Ordinals

I once read about a theory that said that numbers can be described as a common property of two groups that have nothing in common excluding their size. For example the number three is a common property of the following groups - three deers, three stones and three trees.
In modern mathematics we have a sort of an extension to this idea - ordinals. An ordinal is a well ordered set such that if A is an ordinal and x is in A and y is in x then y is in A. The first ordinals are phi (=empty set), {phi}, {phi,{phi}}, {phi,{phi}, {phi,{phi}}}. Those ordinals correspond to 0,1,2,3.
As you probably noticed there is a very simple rule that produces the next ordinal - if A is an ordinal than A(union){A} is the next ordinal. From this we can conclude that: The set of all ordinals is a well ordered set and the union of any number of ordinals is an ordinal.

What makes the ordinals truly interesting for me is the fact that in for them "infinity plus one" is not equal to infinity. This is very simple to see, infinity is the so called least infinite ordinal - w. It can be defined as the union of all finite ordinals. The next ordinal is w+1=w(union){w}. It is rather obvious that the two sets are not equal and therefore w+1 is not equal to w.
Ordinals are not the only example of infinity not being equal to infinity and one, but in my opinion they are extremely intuitive in this regard. After all, all we basically do with ordinals is to constantly "add one". This is the same thing we did with natural numbers long ago, but it appears that the natural numbers don't follow our basic intuition that says that "it is always possible to add one"

In the beginning of the post I told that numbers can be described as a common natural property. This however brings an interesting philosophical question - if our intuition is a product of our world than why do natural numbers that come from it don't follow our intuition after a specific point? A possible answer is that "infinity is not natural" and therefore there is no reason for it to follow our intuition in any way. However, infinity appeared as a concept a lot of time ago. At first it appeared as "many" which basically told that there was no known number large enough.When a new number (or even a number system) where invented the "many" was replaced by an appropriate number. And this brings us to the following thought: Is it possible that we are in the same condition again? That is, should we use ordinals instead of natural numbers? After all, they are pretty much an extension of the natural numbers.