more from this thinker     |     more from this text

Single Idea 23445

[filed under theme 4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / d. Naïve logical sets ]

Full Idea

Naïve set theory is based on the principles that any formula defines a set, and that coextensive sets are identical.

Gist of Idea

Naïve set theory says any formula defines a set, and coextensive sets are identical

Source

Øystein Linnebo (Philosophy of Mathematics [2017], 4.2)

Book Ref

Linnebo,Øystein: 'Philosophy of Mathematics' [Princeton 2017], p.62

A Reaction

The second principle is a standard axiom of ZFC. The first principle causes the trouble.

The 9 ideas with the same theme [sets as defined by absolutely any concept]:

15894 Russell invented the naïve set theory usually attributed to Cantor [Russell, by Lavine]
21695 The set scheme discredited by paradoxes is actually the most natural one [Quine]
10485 Naïve sets are inconsistent: there is no set for things that do not belong to themselves [Boolos]
13441 Naïve set theory has trouble with comprehension, the claim that every predicate has an extension [Hart,WD]
9933 The paradoxes are only a problem for Frege; Cantor didn't assume every condition determines a set [Burgess/Rosen]
9613 Naïve set theory assumed that there is a set for every condition [Brown,JR]
9615 Nowadays conditions are only defined on existing sets [Brown,JR]
23445 Naïve set theory says any formula defines a set, and coextensive sets are identical [Linnebo]
23623 Predicativism says only predicated sets exist [Hossack]