So I was thinking about the meaning of life (or actually, about the discussions about the meaning of life I have heard), and found a nice duality to logic.
The problem with the meaning of life is that you want to have an entity that implies that life actually makes sense. The German phrase "Sinn des Lebens" focuses even more on this, as "Sinn" can be translated to "meaning", but also to "use" and "sense". It is often said that the meaning of life is to make the world a better place for living. If this does the job for you, then accept it, but one might ask what the meaning of the world is at all - which is as hard to answer as the original question. It is also often said that the meaning of life is to serve some deity, be it personal or pantheistic. But then you might question what the meaning of this deity's existence is and question is mostly disregarded, or even prohibited. A dogma - an "axiom" - is introduced that defines an entity that has a meaning, and questioning it is basically not allowed. Just "enjoying life" is another pragmatic way to deal with this problem, and is even more direct: Just ignore the problem.
However, let us look closer at the actual problem. Say we have our life, call it . We ask for some entity that implies that actually makes sense. And then we might question this new entity and get and so on, so we get an infinite ascending chain of entities that imply that the entities below make sense. And then we can define an entity that ensures that all the infinitely many entities we already defined make sense. And since is the smallest ordinal number that is infinite, we call it . And of course, we can define , and so on. In fact, for every ordinal number , we can define . However, itself is a proper class: We cannot talk about it inside our system. We can assume its existence, because we can assume that the stuff we cannot talk about anymore behaves similar to the stuff we can talk about. We talk about "something that is bigger than everything we can talk about", but this does not make sense anymore: As we can talk about it, it is not bigger than everything we can talk about.
Seeking for the meaning of life is now a lot like seeking for : It is desirable, but not possible. Every system we can come up to is either inconsistent, or limited. We can postulate principles which increase the range of numbers we can talk about, like assuming that O# exists. Still, the original problem remains.
However, this just means that We cannot talk about the meaning of life without creating inconsistencies. Whether or not a meaning of life "exists" is a more complicated question, because at the latest when reaching that meta-level, it is a valid question what "existing" actually means.