Then I don't know what I don't know about this topic. Right now I feel like I understand the arguments against my claim.
I think I've made my position clear on why I think you're having difficulty with this topic, so here's a last-ditch attempt to help you. Others have pointed out where your argument fails and if you don't understand the problems they have shown with your claim, then you need to go back and think about it. I won't repeat the same work. Instead, I will take a different tack - I will prove the opposite claim, one that most people just intuitively accept, that there are infinitely many natural numbers. Only one of us can be right, at least if you accept that the axioms underlying all of mathematics are consistent.
Here is a proof that there are infinitely many natural numbers. Fundamentally, it is a property of the
principle of mathematical induction, a property so intrinsic to the concept of number that it is one of the fundamental axioms used to define them. I will actually use an equivalent principle that is more intuitive and easier to state - the
pigeonhole principle. The pigeonhole principle is "if we place n + 1 objects into n categories then at least one category contains at least 2 objects". Restated in terms of functions, it is "If A and B are finite sets where A has more elements than B, then there is no one-to-one function from A to B." I want to emphasize that this principle is how we DEFINE the natural numbers, so it is essentially inviolate. It should also be really intuitively obvious, like good axioms are, but we don't accept it simply because it is intuitive.
First, the definition: A non-empty set S is finite if and only if there is a bijection between S and {1,2,3,...,n} for some number n in N. In that case we say that S has n elements. For the edge case, we'll call the empty set finite as well. An infinite set is a set that is not finite.
Now, the proof. Please pay attention to the structure and syntax of the proof and contrast it with your posts. I don't assume existence or properties for the objects I define. Everything step is explained, using definitions (I left some unstated, but can fill those in if necessary). In the future, you should strive to have your mathematical arguments follow the same pattern.
This is a proof by contradiction - I will assume that the set of natural numbers is finite, that is that there is a bijection f : N -> {1,2,3,...,n} for some number n (since 1 is a natural number, I can discount the empty case). Now, I want to use that assumption to show a contradiction, thereby proving that the set of natural numbers is infinite.
Specifically, look at the bijection f : N -> {1,2,3,...,n}. Since it is a bijection, it is one-to-one and onto. Now, I will define a new function g : {1,2,...,n+1} -> {1,2,3,...,n} by taking g(x) = f(x) for every x in {1,2,...,n+1}. g is called the restriction of f to {1,2,...,n+1}.
First off, g is well-defined - for every number 1,2,...,n+1, f is defined and thus g is defined. Furthermore, g is one-to-one because f is one-to-one, that is, if g(x) = g
then f(x) = f
and so x = y. Now, g is not necessarily onto because we have restricted the domain of the onto function f, but that's OK, we only need one-to-one-ness.
So we have a function g : {1,2,...,n+1} -> {1,2,...,n} that we have proved to be one-to-one. But the pigeonhole principle states that since {1,2,...,n+1} has more elements than {1,2,...,n}, g can't be one-to-one. This is a contradiction. Therefore, the assumption that N was finite is false. That is, the set N of natural numbers is infinite.
