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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]
| 18949 | The universal syllogism is now expressed as the transitivity of subclasses [Putnam] |
| 18951 | For scientific purposes there is a precise concept of 'true-in-L', using set theory [Putnam] |
| 18950 | Physics is full of non-physical entities, such as space-vectors [Putnam] |
| 18955 | Having a valid form doesn't ensure truth, as it may be meaningless [Putnam] |
| 18952 | '⊃' ('if...then') is used with the definition 'Px ⊃ Qx' is short for '¬(Px & ¬Qx)' [Putnam] |
| 18953 | Modern notation frees us from Aristotle's restriction of only using two class-names in premises [Putnam] |
| 18954 | Before the late 19th century logic was trivialised by not dealing with relations [Putnam] |
| 18956 | Asserting first-order validity implicitly involves second-order reference to classes [Putnam] |
| 18957 | Nominalism only makes sense if it is materialist [Putnam] |
| 18958 | In type theory, 'x ∈ y' is well defined only if x and y are of the appropriate type [Putnam] |
| 18959 | Sets larger than the continuum should be studied in an 'if-then' spirit [Putnam] |
| 18960 | Most predictions are uninteresting, and are only sought in order to confirm a theory [Putnam] |
| 18962 | Unfashionably, I think logic has an empirical foundation [Putnam] |
| 18961 | We can identify functions with certain sets - or identify sets with certain functions [Putnam] |