The axiom of infinity cannot be derived from the rest of the axioms of ZFC, if these other axioms are consistent. Nor can it be refuted, if all of ZFC is consistent.ZFC is the minimal set of axioms for modern math to work out. So basically a few things:
- Modern math is just as consistent with or without Infinity. Every finite result in mathematics could have been derived with or without needing to refer to infinity. (Sometimes people do like when then claim things like they are “integrating to infinity”, but technically a reference to infinity isn’t needed to get the same answer)
- No matter how hard you try, Infinity "cannot be derived...Nor can it be refuted" by ZFC meaning you cannot ever infer the existence or non-existence of Infinity from the rest of mathematics.
It would ruin mathematics for a lot of people, especially the mathematicians themselves, since infinity is required to produce all the most interesting stuff. Furthermore, what if infinity does exist and you are removing it just because you can't know either way given everything else? Think of all the intellectual damage done to mathematics if infinity is falsely removed! (There are entire branches of good math if infinity is real!) But then by that same token, I guess you also have to ask if myriads of mathematicians are just wasting their time studying an entity that you can only be axiomatized and no more.
As for me and my house. I for one double majored in mathematics and so am quite aware of the amazing intellectual structures that can be built with the assumption of infinity and to be honest... they are pretty amazing! At some point, the beauty, wonder, and to be quite honest *practical utility* (integrating to infinity is darn helpful sometimes!) of assuming the existence of Infinity outweighs any pros that seem to come from getting rid of armed by nothing more than philosophical arguments.
The reality is Infinity has been around for enough centuries to have proven it's utility, beauty and ability to make many magnificent contributions to mathematics. I can't say there is any evidence beyond philosophical wishful thinking to the claim that math would be better off without Infinity... even if you were to believe Infinity was false! I for one have tasted math containing Infinity and love it, am inclined to believe it, and am pretty sure I will be engaging in math containing/refereeing to infinity for the rest of my life as it seems like the most wise, lovely, productive and correct thing to do.