Single Idea 18959

[catalogued under 6. Mathematics / A. Nature of Mathematics / 5. The Infinite / f. Uncountable infinities]

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]