This paper proves that set cover cannot be approximated efficiently within a ratio of (1-o(1))ln n, unless NP has slightly superpolynomial time algorithms. This result closes the gap between the approximation ratio of the greedy algorithm (which is (1-o(1))ln n) and previous hardness results (log₂n/2 ≈ 0.72ln n). The result is based on a reduction from a new k-prover proof system for NP, designed specifically for this purpose. Set cover is an NP-hard problem, but can be approximated within a ratio of ln n. Lund and Yannakakis showed that it is hard to approximate set cover within a ratio of (log n)/2. The paper extends this result, showing that for any ε > 0, it is hard to approximate set cover within a ratio of (1-ε)ln n. The result is proven under the assumption that NP does not have n^{O(log log n)}-time deterministic algorithms. The paper also discusses related work, including the use of greedy algorithms and linear programming relaxations to approximate set cover within a factor of H(n), the harmonic function. It also discusses the hardness of approximation results for set cover derived from probabilistically checkable proof systems. The paper presents a k-prover proof system for satisfiability of a formula φ, which is used to prove the hardness result. The proof system is based on a reduction from a new k-prover proof system for NP, designed specifically for this purpose. The paper also discusses the construction of partition systems and the reduction from the k-prover proof system to set cover. The paper concludes by noting that the upper bound of ln n is tight under the assumption that NP is not contained in a deterministic time class. It also discusses the implications of the result for other NP optimization problems, such as the minimum p-center problem. The paper also discusses the possibility of proving the same threshold under the weaker assumption that P ≠ NP.This paper proves that set cover cannot be approximated efficiently within a ratio of (1-o(1))ln n, unless NP has slightly superpolynomial time algorithms. This result closes the gap between the approximation ratio of the greedy algorithm (which is (1-o(1))ln n) and previous hardness results (log₂n/2 ≈ 0.72ln n). The result is based on a reduction from a new k-prover proof system for NP, designed specifically for this purpose. Set cover is an NP-hard problem, but can be approximated within a ratio of ln n. Lund and Yannakakis showed that it is hard to approximate set cover within a ratio of (log n)/2. The paper extends this result, showing that for any ε > 0, it is hard to approximate set cover within a ratio of (1-ε)ln n. The result is proven under the assumption that NP does not have n^{O(log log n)}-time deterministic algorithms. The paper also discusses related work, including the use of greedy algorithms and linear programming relaxations to approximate set cover within a factor of H(n), the harmonic function. It also discusses the hardness of approximation results for set cover derived from probabilistically checkable proof systems. The paper presents a k-prover proof system for satisfiability of a formula φ, which is used to prove the hardness result. The proof system is based on a reduction from a new k-prover proof system for NP, designed specifically for this purpose. The paper also discusses the construction of partition systems and the reduction from the k-prover proof system to set cover. The paper concludes by noting that the upper bound of ln n is tight under the assumption that NP is not contained in a deterministic time class. It also discusses the implications of the result for other NP optimization problems, such as the minimum p-center problem. The paper also discusses the possibility of proving the same threshold under the weaker assumption that P ≠ NP.