Re: Continuum hypothesis



Rupert wrote:
herbzet wrote:
Aatu Koskensilta wrote:
herbzet writes:

Does that not differ in different structures?

I'm not sure what you have in mind.

I have in mind, in a fuzzy sort of way, that Cohen(64) shows
there are structures in which (CH) is false.

It's a (rather trivial)
mathematical theorem that the truth of the sentence CH in third-order
arithmetic -- truth for third-order arithmetical sentences is defined
simply by augmenting the inductive definition of truth for
arithmetical sentences with clauses governing higher order
quantifiers, e.g. that "for all sets X of naturals, P" is true iff for
all sets X of naturals, P[X/A] is true and so on -- is equivalent to
every set of reals being countable or of the cardinality of the
continuum. Different structures do not enter into this in any way,
only the naturals, sets of naturals, sets of sets of naturals.

--
Aatu Koskensilta (aatu.koskensi...@xxxxxxxxx)

"Wovon man nicht sprechen kann, daruber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus- Hide quoted text -

- Show quoted text -

You're confusing truth with truth in a particular model.

No doubt. I am quite unclear on the distinction, particularly in
the case of CH.

--
hz
.

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