I have been thinking about the nature of infinity lately. I have no experience with higher mathematics or theorems regarding infinity, so please forgive me if my ideas on this topic are extremely naive or superficial.
Cantor's theorem proves that there are infinities that are larger in size than other infinities (again, forgive me if I have stated this wrong). This reminds me of someone who is given a titanium box, and there is a mysterious object placed inside the box. The person being given the box doesn't know what this object is, but he is allowed to conduct experiments on the box. For instance, he can measure the temperature of the box, its weight, and so on, but he is never allowed to directly (or indirectly) view the contents of the box. What Cantor has proven makes me think that humans to infinity is analogous to the man and the object. We have been able to discern some property of infinities, but we cannot view them directly. This is where the idea I have been thinking about comes in. As humans, we can automatically distinguish the magnitude of 3 and 6, and it is a simple task to see that 3 < 6. However, what if there were creatures on another planet, who found difficulty with this. Their numbers may be extremely small, in the limit as we approach 0 say, and they may be comfortable working with such numbers. They may even have theorems for their numbers, many of them, like we have Fermat's Last Theorem for the integers, and quadratic reciprocity for primes, and so on. In fact, many of our theorems may even break down with the "numbers" of these creatures, because the numbers are so small, that they do not even behave like our numbers. However, when they come to view the integers, or 0.5, or even 0.0001, they cannot view it directly, so they have to do what we did with infinity. They may use the tools they have, and deduce that one of these numbers is bigger than another, and so on. What if infinities are just extremely huge numbers, that they cannot be called numbers anymore. We cannot call them numbers, because you cannot treat $\infty$ as a number since it leads to contradictions, but what if they are really really huge numbers, i.e. that we actually need to define a new set which consists of all really huge numbers.
Question: Could infinity be just what we label as one of those large numbers, because we cannot see them or manipulate them directly? Maybe there are many of these numbers, like we have 1,2,3,..., each one bigger than the other. They may have their own algebra too, which is why plugging one of these large numbers ($\infty$) into our algebra leads to contradictions. Could there be a whole new mathematics just using large numbers, that we have no access to?
Furthermore, is it "correct" to say that, since our mathematics is entirely based on axioms which we view as self-evident because of the universe (for instance, if we lived in a new universe where taking a part from a whole leaves you with 2 wholes instead of "less than the whole", or whatever, then euclidean geometry breaks down completely), there could in theory be another mathematics based on these extremely large numbers, which complies with the physical intuition that the creatures who use it experience in THEIR universe? (which in turn explains why we cannot treat infinities as numbers in our mathematics, but they can)
I didn't want to post this on philosophy xchange, because they may not have the mathematical insight many of you hear have. Please reply genuinely and without shunning me off, I am genuinely interested, and as I said I apologize if this question is very bad, naive, or just plain stupid. I am curious and want to know an answer this way or that. Thanks.