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Full Idea
Sets of a very high type or very high cardinality (higher than the continuum, for example), should today be investigated in an 'if-then' spirit.
Clarification
The continuum is aleph-0
Gist of Idea
Very large sets should be studied in an 'if-then' spirit
Source
Hilary Putnam (The Philosophy of Logic [1971], p.347), quoted by Penelope Maddy - Naturalism in Mathematics
Book Ref
Maddy,Penelope: 'Naturalism in Mathematics' [OUP 2000], p.105
A Reaction
Quine says the large sets should be regarded as 'uninterpreted'.
Related Idea
Idea 18198 Mathematics is part of science; transfinite mathematics I take as mostly uninterpreted [Quine]
18199 | Indispensability strongly supports predicative sets, and somewhat supports impredicative sets [Putnam] |
18200 | Very large sets should be studied in an 'if-then' spirit [Putnam] |
8857 | We must quantify over numbers for science; but that commits us to their existence [Putnam] |