I'm interested in Fuzzy logic. I have read that the compactness theorem holds for predicate Lukasiewicz logic, with semantics over $[0,1]$.
However I found the following question on mathoverflow about Godel logic
Compactness and completeness in Gödel logic
While I realize that the comments under the "EDIT" doesn't apply, it seems to me that you could take a structure, expand the language by adding uncountably many nullary predicates, and $\{P_{a}|a\in{\omega_{1}+2}\}$ and show that the theory $T$ axiomatized by $\{(P_{b}\rightarrow{P_a})\rightarrow{P_b}|a<b\}$ is finitely satisfiable. But this should lead to a contradiction as in the above question; paraphrasing the answer given there slightly;
However, any model of $T $ where $P_{\omega_{1}+1}$ has truth value $x$, with $x<1$ in its ordering, must have the valuation of $P_{a}$ is less than the valuation of $P_{b}$ for $a<b$ (as the linear order has the property that the truth value of $p\implies{q}=1$ iff $p\leq{q}$ where the ordering here is the ordering on the truth value of algebras), and hence the valuation $v$ provides an embedding of $\omega_{1}+1$ into the algebra of truth values, which should be a contradiction.
My questions are:
Where am I going wrong? If I were to hazard a guess, I would say that since we cannot add $\neg(P_{\omega + 1}\leftrightarrow{1})$ to $T$, any model of $T$ over $[0,1]$ would necessarily have all of the $P_{a}$'s having truth value $1$. Am I correct on this?
Am I correct in stating "as the linear order has the property that the truth value of $p\implies{q}=1$ iff $p\leq{q}$ where the ordering here is the ordering on the truth value of algebras"? This is true in $[0,1]$, and I think it should be true in general, but I'm not a 100%.
Edit: Here by compactness I mean the following: If $T$ is finitely satisfiable over $[0,1]$ then $T$ has a model over $[0,1]$.
Edit2: I would also like to know if the following is true in general: If $T$ is finitely satisfiable over $K$, where $K$ is some $MV$-algebra, then $T$ has a model over $K$.