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Single Idea 14091

[filed under theme 7. Existence / C. Structure of Existence / 4. Ontological Dependence ]

Full Idea

We can give an exhaustive account of the identity of the empty set and its singleton without mentioning infinite sets, and it might be possible to defend the view that one natural number depends on its predecessor but not vice versa.

Gist of Idea

There may be a one-way direction of dependence among sets, and among natural numbers

Source

Øystein Linnebo (Structuralism and the Notion of Dependence [2008], V)

Book Ref

-: 'The Philosophical Quarterly' [-], p.73

A Reaction

Linnebo uses this as one argument against mathematical structuralism, where the small seems to depend on the large. The view of sets rests on the iterative conception, where each level is derived from a lower level. He dismisses structuralism of sets.

The 17 ideas with the same theme [things that rely on other things for their existence]:

1699 A thing is prior to another if it implies its existence [Aristotle]
18366 Of interdependent things, the prior one causes the other's existence [Aristotle]
24057 What is prior is always potentially present in what is next in order [Aristotle]
11154 Prior things can exist without posterior things, but not vice versa [Aristotle]
17443 Many of us find Frege's claim that truths depend on one another an obscure idea [Heck on Frege]
17445 Parallelism is intuitive, so it is more fundamental than sameness of direction [Frege, by Heck]
17656 Being primitive or prior always depends on a constructional system [Goodman]
11919 Ontological dependence rests on essential connection, not necessary connection [Molnar]
11151 An object is dependent if its essence prevents it from existing without some other object [Fine,K]
14251 A natural modal account of dependence says x depends on y if y must exist when x does [Fine,K]
14257 An object depends on another if the second cannot be eliminated from the first's definition [Fine,K]
14254 Dependency is the real counterpart of one term defining another [Fine,K]
14261 There is 'weak' dependence in one definition, and 'strong' dependence in all the definitions [Fine,K]
12879 Independent objects can exist apart, and maybe even entirely alone [Simons]
14091 There may be a one-way direction of dependence among sets, and among natural numbers [Linnebo]
17327 Non-causal dependence is at present only dimly understood [Liggins]
17720 There's essential, modal, explanatory, conceptual, metaphysical and constitutive dependence [Jenkins, by PG]