On the other hand that proof system is not effective, which makes it not very exciting. <><><><>Posted by<><> <><>Aatu Koskensilta, University of Calgary Use MathJax to format equations. Would second-order logic then be complete, too? We can even take $T$ to be the empty theory. First of all, is this even the case? ), and so SOL completeness ($N\vDash \phi$ iff $N\vdash \phi$) would mean that any $\phi$ true in the standard model of number theory (i.e. Showing that Val^2 is not r.e. With this definition of completeness, Gödel's Incompleteness result seems not surprising, so why it was back then? If you want to show that Val^2 is not Sigma^0_1(W) this way (as I did above), you need W to be definable in the langauge. $\endgroup$ – Asaf Karagila ♦ Sep 9 at 9:30 What is the disadvantage of using impact sockets on a hand wrench? 7, No. that there are sentences that are true in all models of some theory $T$, and yet still can not be proved from $T$. It only takes a minute to sign up. <><><><>Posted by<><> <><>Aatu Koskensilta, While I'm at it, here's a problem I've been playing with: give a "nice" example of an axiomatizable complete second order theory that is not categorical. Thus Sigma-m-n truth reduces to second order valditiy. Have any other US presidents used that tiny table? (That's why it's not obvious.)b. (In one direction: $\psi(x)$ says "there is a number k which codes a computation of a Turing machine started on input l ? I have no idea where I originally learned this stuff from). That is, there exists some proof system "⊢" in second order logic, such that ϕ can be proved using it. (The function of such an oracle, of course, cannot itself be performed by a Turing machine, since the set of Gödel numbers of true arithmetic sentences is undecidable, and not even r.e.) $N\vDash \phi$ ) is also provable from N, contradicting the incompleteness theorem for SOL. Showing that Val^2 is not Sigma^0_1 or Sigma^1_1 or whatever can be proved using diagonalization. The proof system is allowed to be, let's say, any (decidable) set of logical axioms, that are valid in T. The problem is that there is really no proof system for second-order logic. But any finite subset is consistent. For the case of W = TA, this is the case, but it isn't in general. How do “we” know the incompleteness of second-order logic? I found that a method I was hoping to publish is already known. The problem is that just saying "any proof system" doesn't give you enough to work with. $\begingroup$ But if you're talking about the incompleteness of second-order logic, what does a Gödel sentence have to do with it? I will argue that the 9000 ft.) is 15,000 feet high? MathJax reference. then true arithmetic (the sentences true in the standard model) would be r.e. After Gödel published his proof of the completeness theorem as his doctoral thesis in 1929, he turned to a second problem for his habilitation. (Dec., 2001), pp. But you can only diagonalize if you have a proof/truth predicate in the language. ), and so SOL completeness ($N\vDash \phi$ iff $N\vdash \phi$) would mean that any $\phi$ true in the standard model of number theory (i.e. If you move to second order logic, you have completeness, but what you're reducing to is just as elusive/unknowable as what you're trying to reduce in the first place. How to show incompleteness of second order logic? (For other readers! Calgary, Alberta, Canada It seems that Gödel's incompleteness theorem should apply also to second order logic (does it? It follows from the incompleteness of arithmetic because for any sentences $\phi$ of first-order arithmetic, there is a sentence of second-order logic $\phi'$ which is valid iff $\phi$ is true in the standard model.