RE: How should be (and actually is) Mathematics
When I imagined a way to demonstrate the theorem I did many attempts and I chose the one that sounded the most natural (because saying that the more you go in dept in Mathematics the more cool it becames sounded natural to me), but if you have an alternative solution please tell me! It was the purpose of the article to make all of you think and imagine!
Anyway I know that Mathematical topics are countable but I though that if I would have stated that there are, let's say, 10 topics in mathematics, someone else could argue that there 11 topics because he counted them in another manner; so I preferred to keep it general, without numbers.
In the end, even if all would have accepted that there are 10 topics in mathematics what would you do with that? Basically you have to demonstrate that C(Tn) > 0 ∀ n of N. But I don't see any advantage by knowing the exact amount of topics...
But, as I have already said, if you have another answer I happy to hear that!
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My reply was meant as a non-serious reply. Don't take it too serious. ^-^
The monotonicity comment was about the property C(Tn)> C(Tn-1). Won't there be topics which are equally cool at some point. :P
It is fun to think about how to define an uncountable topic class. Depending on how you define topic you can do this. You could for example define a topic as a property given to something in set which is or is equivalent to the real numbers :o)
And another thing to think about is why induction is true at all. Because the familiar induction that is used in math relies on arithmetic induction :o)
I'm sorry, from time to time, it happens to me that I don't understand sarcasm (maybe it is due to the fact English isn't my native language)
Anyway mine was a friendly argumentation :)
I totally agree with you about my property C(n), I knew that someone could have noticed this logical flaw! But, as I told you, it was the most "natural" (credible) idea I got.
Oh, yes. I know that not all mathematicians like induction because it can't be demonstrated (in fact it is a principle) and the same they think about Numbers in general!
I know that many mathematicians tried to demonstrate the existence of Natural numbers (because you can demonstrate the existence of other bigger sets (Z,Q...) only assuming that N exists) but it wasn't impossible, so all the existence of all other sets of number isn't demonstrated.
So mathematics needs axioms in order to demonstrate everything.
All the logic of math, is based on something that is supposed to exist but we won't never if it actually exist...
To extend the concept:
Do we exist? Are we talking now? XD
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