What would happen if the Pauli exclusion principle did not exist?

The simplest form of this principle states that two (or more) electrons (fermions, spin $\frac{1}{2}$ particles) cannot occupy the same quantum state in an atom. If an arbitrary amount of electrons could occupy say the first energy level in an atom, then all of these atom's higher energy level electrons would also fit into this state. Matter would collapse into a much smaller volume*.

Another consequence would be that, like for bosons, any number of fermions could occupy the same quantum state for any system. So everywhere, all systems that once had restricted particle number due to the Pauli exclusion principle, would allow for unlimited particle numbers in the same state. Stars, planets, everything will begin to collapse.

“Infinite" numbers of particles throughout space will begin to combine into the same state at which point there would be many points or regions with energy density approaching infinity. Eventually regions everywhere would collapse into black holes as explained in the general theory of relativity.

For more about infinite bosons (photons) in a finite region, see this post here.

Virtually “infinitely” many regions of space will have these black holes and these black holes everywhere may begin to merge and eventually the universe itself would collapse into an infinitely dense singularity. I think that is what is meant by in that book stating matter would not exist.

• It is important to note that physicists considered repulsive forces between electrons and between nuclei and attractive forces between the nucleus and electrons and showed that matter will still collapse into a smaller volume if the Pauli exclusion principle did not hold.

Also, there would be no chemistry. All electrons would just be in the lowest orbitals around nuclei.

As already mentioned, there would be no Fermi gas; not in stars, not in metals.

Atomic nuclei would also be totally different: no even-odd alternation in binding energies, no magic numbers.