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calculus - What is infinity divided by infinity? - Mathematics Stack . . . One advantage of approach (2) is that it allows one to discuss indeterminate forms in concrete fashion and distinguish several cases depending on the nature of numerator and denominator: infinitesimal, infinite, or appreciable finite, before discussing the technical notion of limit which tends to be confusing to beginners
limits - Can I subtract infinity from infinity? - Mathematics Stack . . . $\begingroup$ Can this interpretation ("subtract one infinity from another infinite quantity, that is twice large as the previous infinity") help us with things like $\lim_{n\to\infty}(1+x n)^n,$ or is it just a parlor trick for a much easier kind of limit? $\endgroup$ –
elementary set theory - What do finite, infinite, countable, not . . . Clearly every finite set is countable, but also some infinite sets are countable Note that some places define countable as infinite and the above definition In such cases we say that finite sets are "at most countable"
Uncountable vs Countable Infinity - Mathematics Stack Exchange As far as I understand, the list of all natural numbers is countably infinite and the list of reals between 0 and 1 is uncountably infinite Cantor's diagonal proof shows how even a theoretically complete list of reals between 0 and 1 would not contain some numbers My friend understood the concept, but disagreed with the conclusion
What exactly is infinity? - Mathematics Stack Exchange Georg Cantor formalized many ideas related to infinity and infinite sets during the late 19th and early 20th centuries In the theory he developed, there are infinite sets of different sizes (called cardinalities) For example, the set of integers is countably infinite, while the set of real numbers is uncountably infinite "-Wikipedia: Infinity
linear algebra - Definition of Infinite Dimensional Vector Space . . . In the text i am referring for Linear Algebra , following definition for Infinite dimensional vector space is given The Vector Space V(F) is said to be infinite dimensional vector space or infinitely generated if there exists an infinite subset S of V such that L(S) = V I am having following questions which the definition fails to answer :-
Example of infinite field of characteristic $p\\neq 0$ On the other hand, if we had $\overline{\mathbb{F}_p}\subseteq\mathbb{F}_p(T)$, then we would have that there were some $\frac{f}{g}\in \mathbb{F}_p(T)$ such that $\frac{f}{g}\notin\mathbb{F}_p$ and $\frac{f}{g}\in\overline{\mathbb{F}_p}$ (because $\overline{\mathbb{F}_p}$ is infinite and $\mathbb{F}_p$ is finite), and they would have to be
general topology - Why is the infinite sphere contractible . . . Now, the classifying space of the free-living iso is easily seen to be the infinite dimensional sphere So now the categorical equivalence between the free-living iso and a terminal category means the infinite dimension sphere is homotopic to the classifying space of a terminal category, namely a point