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Results tagged with independence-results
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This tag is for questions about proving that some statement is independent from a theory, meaning it is neither provable nor refutable from that theory. Common examples are the continuum hypothesis from the axioms of ZFC, and the axiom of choice from the axioms of ZF.
9
votes
Accepted
Is there a formula phi s.t. phi and not-phi have a stronger consistency?
No, it's impossible for any axiom system. If Σ is consistent, then by the Completeness theorem, it has some model M. In M, φ is either true or false. So M is a model of either (Σ+φ) or (Σ+not φ). So a …
- 5,275
3
votes
Is the existence of a well-ordering on R independent of ZF?
Yes. Here's a sketched example:
Start in L. Let P be the forcing which adds ω1 many Cohen reals, and let G be an L-generic filter for P. Then L(ℝ)L[G] will model ZF, but will have no well ordering of …
- 5,275
34
votes
What are some reasonable-sounding statements that are independent of ZFC?
Another example is certain strong forms of Fubini's Theorem.
If you have a real value function on the product of two closed intervals which is bounded, and which is measurable in either coordinate w …
97
votes
What are some reasonable-sounding statements that are independent of ZFC?
If X is a compact Hausdorff space, and f is an algebra homomorphism from C(X) to some Banach Algebra, must f be continuous?
This question turns out to be independent. The affirmative answer is referr …