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comment What are the most prominent uses of transfinite induction outside of set theory?
I find it the most natural and intuitive way to apply the axiom of choice in just about any context where it's needed. (more precisely, in the form of algorithms with transfinitely many steps)
Jun
27
revised Metrizability, Models, of Non-Standard Reals
added 429 characters in body
Jun
27
answered Metrizability, Models, of Non-Standard Reals
Jun
26
comment Is every tensor an element of a vector space?
@Gary: You should ask your question via "ask question" from the main page, not as a comment in a loosely related thread.
Jun
20
comment Understanding the definition of product of sets
@easy: This is, I think, one of the many instances where the easiest way to prove something is a bijection is to write down its inverse.
Jun
19
comment Why aren't there $+\infty^{+\infty}$ real numbers?
Aside: the mathematical object usually denoted by $+\infty$ has nothing to do with the notion of size you are asking about. (and also, for this object, $(+\infty)^{+\infty} = +\infty$)
Jun
17
comment What is a negative number?
I will make the typical formalist appeal. "What is a negative number?" is the wrong question to ask. "What can I do with negative numbers?" is the right question to ask.
Jun
17
comment A bit confused about definition of set of mappings in Herstein's Algebra
Aside: some people use "one-to-one" to mean "injective", and others to mean "bijective". It's probably better to avoid the phrase.
Jun
15
comment Integrating wrt to same parameter twice
Sometimes people are careless, typically with substituting. e.g. if they've written the equation $1 = \int_0^1 1 \mathrm{d}x$, then later on they have some reason to replace $1$ with such an integral in a context where $x$ already has a meaning, they will blindly substitute $1 \mapsto \int_0^1 1 \mathrm{d}x$ which is confusing because the $x$ in the substituted version has nothing to do with the meaning that $x$ already had, so you have two completely different variables going by the name $x$. The right thing to do is to pick a new letter first: e.g. $1 \mapsto \int_0^1 1 \mathrm{d}y$.
Jun
15
comment Integrating wrt to same parameter twice
The trick is that $\theta$ isn't $\theta$; it's using the same letter to refer to two different things, to confusing effect.
Jun
14
comment Conceptual Understanding behind a Limit
If you're going to try and follow your professor's thinking, you should probably first consider the even simpler example: the behavior of $\frac{x}{x}$ at/near $0$. (for reference, the functions $\frac{x}{x}$ and $1$ are not equal, because the domains of the left and right hand sides are different: the left hand side is not defined at $0$ but the right hand side is)
Jun
14
comment pth root of unity in $p$-adic field
That should be $(p-1)p^{n-1}$, or do I have an off-by-one error?
Jun
14
comment Matrix equation solving
$2X = (2I) X$, not $(2O) X$. @Alex: In matrix algebra, have never seen the notation $A-2$ used to mean anything other than $A-2I$, and that's certainly what's needed here.
Jun
14
comment Why can't epsilon depend on delta instead?
You have the rigorous version wrong: it's not "whether there are $x$'s", it's "whether all nearby $x$'s". The rigorous definition says for whatever definition of 'nearby outputs' you pick, there is a definition of 'nearby inputs' so that 'nearby inputs imply nearby outputs'.
Jun
13
comment Why can't we add functions?
I think your question is missing a lot of important context, but I imagine the answer is that your professor was saying "the relationship you're looking for is given by the derivative", rather than any of the stronger claims you've inferred.
Jun
13
comment Is there ANY possible way to solve this equation?
You probably need to be more flexible about what "solve" means.
Jun
13
answered Where is sheafification in the definition of exact sequence of sheaves?
Jun
13
comment Where is sheafification in the definition of exact sequence of sheaves?
It depends on whether you're working in the category of sheaves or the category of presheaves. Also, in the category of sheaves, images (in the categorical sense) of morphisms generally can't be computed pointwise.
Jun
13
answered What are “instantaneous” rates of change, really?
Jun
13
comment What are “instantaneous” rates of change, really?
$(f(x+h) - f(x)) / ((x+h) - h)$ is not the derivative of $f$: it is merely infinitesimally close to the derivative (assuming the derivative exists).