but many sets can be constructed from it [Enderton] |
| 13201 | ∈ says the whole set is in the other; ⊆ says the members of the subset are in the other [Enderton] |
| 13202 | Fraenkel added Replacement, to give a theory of ordinal numbers [Enderton] |
| 13203 | The singleton is defined using the pairing axiom (as {x,x}) [Enderton] |
| 13204 | The 'ordered pair' <x,y> is defined to be {{x}, {x,y}} [Enderton] |
| 13205 | We can only define functions if Choice tells us which items are involved [Enderton] |
| 13206 | A 'linear or total ordering' must be transitive and satisfy trichotomy [Enderton] |