It is well proven that Aleph one, which is the infinity of the real numbers, is undeniably bigger than the infinity of the natural numbers.
The clause "Aleph one, which is the infinity of the real numbers", is known as the continuum hypothesis, and has a fascinating background in itself.
First, the existence of Aleph one in axiomatic Zermelo-Fraenkel set theory depends (surprisingly) on the Axiom of Choice. If you reject AC, we can't show that there exists a unique Aleph one.
It gets weirder. If you accept ZFC (ZF set theory, plus the Axiom of Choice), we can prove that both Aleph one and the cardinality of the reals are greater than Aleph nought. However, Gödel proved in 1940 that Aleph one cannot be proven to be equal to the cardinality of the reals, given ZFC. In fact, none of the main ZFC axioms constrain the continuum hypothesis--there are some proposed axioms, like constructability, which imply CH, but nobody is really sure whether we should accept them.
This is very much a philosophical problem in mathematics: having proven we cannot decide on the basis of the axioms we widely accept, it's now up to us to choose which branch (or both) of mathematics is more useful or epistemologically satisfying--or find other axioms we can agree on that in turn constrain CH.
[edit] derp, just read to the bottom, and it's comment #1. Right then, carry on. :)
I feel like every time I get deep enough into a topic of mathematics, Gödel inevitably shows up to say, "things beyond this point are provably unprovable!" It's simultaneously vexing and fascinating.
It's precisely this conundrum and others like it that have lead me to take two controversial opinions that are far less firm than the real thought that your describing: 1) mathematics and what is taught as physics are not "real," but merely leaky abstractions. Even though they can capture reality very well, they are not reality itself. There is no perfect circle that exists in physical space, and not necessarily a pi represented anywhere in physical space. You'll notice that a lot of the wonder that is expressed in this post comes from thinking that 3sin(57) is something that is exactly* reality. And 2) all of mathematics is invented, not discovered. The particular math we use may be discovered independently by different, unconnected, civilizations, because there are certain thought processes that fit well with human intelligence. If we discover other intelligent life forms, there may be some small overlap by coincidence in our mathematics, it's likely that it could be quite different. Perhaps even arithmetic could be considerably different.
We are taught from day one in class that mathematics is some sort of ideal plane of existence, pure, and real. However I see it only as technology for our squishy gray matter to help navigate a mysterious universe. I get huge huge resistance on this from engineers and some young scientists; they see the textbook science and math where everything has a nice closed-form answer you can look up in the back of the book. More mature scientists object less, but that may just be because they think it's pointless to discuss these things with me.
Trying to use mathematical technology in the real world in new areas makes one realize that only a tiny percentage of questions are answerable with the tools they teach us in school. The lack of a closed solution to the 3-body problem is not due to a lack of cleverness in answering, but a lack of cleverness in questioning. And further study leads to the paradoxes and the holes that were discovered in the previous century, at which point, most people's faith in math's fidelity to reality begins to become less than absolute.
The clause "Aleph one, which is the infinity of the real numbers", is known as the continuum hypothesis, and has a fascinating background in itself.
First, the existence of Aleph one in axiomatic Zermelo-Fraenkel set theory depends (surprisingly) on the Axiom of Choice. If you reject AC, we can't show that there exists a unique Aleph one.
It gets weirder. If you accept ZFC (ZF set theory, plus the Axiom of Choice), we can prove that both Aleph one and the cardinality of the reals are greater than Aleph nought. However, Gödel proved in 1940 that Aleph one cannot be proven to be equal to the cardinality of the reals, given ZFC. In fact, none of the main ZFC axioms constrain the continuum hypothesis--there are some proposed axioms, like constructability, which imply CH, but nobody is really sure whether we should accept them.
This is very much a philosophical problem in mathematics: having proven we cannot decide on the basis of the axioms we widely accept, it's now up to us to choose which branch (or both) of mathematics is more useful or epistemologically satisfying--or find other axioms we can agree on that in turn constrain CH.
[edit] derp, just read to the bottom, and it's comment #1. Right then, carry on. :)