@LeakyNun But then without axioms what allows you to write anything down?

premise is not the same as axiom

@LeakyNun oh I see it's precisely that awkwardness of bar(A+B) that suggests the correct definition of addition in a Boolean algebra

@LeakyNun I know, but what I mean is that in a usual Hilbert style proof, the things you can write down are either: 1. From your premise set, 2. Any of your axioms, or 3. Using your rule of inference / modus ponens on two previous lines in the proof

right

so in natural deduction there are no axioms, just rules of inference you can use on previous lines

so where do these previous lines eventually come from if not ultimately the premise set

In terms of mod 2, bar isn’t acting like a minus sign but as +1

@LeakyNun if A/I is noetherian as A module then how to show it is noetherian as A/I module?

that's half the truth

it's mainly Hilbert vs Fitch

@Ninjahatori use the finitely generated condition

Trying to get spherical harmonics a faster way - take a homogeneous polynomial of degree $l$ in $u_l = \sum \xi^a \eta^b z^c$ for $a+b+c=l$ and $\xi = x + i y, \eta = x - i y$, that satisfy $\nabla^2 u = (4 \frac{\partial^2}{\partial \xi \partial \eta} + \frac{\partial^2}{\partial z^2})u = 0$, why would you think to say $u_l$ separates into a sum of terms $\sum u^m$ for $m = -l,\dots,0,\dots,l$ where the $a - b = m$?

@LeakyNun is [0,1] is infinite compact hausdorff space?

yes

is it infinite I know it is compact and hausdorff ? But infinite in what sense they talking about?

As a set

ok fine thank you sir.

@bolbteppa my preferred fast way is "look them up in a table"

(I actually have no grounds to make that joke, given how obsessed I've been lately about Wigner-d matrix elements)

@LeakyNun Half the truth?

well you can say that Fitch has no axioms

but Hilbert surely has axioms

Yeah they come up too often as if you're supposed to know them since birth :\

i'm talking about natural deduction though vs. hilbert

i'm not familiar with fitch

i'm asking about how you can even write a proof if you don't have any premises or axioms, just inference rules

I'm saying you have axioms if you use hilbert

yes\

i'm talking about ND though

Recall: If $S$ is a subset in some topological space, then the boundary, denoted by $\partial S$, is $\overline{S}\setminus int(S)$, where $\overline{S}$ is the closure of $S$, $int(S)$ the interior of $S$. Claim: If $\partial S \subseteq S$, then $S$ is closed. Proof: Clearly $\overline{S} = int(S) \cup \partial S$, so $\partial S \subseteq S$ implies $\overline{S} = int(S) \cup \partial S \subseteq int(S) \cup S = S$.

How does that sound? Seems more like an elementary set theory prob. than a top. prob.

topology is just applied set theory

I suppose one could view topology from that perspective...So am I justified in thinking my solution is right, given that you didn't give any corrective remarks?

I didn't check your work

I just wanted to make a joke

Oh, ha! Okay.

Anyone else mind verifying my solution (given a few posts above this one)?

you should probally check your definitions

this isnt something you should be able or have to prove it should follow directly from the definitions

a common defintion of a closed set in topology is the smallest closed set containing its limit points

or the smallest closed set containing your set

in which case theres something to prove

no that's the closure

yeah

a set is closed if its equal to its closure which is that