INFINITY

Let's say you have a bag with infinite apples inside. You begin taking out the apples randomly. There is one (maybe more) particular apple that will never be taken out.

>>12
If that's how you want to define it, it's fine, although the definition is something along these lines (I used google's definitions):
1. The belief in and worship of a superhuman controlling power, esp. a personal God or gods.
3. A particular system of faith and worshipNow if arithmetic is a religion, it's only one which asserts that it's consistent (contains no contradictions) when extended to infinite length (as opposed to any finite partition, such as with digital computers(although that's actually modulo n, not limited at some specific number), which is free from these issues). What it has been proven is that a finite consistency proof for an infinite system like peano arithmetic is not possible within itself, which actually makes perfect sense. You could of course just do a proof by induction, erm, transfinite infuction up to Aleph Null cardinality, but wether that is better or not is questionable as set theory is stronger than arithmetic and is more likely to have issues.

I can have "faith" (if we're using religion's terms here) in arithmetic's consistency, but I'm not so sure about set theory as it's a lot more varied in its axioms.
The actual "faith" part does make sense. You have provable and true statements in arithmetic, they apply to finite statements which can be proven in finite steps. And you may have true, but unprovable statements, such as the consistency of the entire system - this is unprovable as it would require an infinite amount of steps to prove it within arithmetic itself (without adding any axioms or asking some stronger system for help, such as the transfinite induction example earlier). Of course, one should be very careful here: asserting truth of an unprovable statement can lead to contradiction if done within arithmetic itself (and thus it leads to contradiction and falsity), so while we can intuit that arithmetic is consistent (or even assert it within some stronger axiomatic system), we can never make that claim within arithmetic itself, lest is inconsistent.

Either way, the stranger thing is wether this is true within set theory:
http://upload.wikimedia.org/math/3/b/7/3b7eae31da752e0d91c10da0d3b489f4.png (2^aleph_0=aleph_1)
It is both impossible to prove and disprove its truth value, without adding additional axioms to the system.

Either way, while I think it's not that troublesome to be a finitist as far as uncountable infinities of infinite objects (as real numbers are) are concerned and regard them as useful fictions, but I cannot ever take the ultrafinitist approach for purely philosophical reasons which would make us impossible to exist if we took that approach, but I'm not going to elaborate on that as I'm guessing you equate any philosophical dabbling with religion (despite that I lack any belief in a "Supreme Being", the most unusual belief that I have is that of Arithmetical Platonism, but like all beliefs, it's updatable, it's not faith, it's merely what what I consider most probable for now, and if something arises that would introduce some serious contradictions that would make me change this belief, I can do it (I've examined the current arguments against this position and they are unconvincing to me)).

Maybe length is the wrong way to say it. It is merely that there is an infinite number of natural numbers. If you don't like saying the word "infinite", look at it this way, assume N is a natural number, then by peano's axioms, N's successor (what we write N+1), exists and is greatar than N, thus we achieve a contradiction, which means there is no greatest number N (which is what would be called an infinite amount of natural numbers. I'm not involving cardinalities here, despite them making sense to a certain degree when talking about infinities).

Oh, I'm also not of the race that you claim I am, however it's pointless to argue such details over the Internet.