Tagged Questions
22
votes
6answers
795 views
Why is the Continuum Hypothesis (not) true?
I'm making my way through Peter J. Cameron's seminal text "Sets, Logics and Categories" where he makes the statement that the Continuum Hypothesis (There does not exist a set with a cardinality less ...
0
votes
2answers
157 views
What is actually “relatively consistent”?
Gödel's incompleteness theorem states that: "if a system is consistent, it is not complete." And it's well known that there are unprovable statements in ZF, e.g. GCH, AC, etc.
However, why does this ...
7
votes
3answers
274 views
Can ZFC decide number theory?
Among the versions of the Incompleteness Theorem that I've seen are the following:
Assuming the Peano axioms are consistent (which they are, if we accept the existence of the natural numbers), there ...
2
votes
1answer
124 views
Why does the existence of independent statements not prove completeness?
I've read before that, by the Principle of Explosion, if a theory is inconsistent, then absolutely any statement can be proven within it.
Obviously, there are statements which are independent of ZFC ...
9
votes
2answers
242 views
How to show the existence of an infinite set of independent undecidable sentences?
How to show the existence of an infinite set of independent undecidable sentences?
By "independent" I mean that no implication between any two elements is provable. A finite set that satisfies the ...
