I hope I can communicate my concerns effectively, so I can reach an understanding about a topic that I've been reflecting and researching intensely on for a few days. I am thinking about actually infinity in mathematics, specifically set theory with respect to the axiom of infinity, and the set of natural numbers as a completed infinity.
I understand that there will not be answers to this post that will solve the question as to whether or not an actual infinity exists theoretically, but I hope to have answers that will spark insight in my mind.
- Is the axiom of infinity truly an axiom?
What I mean is that axioms are usually taken as self evident truths that need no proof, but I don't see how the axiom that an actual infinite set exists is truly self evident. It appears that it was created solely for allowing the possibility of infinite sets regardless of how self evident it may or may not be, which seems inconsistent with how axioms are meant to be used.
- Are there plausible reasons to believe in the theoretical existence of a completed infinity?
I believe an actual infinite set does theoretically exist, like the set of natural numbers. I see nothing intrinsically wrong or ambiguous with the definition of the set of natural numbers, or with the inductive set used in the axiom of the infinite set. Further, I also believe that if a set is defined, all the elements satisfying the definition or property of that set already exist in that set even though we may not be able to enumerate them all even given infinite amount of time. Consequently, all the natural numbers are in the set containing them, which means this set is a completed infinity. Somehow I find it plausible to have a collection of infinitely many objects within a collection, but I am curious if anyone has found reasonable arguments to bolster his or her confidence that accepting this axiom is not just a leap of faith, but a rational thing to do.
I do hope that this question is philosophical enough in nature, or at the very least have the potential to spark philosophical discussion.
Thank you all in advance for any feedback.