The Continuum Hypothesis is an Open Problem
I adhere to Bourbaki’s set theory and terminology (see [1]).
I think that, for any relation R, there is no Bourbaki’s “assemblage” corre-
sponding to the statement “R is undecidable”. Thus, there can be no proof of this
statement.
More generally there is, in texts of people like Gödel and Cohen, something I
find puzzling.
Recall Hilbert’s quotation: “it must be possible to replace in all geometric
statements the words point, line, plane by table, chair, mug”; take your favorite
text of Gödel or Cohen (like [2] or [3]); and replace the words “integer” and “set”
(when these words are employed in their mathematical sense) by, say, “word1”
and “word2”.
Many words have at the same time a usual meaning and a mathematical one
— and it’s vital to distinguish these two meanings. In most of the cases this dis-
tinction is easy to make: for example for words like “group”, “ring”, “field”, . . .
For words like “integer” and “set”, the distinction is not as easy to make, but not
less important. (Some people would say that, in mathematics, the word “set” is
a “primitive” one and has no definition. Indeed it’s not so easy to find a mathe-
matical definition of the word “set”. The only one I know is that of Bourbaki [1],
p. II.1.) Strictly speaking, we have no more reasons to expect mathematical inte-
gers and mathematical sets to behave like “real world” integers and “real world”
sets, than we have to expect mathematical rings to behave like “real world” rings.
(Let’s not be afraid of Virginia Woolf!) — Of course this also applies to many
other words (or strings of words) like “equal”, “proof”, “there exists”, “for all”,
“and”, “or”, “not”, . . . (This is crucial for instance to understand why different
sets can be equal — although intuitively two sets are equal only if they are the
“same” set.)
People like Gödel and Cohen believe that some statements about integers and
sets are true in an absolute sense, independently of any axiom. (Such would be
the case, I presume, of a statemen