Microwave kinetic inductance detectors (MKIDs) are the most attractive radiation detectors for far-infrared and sub-mm astronomy. They combine ultimate sensitivity with the possibility to create very large detector arrays, in excess of 10000 pixels. This is possible by reading-out the arrays using RF frequency division multiplexing, which allows multiplexing ratios in excess of 1000 pixels per readout line. We describe a novel readout system for large arrays of MKIDs, operating in a 2-GHz band in the 4-8-GHz range. The readout, which is a combination of a digital frontand back-end and an analog upconverter and downconverter system, can read out up to 4000 detectors simultaneously with 1-kHz data rate. The system achieves a readout noise power spectral density of -98 dBc/Hz while reading 1000 carriers simultaneously, which scales linear with the number of carriers. We demonstrate that 4000 stateof-the-art aluminium-NbTiN MKIDs can be read out without deteriorating their intrinsic performance.