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.
 A: To conclude some of your answers:

*

*Mathematics is different from Physics. We can't completely compare them with each other. Mathematics is an abstract science while Physics is natural science. So they are different types.


*We should choose an axiomatic system before talking about a theorem. ( like ZFC ).


*Abstract concepts are a part of mathematics, not Physics. We can imagine that there is a number bigger than any number. That number is Infinity ! ... I still don't know if it exists in physics or not. For example "is world infinity or not !?"... "Is there a new world inside a black hole or not!" ... But in mathematics, we can imagine it. So that zero and other abstractions.
I still would love to hear more answers and comparisons with nature.
A: Concerning infinity, here is my take. When we say, for example, that there is an infinity of natural numbers, we mean only that there is no largest number. Whenever you add 1 to a number, you will get another number that is greater still. Nothing very profound there. To say otherwise would greatly complicate even basic arithmetic.
It may even be possible to do away with the use of the word "infinity" (or any equivalent notion) in most if not all of mathematics. It may be nothing more than a convenient shorthand. When we say, for example, the limit of some expression as x tends to infinity, we could as easily have said, the limit as x increases without bound. The latter would have the advantage that there would be no confusion about there being some kind of endpoint called "infinity."
A: 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!
A: As an indirect way of answering this question, I'd like to bring up an example from wave mechanics that I was told earlier this semester.  Physically, when an electromagnetic wave hits a perfect conductor, say from the left, the wave disappears to the right of the conductor.  What happens mathematically is the following: the charges emit a wave that is the exact mirror of the original one, cancelling it for all time and space to the right of the conductor.  Now, a natural question in this scenario is, are the two waves to the right of the conductor real, actually extending infinitely, or is there really no wave there any more?  
The answer to this question is: the question does not make physical sense.  Physics is concerned with measurable quantities; you can ask "what is the electric field to the right", and both interpretations of what's 'really' happening will give you the same (correct) answer.
In the same way, mathematics is built on abstractions.  The question of these abstractions being 'real' or not is not mathematical (nor logical), but philosophical; mathematics is the foundation of our technological society, which works quite well, which is the proof that mathematics is valid in the real world.
A: I think your questions and struggles with these difficult concepts are completely understandable, but your presentation is terrible. You are unnecessarily confrontational and aggressive. Having difficulties with these concepts is completely all right. Assuming that during hundreds of years mathematicians conspired to build a make-believe world makes you a crackpot. Hence the series of down-votes.  I think the first thing you should do is to learn humility and present your ideas with respect. Don't assume that just because something does not make sense to you it necessarily does not make sense. 
So, putting aside my initial discomfort with your attitude, let me try to say a few words towards resolving your issues. 
1) infinity --- This is indeed something hard to grasp and there are grown mathematicians who are advocating similar views. They are a bit more sophisticated and a lot less arrogant though. As for smallest distance, it has already been pointed out that that would be a statement in physics and not in math. On the other hand, if you believe in dividing integers by integers, you must realize that there are infinitely many numbers between any two numbers or infinitely many points between any two points, just divide the distance by $n$ where $n$ runs through $\{1,2,3,\dots\}$. But of course, you could try to argue that it does not make sense to talk about arbitrarily large numbers such as $2^{2011}$ as they have no practical applications. I can't say anything to that.
2) zero --- I always thought that whoever first invented zero and the empty set had to be a genius. These are indeed difficult concepts. However, you are confusing a few things here. You say 

..."nothing" is a member of a set...so there is no set with size "zero". There is always a "nothing" or "null" element in there. Am I right ?

No, you are not right.  First off, what do you mean by "nothing"? The only sensible mathematical meaning of that is the empty set. So, you are talking about a set that consist of one element, which is the empty set. That's different from the empty set that has no elements at all.
You also say

You wanna tell me that "there is 3 apple on the desk, if you take all of them there is 0 apple on the desk ? " I still don't see number "0" ?

Yes, you are right! There is $0$ apple on the desk and you don't see the number $0$. There are many other things you don't see. For example you don't see 
the dark side of the moon, but you would not doubt that it is there. You don't see the Sun at night but you know it exists even at night.
What about negative numbers? Those don't exist either? So when you take out a loan and you owe the bank a ton of money, it is not real? If you accept negative numbers, you have to accept zero as well.
3) uncountable sets --- I guess this is a more sophisticated version of your problem with infinity. So, can we accept the existence of (non-zero ;) natural numbers? Say $1,2,3,\dots$? If so, then it is easy to show you an uncountable set: take the set that consists of all the subsets of $\{1,2,3,\dots\}$.
A: In reading through these answers, I'm pretty sure that most everything has been answered - except for the treatment of the empty set. By now, I think that we can say that $\infty$ is not a number, but instead a convenient convention referring to an unbounded convention. $0$ is a number and it refers to a very concrete thing - but no one (I think) has yet said anything of how number systems came into being. While I think that many people know that our current positional number systems (like binary or hex for programmers, decimal for day-to-day life) are relatively recent developments. But many number systems do not have a symbol for 'zero.' Why would they? We use it in our positional system all the time - when we write 102, we refer to 1 100, 0 10's, and 2 1's. We rely on zero all the time. But without relying on a positional number system, the only time one would need zero would be when referring to... nothing. There are no apples on the desk and there are zero apples on the desk. Does that deserve its own number?
We think it does, and it's useful for it. 
But this is very different than the empty set. We think of sets of things all the time, even if we don't explicitly call it a set. We have a box of chocolates, for example. This is like saying that we have a set of chocolates, or maybe we have a set of boxes that can contain chocolates with exactly one element. What happens when we eat all the chocolates (or as happens upon occasion, the chocolates are not tasty, and so we throw them away)? Then now or 'set of chocolates' collection is empty. But until we throw away the box, our 'set of boxes' still has one thing.
I think that maybe the problem you're identifying with the empty set is related to the concept of vacuously true statements. For example, the set of flying pink elephants that contributed to both Nephireti's and Hitler's rule is empty. And all such elements are right in front of me. All such elephants are also versed in calculus. And so on. The problem with vacuous things is that there is no substance to them - there can be no counter examples found within the set if it's empty. It's very troublesome.
But just like the number zero, the empty set is useful. We use it all the time - it's a useful idea. This is similar to the number 7 - it's a useful idea that may not be entirely obvious. For a while, people had trouble distinguishing between a number and a number of objects. For example, is the '7' in '7 sheep' somehow different than the '7' in '7 cows?' Does it make sense to talk of this '7' apart from some sort of counting reference? We do it all the time, but it's not so obvious to do so.
That's my take on the subject, anyway.
A: (Disclaimer: I am neither a trained linguist nor a psychologist, so take my claims with a grain of salt)
Part of your concern is not about mathematics, or of physics, but actually of the English language. There is a famous idea that "language constrains thought".
One example of this is the notion of degeneracy. In English, one can say
There is/are <quantity> apple(s) on the desk.

where  is a positive integer. But in natural English, one cannot say
There are zero apples on the desk.

Instead, one expresses this degenerate case with a different linguistic construct, e.g.
There are not any apples on the desk.

("there are no apples" is an abbreviation of this: see here). In this situation, one does not answer the question "how many apples are on the desk?" but instead rejects the premise that there are any apples at all.
Because language treats the cases fundamentally differently, this shapes peoples' thought about quantity and makes it difficult for them to think of "zero" as an answer to a "how many" type of question.
This gets confounded further with words like "nothing". When the phrase
There is no thing on the desk.

is shortened to
There is nothing on the desk.

people still think of the latter being the same as the former: a rejection of the claim that there are any things on the desk. The danger is when one gets the idea that we can use a word like "nothing" to answer the question "what is on the desk?", but is still believes answering the question without rejecting it means accepting the implicit premise that there is something on the desk.
Such a person then has an incredible amount of trouble sorting through all of the classic "nothing is something" wordplay.
A: It's not mathematics but rather physics which is based on these "lies". In mathematics, we assume (if we're Platonists) that objects like the real numbers "really exist", just not in the physical world, and then everything makes sense. Some people pretend that when they're doing mathematics, they're just combining axioms and derivation rules to prove theorems; these people have to take as an article of faith the fact that their chosen axiom system (e.g. ZFC) is consistent, otherwise all their toil makes no sense.
When mathematics is applied to the real word, e.g. in physics, then often some approximations are made, like the fact that (at least in classical physics) a Cartesian coordinate system describes space, and the real line describes time. These approximations can be explained mathematically: taking discreteness into account in general will only slightly alter the results; but it complicates everything greatly.
Some mathematics is not like that, for example when a poll is being taken, statisticians will calculate the standard deviation - this makes sense even if you don't believe in the real numbers; they are just a theoretical construct introduced to understand discrete phenomena. At a final count, everything reduces to finitistic reasoning, whose validity however rests on some unfounded belief in the consistency of some axiom system.
Last but not least, why do you oppose zero? If you take a ruler and mark a peg every inch, then you ask yourself "how many pegs do I need to jump from 2in to 3in"? The answer is $1$. Then "how many pegs do I need to jump from 2in to 2in"? The answer is zero. You can understand negative numbers this way.
Also, zero apples is the number of apples that remain after you've eaten all of them. And there are many more examples, in fact an entire book was written on the subject.
A: It seems like your premise for mathematics being a 'lie' is that its theoretical background cannot be completely applied to the real world (e.g. having the smallest distance in the real world, but not in mathematics). If that's how you define a lie, sure, but mathematics is an abstract science and I am not sure why you think it's supposed to hold certain properties that are true in the real world -- abstraction is actually the whole point. 
Solutions to Zeno's paradox have been offered, I am not sure what you mean by "but i don't talk about solutions. I'm saying "this is LOGICAL". But it's not TRUE." In fact, Zeno's paradox shows the lie of the real world, as Zeno contested that motion does not exist and is an illusion.
"You wanna tell me that "there is 3 apple on the desk, if you take all of them there is 0 apple on the desk ? " I still don't see number "0" ?" -- true, but neither you see number 3 when there are 3 apples on the desk, correct? 0 is used to denote lack of objects, and it's definition is a tad more complicated than definition of other integers, and for a reason (it can be used differently in different branches of mathematics). Also, if you want a real-life definition of zero, think of not moving, not gaining weight, not making any progress -- all of this can essentially be seen as addition of zero.
An empty set is a set (which, again, is an abstract concept) that has no elements. Careful with the words -- it's not the case that it has 'nothing' as an element, it's rather the case that it has no elements at all. Here, it comes down to semantics of the definition, 
but you once again seem to be drawing parallels to real life and trying to find a corresponding entity. I do not quite see why that should be the approach -- we are talking about mathematics, not physics.
With respect to the set of Real numbers, again, same story. I do not know if you can find an uncountable set in the 'real world' (though I suppose you could contest that the set of points of any object is uncountably infinite, because as you keep zooming in you will be getting more and more particles, and it's our limitations that do not allow us to see past atoms). But that is not the point. Mathematics is based on some abstract ideas, and it's true that not all of them are necessarily reflected in nature. That does not, however, make mathematics based on 'lies.'
A: 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 <, =, 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). 
