There is some kind of analytic continuation theorem in complex analysis I don't understand, something about if a holomorphic function isn't identically zero, then it has an infinite countable set of zeros. I don't see a name for the theorem, I've only been told it in spoken language and then maybe have found similar theorems on the internet like here https://math.berkeley.edu/~vvdatar/m185f16/notes/Lecture-19_Zeroes.pdf.
But $e^{z}$ is holomoprhic and yet I have been told by several professors it has no zeros in the complex plane, so how is this not a contradiction with the alleged theorem? What is it actually saying?
It's related to this other question Holomorphic function has at most countably zeros but I am trying to understand it from this zero set argument. The question has "at most" but I don't see an "at most" in the PDF I linked to nor was I told "at most" in real life, just that the set of zeros of a holomorphic function is countable if it is not identically zero, which was used as a lemma to the fundamental theorem of algebra and that if there is always one zero of a power series of a holomorphic function for any order, then it must be polynomial.
There is no such theorem and neither the word “infinite” nor “countable” appear on that text that you linked. I suppose that there is a language issue here. For some authors, a set is countable if its cardinal is smaller than or equal to the cardinal of $\mathbb N$. But, if this is so, then finite sets are countable and, in particular, the empty set is contable. So, one can perfectly say that a function without zeroes (such as the exponential function) has finitely many zeroes.