Do infinity and zero really exist? [closed]

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!

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 $\{1,2,3,\dots\}$.

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.