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Single Idea 18169

[filed under theme 4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / p. Axiom of Reducibility ]

Full Idea

The Axiom of Reducibility states that every propositional function is extensionally equivalent to some predicative proposition function.

Gist of Idea

Axiom of Reducibility: propositional functions are extensionally predicative

Source

Penelope Maddy (Naturalism in Mathematics [1997], I.1)

Book Ref

Maddy,Penelope: 'Naturalism in Mathematics' [OUP 2000], p.11

Related Idea

Idea 18168 'Propositional functions' are propositions with a variable as subject or predicate [Maddy]

The 26 ideas from 'Naturalism in Mathematics'

18182 The extension of concepts is not important to me [Maddy]
18163 Mathematics rests on the logic of proofs, and on the set theoretic axioms [Maddy]
18169 Axiom of Reducibility: propositional functions are extensionally predicative [Maddy]
18168 'Propositional functions' are propositions with a variable as subject or predicate [Maddy]
18172 Infinity has degrees, and large cardinals are the heart of set theory [Maddy]
18175 For any cardinal there is always a larger one (so there is no set of all sets) [Maddy]
18164 Frege solves the Caesar problem by explicitly defining each number [Maddy]
18167 We can get arithmetic directly from HP; Law V was used to get HP from the definition of number [Maddy]
18171 Cantor and Dedekind brought completed infinities into mathematics [Maddy]
18177 In the ZFC hierarchy it is impossible to form Frege's set of all three-element sets [Maddy]
18187 Theorems about limits could only be proved once the real numbers were understood [Maddy]
18186 Identifying geometric points with real numbers revealed the power of set theory [Maddy]
18188 The line of rationals has gaps, but set theory provided an ordered continuum [Maddy]
18184 Making set theory foundational to mathematics leads to very fruitful axioms [Maddy]
18185 Unified set theory gives a final court of appeal for mathematics [Maddy]
18183 Set theory brings mathematics into one arena, where interrelations become clearer [Maddy]
18190 Completed infinities resulted from giving foundations to calculus [Maddy]
18191 Axiom of Infinity: completed infinite collections can be treated mathematically [Maddy]
18193 The Axiom of Foundation says every set exists at a level in the set hierarchy [Maddy]
18194 'Forcing' can produce new models of ZFC from old models [Maddy]
18195 A Large Cardinal Axiom would assert ever-increasing stages in the hierarchy [Maddy]
18196 An 'inaccessible' cardinal cannot be reached by union sets or power sets [Maddy]
18204 Scientists posit as few entities as possible, but set theorist posit as many as possible [Maddy]
18205 The theoretical indispensability of atoms did not at first convince scientists that they were real [Maddy]
18206 Science idealises the earth's surface, the oceans, continuities, and liquids [Maddy]
18207 Maybe applications of continuum mathematics are all idealisations [Maddy]