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 Reference
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]