6
votes
Accepted
Confusion about the consistency of $\mathsf{ZFC}+\neg\mathsf{Con}(\mathsf{ZFC})$
Yeah, you're mixing up the levels.
We formalize it as $$ \sf \lnot Con_{ZFC}\to \forall p\in Sentences_{LST}\; Prov_{ZFC}(p),$$ where $\sf Prov_{ZFC}$ is the same provability predicate that goes into $...
- 55.4k
6
votes
Is "non-rigid" first-order axiomatisable?
Let $L' = L\cup \{\sigma\}$, where $\sigma$ is a unary function symbol not in $L$. Let $T'$ be $T$ together with axioms asserting that $\sigma$ is a non-trivial-automorphism (non-triviality is $\...
- 72.3k
3
votes
Accepted
How to prove $\phi \models \psi$ iff $\phi \cup \{\neg \psi\}$?
I want to prove $\phi \models \psi$ iff $\phi \cup \{\neg \psi\}$ unsatisfiable.
What's the meaning of the union operator w.r.t. satisfiability in FOL?
We have to be careful with symbols. $a\models b$...
- 12.4k
2
votes
Is "non-rigid" first-order axiomatisable?
Yes, you can add a new unary function symbol $g$ and then write down a schema that says "$g$ is a nontrivial automorphism" and add that to the schema saying "$M$ is infinite" and ...
- 55.4k
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