The empty set is a set with no elements. It contains nothing. If you can construct a set of no elements, then the empty set exists (in the platonic sense). Surely I can construct a set with nothing in it. I just did. So did you. It's an axiom because we can't "prove" it, but it doesn't really make sense to think that it may not be true.

In your second question, existence is also in the platonic sense. If the form of, say, a triangle did not exist, would one be able to construct a triangle? The natural numbers can be constructed set-theoretically, and the rationals from those, and a good number of real and complex numbers can be algebraically constructed. If we can construct a number algebraically, then clearly that number

*exists *in the platonic sense.

The more interesting thing, at least as far as algebra goes, is the vast array of transcendental numbers which indisputably exist, but cannot be constructed The most famous transcendental is pi, but really most of the real number line is transcendental. If you were to throw a dart at the real number line, the probability of that dart landing on an algebraic number (that is, not transcendental) is

*zero*. In fact, most of the real number line is even weirder - they are

*indescribable*. Even beyond transcendental, most real numbers can't even be described. Here is a layman's-speak version of the proof the indescribable numbers exist, that also goes into some of the different 'kinds' of infinity which I've mentioned to you before.

http://blog.ram.rachum.com/post/54747783932/indescribable-numbers-the-theorem-that-made-meEDITED TO ADD: In an even more "first principles" kind of way, here is the more complete set theoretic construction of the natural numbers. It goes into something called the Peano axioms, which is pretty abstract stuff about the properties of the natural numbers.

http://web.math.ucsb.edu/~padraic/ucsb_2014_15/ccs_proofs_f2014/ccs_proofs_f2014_lecture4.pdf