Indoor wireless simultaneous localization and mapping (SLAM) is considered as a promising technique to provide positioning services in future 6G systems. However, the accuracy of traditional wireless SLAM system heavily relies on the quality of propagation paths, which is limited by the uncontrollable wireless environment. In this paper, we propose a novel SLAM system assisted by a reconfigurable intelligent surface (RIS) to address this issue. By configuring the phase shifts of the RIS, the strength of received signals can be enhanced to resist the disturbance of noise. However, the selection of phase shifts heavily influences the localization and mapping phase, which makes the design very challenging. To tackle this challenge, we formulate the RIS-assisted indoor SLAM optimization problem and design an error minimization algorithm for it. Simulations show that the RIS assisted SLAM system can decrease the positioning error by at least 31% compared with benchmark schemes.