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Results tagged with independence-results
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user 3106
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.
7
votes
What's the earliest result (outside of logic) that cannot be proven constructively?
A somewhat different type of example, not as early as the ones in Andrej Bauer's answer, but perhaps a bit more resistant to "moving the goalposts," is an ineffective result in number theory.
For exam …
- 82.6k
67
votes
What are some reasonable-sounding statements that are independent of ZFC?
Harvey Friedman has devoted a large portion of his career to finding "natural" statements that are unprovable in ZFC. One example is given at the end of Martin Davis's article "The incompleteness the …
- 24.2k
7
votes
Formal proof of Con(ZFC) => Con(ZFC + not CH) in ZFC
You have almost answered your own question; it seems that the only part you are confused about is whether "the reflection principle operates outside ZFC."
One must, as always, distinguish between the …
- 82.6k
7
votes
Accepted
Natural statements independent from true $\Pi^0_2$ sentences
I passed this question on to Harvey Friedman, who provided the following information. Friedman has shown that the following statement is equivalent to the 2-consistency of PA:
For every recursive …
- 82.6k
13
votes
Accepted
Is there an "undecided" assertion of which a proof that it's not undecidable is known?
If it's known that some statement $S$ is decidable in ZFC, then you can just run a computer program that enumerates all ZFC-proofs and stops when it finds a proof of $S$ or a proof of $\neg S$. By hy …
- 82.6k
19
votes
Accepted
"Simpler" statements equivalent to Con(PA) or Con(ZFC)?
The discussion in the comments has helped clarify your question for me. I believe that it is closely related to the following remark by Harvey Friedman:
I am convinced that trying to take consist …
- 82.6k