Mr Indeterminate said:
Thus Salman effectively states, he has provided the reason as to why higher levels of math will refer to 0/0 as having an indeterminate result, rather than just being undefined.
Is this incorrect or does it have a basis?
By what standard do you wish to judge correctness?
As mentioned before, within the assumptions and definitions for the real numbers, there are no theorems or axioms that say "0/0 is undefined because...". There is no definition that defines what it means for a
number to be "indeterminate".
To talk about "0/0", we are not talking about a
number. We are talking about a string of symbols. What set of definitions and axioms are we using to talk about strings of symbols?
It is possible to talk about strings of symbols in a rigorous manner. For example, one can study computer programming languages formally. However, the video is not doing this. The video is an informal discussion of the problems that a person would face in trying to extend the definitions of the real numbers so that "0/0" would be defined to be a number. As an informal discussion, I agree with the ideas in the video. However, the video isn't giving a formal proof of anything. It also isn't giving a formal definition of what it means for a string of symbols to be "indeterminate".
There are definitions in mathematics that use the word "indeterminate" in various contexts. Mathematical definitions, when written precisely, do not define single nouns or adjectives. Mathematical definitions define
statements, i.e. complete sentences.
In a calculus book, some authors might define the statement:
"The expression ##lim_{x \rightarrow a} \frac{f(x)}{g(x)}## has indeterminate form 0/0"
to mean the statement:
" ##\lim_{x \rightarrow a} f(x) = 0## and ##lim_{x \rightarrow a} g(x) = 0##".
Such a definition does not define the symbols "0/0" to be a number. (it also does not define the adjective "indeterminate" as an isolated concept.)
For example, the expression ##lim_{x \rightarrow 1} \frac{ 2(x-1)}{x-1}## has indeterminate form 0/0, but the limit is equal to 2. So, in the context of calculus, "the expression has indeterminate form 0/0" is not the same as saying "the limit denoted by the expression does not exist" or "the limit denoted by the expression is not equal to any particular number".