If the exponential function fails to be total in M, then M's arithmetic is too weak to interpret standard complexity-theoretic distinctions, undermining the claim that M's polynomial-time class is the complexity-theoretically relevant one.

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Arithmetic: Arithmetic is the branch of mathematics dealing with basic number operations—addition, subtraction, multiplication, and division. It's the foundation of math that you use in everyday life, like calculating change at a store, splitting a bill, or figuring out measurements. Essentially, it's the practical math skill everyone learns early in school to work with numbers in simple, straightforward ways.
Complexity-theoretic(in computer science): Related to the study of how difficult or resource-intensive it is to solve a computational problem (like how much time or memory a computer program needs).
Exponential function(in mathematics and computer science): A mathematical function where a number is repeatedly multiplied by itself a certain number of times, like 2×2×2×2. It grows extremely fast compared to other types of functions.
Interpret (in logic)(in mathematical logic): To set up a system so that its symbols and rules match up with another system's symbols and rules, allowing the second system to express the first system's ideas.

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