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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-one
Gist of Idea
Sets larger than the continuum should be studied in an 'if-then' spirit
Source
Hilary Putnam (Philosophy of Logic [1971], Ch.7)
Book Ref
Putnam,Hilary: 'Philosophy of Logic' [Routledge 1972], p.56
A Reaction
This attitude goes back to Hilbert, but it fits with Quine's view of what is indispensable for science. It is hard to see a reason for the cut-off, just looking at the logic of expanding sets.
Related Idea
Idea 18958 In type theory, 'x ∈ y' is well defined only if x and y are of the appropriate type [Putnam]
| 15896 | Cantor needed Power Set for the reals, but then couldn't count the new collections [Cantor, by Lavine] |
| 10112 | The naturals won't map onto the reals, so there are different sizes of infinity [Cantor, by George/Velleman] |
| 18959 | Sets larger than the continuum should be studied in an 'if-then' spirit [Putnam] |
| 10483 | Mathematics and science do not require very high orders of infinity [Boolos] |
| 15909 | 'Aleph-0' is cardinality of the naturals, 'aleph-1' the next cardinal, 'aleph-ω' the ω-th cardinal [Lavine] |