This paper examines whether the ability to acquire costly signals during trading can make markets more efficient at aggregating information. This question becomes more relevant as the continuous improvements in information technology have created an abundance of available information, which is now cheaper than ever to acquire, analyze, and act upon.

We provide a complete and sound ax-iomatization of the set-theoretic model of Galanis [2007]. By constructing a syntax with several knowledge modalities, one for each sub-language, we are able to allow for agents to make mistakes about the knowledge of others without discarding the truth axiom. Comparing the present axiom system with that of Heifetz et al. [2008a] we find that neither is a generalization of the other.

This paper provides a set-theoretic model of knowledge and unawareness, in which reasoning through theorems is employed. A new property called Awareness Leads to Knowledge shows that unawareness of theorems not only constrains an agent's knowledge, but also, can impair his reasoning about what other agents know. For example, in contrast to Li (2006), Heifetz, Meier, and Schipper (2006) and the standard model of knowledge, it is possible that two agents disagree on whether another agent knows a particular event.