For example, in this question: Polynomials, finite fields and cardinality/dimension considerations
Bonus question: More generally, I sometimes see the word "consideration" being used in various mathematical contexts. Here are examples: linear space - considerations about dimensions and max degree polynomial for time complexity considerations Is there a shared meaning for the word "consideration" in all of these contexts?
Thanks!
It's not a technical term - when we say "by cardinality considerations, [stuff]," all we mean is "considering the cardinalities of the objects involved, we can see that [stuff]."
For an example of why thinking about cardinality can be enough to solve a problem, consider the following proof that $\mathbb{R}$ is not finite-dimensional as a $\mathbb{Q}$-vector space:
$\mathbb{Q}$ is countable.
Any finite-dimensional vector space over $\mathbb{Q}$ is a finite product of countable sets, hence countable.
But $\mathbb{R}$ is uncountable - so we know $\mathbb{R}$ can't be a finite-dimensional $\mathbb{Q}$-vector space.
More broadly, it's often quickest to show that two mathematical structures are different by showing that they have different cardinalities, and calculating their cardinalities usually doesn't take too much work.
