A finite number of smaller lengths, added together will exceed a longer length. whats this theorem called? Like, you have a small string, if you put lots of these small strings end to end, you will exceed the length of any string after doing it a finite number of times.
I think this was Galileo, euclidean, Pythagoras? It was named after some famous mathematician from like a thousand years ago. whats it called again?
 A: This is the Archimedean axiom.
In some contexts it's an axiom and in some it's a theorem.
It rules out "infinitesimals" other than zero, i.e. positive numbers $\varepsilon$ for which $\displaystyle \underbrace{\varepsilon+\cdots+\varepsilon <1}_{n\text{ terms}}$ remains true no matter how big the (finite) number of terms.
Archimedes lived in the third century BC in Syracuse, Sicily, which was then inhabited by Greek-speaking people.
Take a look at this image:
https://upload.wikimedia.org/wikipedia/commons/thumb/d/d7/Parabolic_Segment_Dissection.svg/503px-Parabolic_Segment_Dissection.svg.png
Archimedes considered an arbitrary secant line of a parabola and drew a line through its midpoint parallel to the axis, forming a triangle whose three vertices are the endpoints of the secant and the point where the last mentioned line intersects the curve.  He claimed that the area bounded by the curve and the secant is $4/3$ of the area enclosed in the triangle.  He did that by iterating the construction of the aforementioned triangle, using the new secant lines created as you see in this picture, and he showed the sum of the areas of the new triangles created at each step is $1/4$ of the areas at the previous step, so you have $1+\frac 1 4 + \frac 1 {4^2} + \frac 1 {4^3}+\cdots = \frac 4 3$.  Now the punch line: This doesn't work if there are infinitesimals, because it then only shows that the area differs by a (possibly nonzero) infinitesimal) from $4/3$.
Perhaps, strangely, he also wrote a different, and brilliant, argument using infinitesimals to prove the same proposition, and said explicitly that it falls short of being a proof because it uses infinitesimals.
