Do infinity and zero really exist? From the first day that I entered college, I was wondering about the relationship between some basic mathematical abstract concepts and nature. I'm going to explain them here and you may find them a little bit opinion-based, but it's not. So please guide me in case you have good scientific-based answers/examples.

Infinity:
Does infinity exist in reality? We all know it is an abstract concept but what I'm curious to know is if there are any physical phenomena out there that can show, stimulate or somehow help us understand the concept of infinity in reality/nature/the physical world.

Zero:
What about Zeno's paradox? In nature (our physical world) there is a "smallest distance". It's about $1.6 \times 10^{-35}$ meters. Another example would be when someone says "there are three apples on the desk, if you take all of them there are $0$ apples on the desk". Obviously, it's an abstract concept, but what I would like to know is that if there is any observable physical event that can anyhow show us the concept of nothingness or absolute zero. E.g., we don't have absolute zero temperature in thermodynamics, absolute zero distance between 2 points in mechanics, or absolute zero gravity in a given space, etc.

Axiom of empty set: This axioms states:


  "There is a set such that no element is a member of it."


One can imagine an empty set in nature as an absolutely empty box. But one can see it the other way around: the element "nothing" is there (quoting from Prof. Lawrence Krauss's book "nothing is something"). So, the same way one can say the element "nothing" is a member of a set. Obviously, it's an abstract concept and this may look like a play on words, but it's also an interesting paradox. 

$\mathbb R$?! (the set of real numbers): There are millions of mathematical theorems which are based on $\mathbb R$! I wonder if there are any non-countable physical phenomena in the real world/nature? 

Again, I know these mathematical concepts are just abstractions and help us to solve real-world problems! However, I'm more interested in the relation between these abstract concepts and our physical world/nature.
Claim: At least one of $0$, $\infty$ exists.

Proof: If $\infty$ exists, we are done.  If not, the number of instances of $\infty$ is $0$.

I am actually somewhat serious.  This is an adaptation of a certain philosopher's (unfortunately I cannot remember who) ontological proof of the existence of the empty set, which is itself an extraordinarily perceptive and witty riff on St. Anselm's ontological argument for the existence of God.  Strangely, while theists and atheists alike tend to find St. Anselm's argument intriguing but not convincing (I seem to recall that Anselm's paper begins by apologizing -- to God -- for making the argument), when you invert it to prove the existence of nothingness, it seems quite convincing!
The empty set and 0 are easy. The empty set is a set which has no proper subsets and its cardinality is 0. The important concept here is that if you have 0 apples on your desk I can not take an apple from your desk, and I shall be able to see that.

If you had 5 apples on your desk there would be a 1 to 1 mapping between the set of digits on my hand and the set of apples on your desk, with 0 elements left in either set. These sets would both have cardinality 5. You can define &lt;, =, and > by considering which, if either, set has elements left over after mapping.  I can now take an apple from your desk and when you map the proper subset I leave against your hand you will see you are missing an apple (or you have grown an extra digit).

Infinity is trickier. The set of even integers is a proper subset of the set of integers and there is a 1 to 1 mapping between the set of integers and the set of even integers, both sets have the same cardinality. This property defines infinite, and in this case both are countable.

There is a 1 to 1 mapping between the set of integers and:
  the set of rational numbers;
  the set of irrational numbers which are the root of an integer polynomial, so these are also countable.

The set of transcendental numbers is much larger than any of these sets and is called uncountable (i.e. there is not a 1 to 1 mapping between the set of integers and the set of transcendental numbers, there are transcendental numbers left over when the set of integers is exhausted). The set of real numbers is made up of the sets of integers, rational numbers, irrational numbers which are the root of an integer polynomial , and transcendental numbers and so it is uncountable.

The next known larger infinity is the set of all single value functions (see http://www.math.utah.edu/~pa/math/sets/FgtR.html).