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Single Idea 18177

[filed under theme 6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / c. Fregean numbers ]

Full Idea

In the ZFC cumulative hierarchy, Frege's candidates for numbers do not exist. For example, new three-element sets are formed at every stage, so there is no stage at which the set of all three-element sets could he formed.

Gist of Idea

In the ZFC hierarchy it is impossible to form Frege's set of all three-element sets

Source

Penelope Maddy (Naturalism in Mathematics [1997], I.2)

Book Ref

Maddy,Penelope: 'Naturalism in Mathematics' [OUP 2000], p.23

A Reaction

Ah. This is a very important fact indeed if you are trying to understand contemporary discussions in philosophy of mathematics.

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]