Consider a Euclidean line segment AB. "Euclidean" means "as defined by Euclid in his book: The Elements of Geometry. If we are to take numbers as cuts in a line, as per Euclid and 2500+ years of tradition in mathematics, then for every x lying x dimensionless units away from A, there is another number x' lying x dimensionless units away from B. We can take a finite chart y on AB to be such that y runs from 0 to pi/2 between AB. This chart covers every possible cut in AB with some value in (0,pi/2), endpoints excluded. If the x chart is such that x=tan(y), then the x chart runs from 0 to INF between A and B such that every possible cut in AB takes some value in (0,INF)=R^+. By the above stated reasoning, the existence of the number 5 situated five dimensionless units of distance away from A (assuming the Euclidean metric on the Euclidean line segment, obviously), then there must also be some other number that is five dimensionless units of distance away from B which is associated with the extended real number INF. Let the symbol J be such that J-x denotes the number that is x dimensionless units away from INF@B. Let us call J-5 a number in the neighborhood of INF. This number must exist in the Euclidean system of R which supposes that real numbers are cuts in lines. J-5 does not exist in certain modern constructions of R, however. Since these modern constructions were only adopted when they were thought to exactly replicate and encapsulate the Euclidean system of R, the modern definitions should be updated to reflect the historically motivated existence of numbers such as J-5. Speaking of those updates, I have already done exactly that in my nice paper. I also show that if numbers such as J-5 exist, which they do, then the Riemann hypothesis is false.

Fractional Distance: The Topology of the Real Number Line with Applications to the Riemann Hypothesis
https://vixra.org/abs/1906.0237