A three-dimensional, field-programmable, non-volatile memory includes multiple layers of first and second crossing conductors. Pillars are self-aligned at the intersection of adjacent first and second crossing conductors, and each pillar includes at least an anti-fuse layer. The pillars form memory cells with the adjacent conductors, and each memory cell includes first and second diode components separated by the anti-fuse layer. The diode components form a diode only after the anti-fuse layer is disrupted.