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elementary set theory - What is the definition for an infinite set . . . However, while Dedekind-infinite implies your notion even without the Axiom of Choice, your definition does not imply Dedekind-infinite if we do not have the Axiom of Choice at hand: your definition is what is called a "weakly Dedekind-infinite set", and it sits somewhere between Dedekind-infinite and finite; that is, if a set is Dedekind
What is the difference between infinite and transfinite? Infinite simply means "not finite", both in the colloquial sense and in the technical sense (where we first define the term "finite") There is no technical definition that I am aware of for "transfinite" Nevertheless, I can attest to my personal use Transfinite is good when there is a notion of order, so "transfinite ordinal", or when you want to talk about non-standard real numbers which
Finding a basis of an infinite-dimensional vector space? For many infinite-dimensional vector spaces of interest we don't care about describing a basis anyway; they often come with a topology and we can therefore get a lot out of studying dense subspaces, some of which, again, have easily describable bases
One divided by Infinity? - Mathematics Stack Exchange Infinite decimals are introduced very loosely in secondary education and the subtleties are not always fully grasped until arriving at university By the way, there is a group of very strict Mathematicians who find it very difficult to accept the manipulation of infinite quantities in any way
linear algebra - Invertibility of infinite-dimensional matrix . . . As far as I can tell, the "infinite matrix" representation of a linear operator is not that popular, especially in non-Hilbert contexts There are many technicalities to address, as Jesko rightfully points out Anyway, I remember that I have seen some information on this point of view on the book "Basic operator theory" of Gohberg and Goldberg
How can Cyclic groups be infinite - Mathematics Stack Exchange I am a little confused about how a cyclic group can be infinite To provide an example, look at $\\langle 1\\rangle$ under the binary operation of addition You can never make any negative numbers with