When two uncharged plates with negligible mass are very close to each other. The plates are actually attracted to each other due to what is called the Casimir Effect. This is a result from quantum vacuum fluctuations of the electromagnetic field.
Electromagnetic waves are "popping up" all throughout the quantum vacuum but between the plates only very specific waves can exist, those with a wave number that causes the wavelength of the wave to be multiples of 1/2 and only 1/2, no more and no less. Outside of the gap, any wavelength can exist, an infinite number of any wavelength to be precise. But between the plates only a subset of the infinite number of waves can exist.
Granted, yes, there can be an infinite number of waves between the plates as they can be infinitesimally small but the number of possible waves is still less than the possible waves outside. The difference between the two "infinities" creates a "quantum pressure" that pushes the two plates together.
This force is not gravity, and it's not by definition an electromagnetic force as the two plates are electromagnetically neutral. Since its inception over half a century ago, a lot is still unknown about its origins.
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