Is uniqueness a fundamental property of nature? [closed]

The Laws of nature are universally applicable and at every point in force. Together they shape our universe but are all "shapes" unique?

For example, is it possible that there is a second identical earth somewhere in the universe?

Or is it possible that there exists an exact copy of a tree somewhere sometimes on earth? And what about a grain of sand; can one find two identical grains of sand?

Or going to even smaller scales; can it be that two free electron are identical? I know that two bounded electrons cannot occupy exactly the same state because of Pauli's principle and thus are not identical. I also know that I can replace an electron with another electron without causing any change; an electron is an electron. But are they actually indistinguishable? As an analogy:

Imagine there are two workers with the same skills. Worker 1 is controlling device X. If worker 1 is replaced with one of his colleagues, the device will work as before. If a third person A is observing the device, he will notice no change. He is unable to say who controls the device right now (worker 1 or 2). For A it makes no difference what worker is controlling the device; the device keeps its functionality. For A it is as if there is just one worker. Now imagine you are one of the workers. Again each worker is interchangeable without altering the functionality of the device. But from your perspective the workers are unique. They just perform the same task. For A all workers are identical and he is not able to distinguish the workers from each other because the replacement of one worker with another is not changing the function of the device. However, a worker is able to distinguish the workers and thus for him they are not identical.

I want to know: If I have an object A, will a second object exists that is an exact copy of A and cannot be distinguished from the original?

• All electrons, protons, atoms etc. are exactly the same. They are so much the same that we have to symmetrize/anti-symmetrize our equations to get the correct answers. As you go up in scale the number of possible combinations of these identical objects increases exponentially and the probability of finding the same "thing" twice becomes vanishingly small. There is, if you like, "no chance in hell", that there is a second earth or even anything remotely close. Jun 24, 2016 at 19:53
• You have Pauli's exclusion principle backwards: it is because electrons are all identical that they cannot have the same state. Jun 24, 2016 at 20:07

Note that a grain of sand is a macroscopic object. It contains in the order of $10^{19}$ atoms, which means a staggeringly high number of possible ways to assemble them. There are no two identical grains of sand on Earth.