Re: Continuum hypothesis



Rupert wrote:
herbzet wrote:
Rupert wrote:
herbzet wrote:
Rupert wrote:
One example of a sentence in the second-order language of field theory
whose validity is equivalent to the truth of the continuum hypothesis
is as follows. It's a conditional sentence, the hypothesis is the
conjunction of the second-order axioms for a complete ordered field,
and the conclusion is "every subset of the domain of discourse can
either be injected into N (where N is the smallest subset of the
domain of discourse containing 0 and closed under the addition of 1)
or else is equipollent to the entire domain of discourse." If the
continuum hypothesis is true, this sentence is true in all models. If
the continuum hypothesis is false, this sentence is false in all
models. But in ZFC we can't decide which.

[...]

Are all the models of the hypothesis isomorphic?

Yes, all complete ordered fields are isomorphic.

OK.

You said before "[CH] is a statement in the third-order language
of arithmetic. It says "Any infinite set of sets of natural numbers
is equipollent either to the set of natural numbers or the set of
sets of natural numbers."

So now we have that the above conditional is valid iff this
sentence of third-order arithemetic is true.

What do you mean by saying this sentence (CH) is true? True
in all models of third order arithmetic? True in all structures
of the language of third-order arithmetic? True in the "standard
model", whatever that means in a third-order context?

Thanx in advance.

--
hz
.

Relevant Pages

  • Re: Continuum hypothesis
    ... It's a conditional sentence, the hypothesis is the ... domain of discourse containing 0 and closed under the addition of 1) ... continuum hypothesis is true, this sentence is true in all models. ... sentence of third-order arithemetic is true. ...
    (sci.logic)
  • Re: Continuum hypothesis
    ... It's a conditional sentence, the hypothesis is the ... domain of discourse containing 0 and closed under the addition of 1) ... continuum hypothesis is true, this sentence is true in all models. ... sentence of third-order arithemetic is true. ...
    (sci.logic)
  • Re: Continuum hypothesis
    ... It's a conditional sentence, the hypothesis is the ... conjunction of the second-order axioms for a complete ordered field, ... domain of discourse containing 0 and closed under the addition of 1) ... continuum hypothesis is true, this sentence is true in all models. ...
    (sci.logic)
  • Re: Continuum hypothesis
    ... conjunction of the second-order axioms for a complete ordered field, ... domain of discourse containing 0 and closed under the addition of 1) ... continuum hypothesis is true, this sentence is true in all models. ...
    (sci.logic)