Is undecidability strong enough to give us truth/falsehood, even though we cannot prove it inside any given theory?//<![CDATA[ aax_getad_mpb({ "slot_uuid":"f485bc30-20f5-4c34-b261-5f2d6f6142cb" }); //]]>

Consider the continuum hypothesis. It is undecidable, so, clearly, there's no counterexample of a set with cardinality between the integers and the reals (in standard ZF). Otherwise, such a set would disprove the CH. Is this strong enough for us to argue (as a metatheorem) that the CH is actually true?

What does this imply for ZF with the negated CH as an axiom? Clearly, it can't be inconsistency, because the CH was undecidable, but something "feels wrong."

(Please feel free to get arbitrarily technical, btw.)

**Physics Forums - The Fusion of Science and Community**

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Undecidability and truth

**Physics Forums - The Fusion of Science and Community**