The paper presents a proof that $(1 - o(1)) \ln n$ is a threshold below which set cover cannot be approximated efficiently, unless NP has slightly superpolynomial time algorithms. This result closes the gap between the approximation ratio achievable by the greedy algorithm ($(1 - o(1)) \ln n$) and previous hardness results by Lund and Yannakakis, which showed hardness within a ratio of $(\log_2 n)/2 \simeq 0.72 \ln n$. The proof is based on a reduction from a new $k$-prover proof system for NP, designed specifically for this purpose. The authors extend the hardness result to show that it is hard to approximate set cover within a ratio of $(1 - \epsilon) \ln n$ for any $\epsilon > 0$, under the assumption that NP does not have $n^{O(\log \log n)}$-time deterministic algorithms. The paper also discusses related work, including previous hardness results and improvements in approximation algorithms, and provides an overview of the main ideas in the proof.The paper presents a proof that $(1 - o(1)) \ln n$ is a threshold below which set cover cannot be approximated efficiently, unless NP has slightly superpolynomial time algorithms. This result closes the gap between the approximation ratio achievable by the greedy algorithm ($(1 - o(1)) \ln n$) and previous hardness results by Lund and Yannakakis, which showed hardness within a ratio of $(\log_2 n)/2 \simeq 0.72 \ln n$. The proof is based on a reduction from a new $k$-prover proof system for NP, designed specifically for this purpose. The authors extend the hardness result to show that it is hard to approximate set cover within a ratio of $(1 - \epsilon) \ln n$ for any $\epsilon > 0$, under the assumption that NP does not have $n^{O(\log \log n)}$-time deterministic algorithms. The paper also discusses related work, including previous hardness results and improvements in approximation algorithms, and provides an overview of the main ideas in the proof.