We discussed briefly Church’s and Trakhtenbrot’s theorems: The set of logically valid sentences and the set of sentences valid in all finite models are undecidable. More precisely: If is a language that contains either a binary symbol or the language of arithmetic, then the set of theorems of reduces to the set of validities of , so this set is (r.e. and) undecidable. For the same , we can axiomatize the behavior of a given Turing machine on a given input (by a similar argument to the association of a semi-Thue system to each Turing machine) in such a way that this axiomatization has a finite model iff the machine converges on this input, thus the halting problem reduces to the complement of the set of finite validities, so this last set is (co-r.e. and) undecidable.

We defined Post correspondence systems and showed that the problem of determining whether such a system admits a solution is unsolvable. Two applications to the theory of formal languages follow: There is no algorithm to test whether 2 given context-free grammars have disjoint languages, and there is no algorithm to test whether a given context-free grammar is ambiguous.

This entry was posted on Friday, March 2nd, 2007 at 2:11 pm and is filed under 117b: Computability theory. You can follow any responses to this entry through the RSS 2.0 feed.
You can leave a response, or trackback from your own site.

As suggested by Gerald, the notion was first introduced for groups. Given a directed system of groups, their direct limit was defined as a quotient of their direct product (which was referred to as their "weak product"). The general notion is a clear generalization, although the original reference only deals with groups. As mentioned by Cameron Zwa […]

A database of number fields, by Jürgen Klüners and Gunter Malle. (Note this is not the same as the one mentioned in this answer.) The site also provides links to similar databases.

As the other answer indicates, the yes answer to your question is known as the De Bruijn-Erdős theorem. This holds regardless of the size of the graph. The De Bruijn–Erdős theorem is a particular instance of what in combinatorics we call a compactness argument or Rado's selection principle, and its truth can be seen as a consequence of the topological c […]

Every $P_c$ has the size of the reals. For instance, suppose $\sum_n a_n=c$ and start by writing $\mathbb N=A\cup B$ where $\sum_{n\in A}a_n$ converges absolutely (to $a$, say). This is possible because $a_n\to 0$: Let $m_0

Consider a subset $\Omega$ of $\mathbb R$ of size $\aleph_1$ and ordered in type $\omega_1$. (This uses the axiom of choice.) Let $\mathcal F$ be the $\sigma$-algebra generated by the initial segments of $\Omega$ under the well-ordering (so all sets in $\mathcal F$ are countable or co-countable), with the measure that assigns $0$ to the countable sets and $1 […]

Sure. A large class of examples comes from the partition calculus. A simple result of the kind I have in mind is the following: Any infinite graph contains either a copy of the complete graph on countably many vertices or of the independent graph on countably many vertices. However, if we want to find an uncountable complete or independent graph, it is not e […]

I think that, from a modern point of view, there is a misunderstanding in the position that you suggest in your question. Really, "set theory" should be understood as an umbrella term that covers a whole hierarchy of ZFC-related theories. Perhaps one of the most significant advances in foundations is the identification of the consistency strength h […]

I'll only discuss the first question. As pointed out by Asaf, the argument is not correct, but something interesting can be said anyway. There are a couple of issues. A key problem is with the idea of an "explicitly constructed" set. Indeed, for instance, there are explicitly constructed sets of reals that are uncountable and of size continuum […]

The question seems to be: Assume that there is a Vitali set $V$. Is there an explicit bijection between $V$ and $\mathbb R$? The answer is yes, by an application of the Cantor-Schröder-Bernstein theorem: there is an explicit injection from $\mathbb R$ into $\mathbb R/\mathbb Q$ (provably in ZF, this requires some thought, or see the answers to this question) […]

If a set $X$ is well-founded (essentially, if it contains no infinite $\in$-descending chains), then indeed $\emptyset$ belongs to its transitive closure, that is, either $X=\emptyset$ or $\emptyset\in\bigcup X$ or $\emptyset\in\bigcup\bigcup X$ or... However, this does not mean that there is some $n$ such that the result of iterating the union operation $n$ […]