Paradoxical Proof?

[Bridge]

My point is there is no such thing as a "smallest" number that is measurable. We use negative infinity to approximate it because there is absolutely no way to know what it is, and infinity doesn't really mean anything on its own. Set theory is designed to work with finite numbers and the infinity abstract, but it doesn't take into account an infinitely small measurable number because it was not designed that way (and it is impossible). Therefore it is impossible to conclude there is definitely such a thing as a "smallest number" using a system that doesn't know what that is.

EDIT: To further illustrate, the reason why the proof tells you there is a smallest number is because it expects it. I do not believe we have sufficient evidence at this point to ascertain if infinity can be followed through to a definite, finite number and therefore we cannot use the tools we made to say that there is. We made it to work with finite numbers - it doesn't work with unknowables.