Re: Continuum hypothesis

On Aug 16, 8:41 am, Peter_Smith wrote:
The Continuum Hypothesis can be formulated as a statement in the
language of pure second-order logic.

Indeed it can, in the sense that there is a sentence CH* in the
language of pure second-order logic that is valid iff CH is true.
However, this probably is not what the original poster was wondering
about. A more relevant answer is that CH is a statement in third order
arithmetic, that is, there is a sentence CH' in the language of third
order arithmetic that is true just in case CH is true. In general,
given an interpreted second order language there is no guarantee that
there is a sentence the truth of which is equivalent to CH.

The result the original poster probably has in mind is that both ZFC
+CH and ZFC+~CH are conservative over ZFC for arithmetical statements,
i.e. statements in which the quantifiers range over the hereditarily
finite sets (these are equivalent to statements in the first-order
language of arithmetic, of course). The conservativity extends to
Pi^1_4 sentences (in the arithmetical hierarchy), if I recall
correctly.

--
Aatu Koskensilta (aatu.koskensilta@xxxxxxxxx)

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

.

Relevant Pages

  • Re: Lets you & him fight
    ... I'm defining a new language. ... And why are you doing this in ZFC ... Because the point of the exercise is to give an interpretation of NBG ...
    (sci.logic)
  • Re: axioms of mathematical logic
    ... i base it on predicate logic and ZFC. ... manipulate expressions and objects without first setting the rules. ... first was taught the language of predicate logic, ... but then saying they can chose the axioms to use only after that. ...
    (sci.logic)
  • Re: axioms of mathematical logic
    ... first was taught the language of predicate logic, ... then given the list of ZFC axioms. ... expressions as if they are themselves some objects they can manipulate. ... but then saying they can chose the axioms to use only after that. ...
    (sci.logic)
  • Re: A set theory equivalent to ZFC.
    ... to the language of ZFC. ... language of your theory in the language of ZFC. ... Axiom schema of Relation 7) Axoim schema of Size Comprehension 8) ...
    (sci.math)
  • Re: axioms of mathematical logic
    ... first was taught the language of predicate logic, ... then given the list of ZFC axioms. ... expressions as if they are themselves some objects they can manipulate. ... but then saying they can chose the axioms to use only after that. ...
    (sci.logic)