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12:55 AM
@Phyllotactic I made the extended version btw
 
1:07 AM
@Deedlit If the above actually makes sense, any comparison to other ordinal functions?
 
 
5 hours later…
6:29 AM
@SimplyBeautifulArt It's... rather strange. Most of the definitions are what I would expect, some of them seem suboptimal (like $\varphi (\alpha;0,\ldots,\beta_t,\ldots,\beta_m)[n] = \varphi(\alpha;0,\ldots,\beta_t [n],\ldots,\beta_m)$ and $\varphi (\alpha;0,\ldots,\beta_t + 1,\ldots,\beta_m) = \varphi(\0;0,\ldots,\alpha,\beta_t,\ldots,\beta)$). But then when we get to $\psi$, we start throwing in $\Omega$ in there, in a way I don't quite get.
Also, using $g(\Omega)$ for in place of $\alpha$ seems ambiguous; you probably want to replace that with something more explicit (like writing an ordinal in Cantor Normal Form base $\Omega$).
 
 
3 hours later…
9:18 AM
@amWhy @Deedlit or anyone else: If you could spare 2 points to delete this (literally going in circles) and this (not answering the question), I would appreciate it! I'm ambivalent about asking for help because I would very much prefer the community auto-corrects, but somehow that never works well. =S
 
9:32 AM
morning @user21820
 
@shredalert: Hello!
@Kyoma: Hello! What brings you here?
@shredalert I love the notes (at least the discrete maths one on induction) in your link!
> any ''proof'' in mathematics that begins with what you want to prove and
ends with TRUE is at best badly written and at worst incorrect and **drives me crazy**.
Although frankly it's still not up to my preferred standard of precision haha..
But I like the way he emphatically warns against some of the annoying logical errors that are frequently made due to an inadequate grasp of induction. =)
Very sad that some car driver was careless and hit him on a cross-walk. =(
 
I believe the discrete notes are his wife's. And sad indeed @user21820 :(
They both write very nicely and their main aim is to make the student understand. I'm so glad I came across them. I have Rob Ash's Abstract Algebra book, that's where I heard of him and looked up his books and came across Carol Ash's notes too.
 
9:49 AM
Ah I see. Didn't realize that the two webpages were for two people.
And in fact, Robert's notes on field theory are one of the references I went to when studying Galois theory.
 
:D
I finished the first chapter of the logic book a few minutes ago btw
started working through it as soon as I woke up
taking a little breather now, then I'll do a bit of linear algebra.
 
 
1 hour later…
10:59 AM
@Deedlit thanks man. Will check it out...
 
 
4 hours later…
2:38 PM
Also does anyone know what a Pole is and why is it important to Contour Integration
 
3:30 PM
@Zophikel Can't tell you much sorry, I know it has to do with complex analysis and residues... check simple poles
 
@amWhy would you please be so good as to help me understand the reasons for deleting math.stackexchange.com/questions/2204335/… ? Was it for mentioning a notoriously hard problem?
 
@Phyllotactic thanks don't worry I got an answer to my intial question in another chat room
@Phyllotactic but Calculus of Residues is an extremely powerful tool
 
@Zophikel Nice, glad to hear that! Yeah I'm building my understanding towards that (not there yet), I really love calculus!
 
@Phyllotactic correction you love analysis
 
Haha correct
I love both Analysis and number theory
 
3:34 PM
i'm found a video on Contour Integration through goes a little bit too fast :( for my speed
 
@Zophikel I'm saving MIT-18.04 Complex Variables for the summer
maybe there's some useful things in there for you
 
@RobertFrost Many of the problems I see with your question are pointed out in comments. The fact that you ran circles around them and never fully addressed them, (not counting personal attacks given in a post or two), I've learned with respect to this question, at least, that nothing I say to you directly would move you to even the simple, intermediary action of seriously considering of my/other's feedback.
@SimplyBeautifulArt Wherefore art thou?
 
3:51 PM
@amWhy: Hello!
 
@user21820 Hello to you too, two times
 
Haha. =)
 
Don't mind me. I was saying hello, and fixed on the alliteration of "to...."
@user21820 Glad you got it!
 
I only half got it. Language and poetry are funny things.
Anyway, how are things?
 
@user21820 was able to downvote each, but there's no current option to delete the first. However, the second, after having downvoted it, it became eligible for deletion, and will require two more delete votes.
 
4:01 PM
@amWhy Strange; the first was available for deletion and I've already cast my vote.
Anyway thanks!
 
You can vote to delete as many times as you want (meaning, if the count resets), you can vote again to delete.) That's different than open/close voting.
 
Well the option to delete the answer does not appear automatically after downvoting to negative score; one has to refresh to see it.
 
OOps @user21820 there are two delete votes at the first (yours and mine), and two delete votes on the second.
 
And the number of delete votes don't refresh automatically either. I had voted on both already. =)
It's a bit strange because upvotes/downvotes refresh automatically, but close/delete votes don't.
 
@user21820 that actually happens >
 
4:11 PM
All each of them need is one more delete, at which point, they will be deleted.
 
Yeap. Since it's now in the moderation queue, I'll just leave it to the frequent reviewers to finish them.
It's always satisfying to get rid of nonsense (the circular one), despite the uphill battle.
 
Speaking of nonsense, I was just called a "globetard" for trying to defend the apparently unsighted position that the earth is round... This is hilarious in a sad way :-/
 
4:26 PM
@Phyllotactic: Lol! By the way, there's actually a mathematical way of verifying to some degree that the earth is not flat.
25
A: Why do objects appear smaller when viewed from a distance?

user21820It is because light travels in more-or-less straight rays. Let's assume for simplicity that your eye is like a pinhole camera; it has a pinhole in front and a screen at the back. Then an image forms by the light rays that pass through the pinhole. (from https://commons.wikimedia.org/wiki/File...

On the globe you can actually observe the 'true horizon' as well as the slightly curved surface if you look at the ocean.
Of course, there are ways to 'reproduce' that on a flat disc-world.
But the easiest is to take two plane flights to go around the globe...
 
@user21820 Beautiful explanation! I upvoted your answer and favorited the question to find it quick, it may come in handy to have near by ;)
What is a bit concerning to me is that this silly discussion happened in a Canadian university corridor in 2017...
 
What?!
That's.....
Is there a possibility the other party was trolling? =P
 
I took that into account and cut the argument short, although the other partIES are know to be "conspiracy afficionados" and will often make me mad defending ridiculous sounding theories against all common sense
 
Oh.
 
4:47 PM
Don't get me wrong I love my silly friends but some of them suffer serious cases of "being on the wrong side of the Dunning-Kruger effect curve" when it comes to science stuff
 
Haha..
I've been saying that about logic for a long time, but never knew the effect had a name.
However, there's a slight difference for logic; once one has complete grasp of formal systems, one will be able to know accurately what one grasps and what one doesn't, unlike for non-mathematical or even non-scientific knowledge.
But it's frequently frustrating trying to convince people to re-evaluate.
 
@user21820 how important is logic as a tool
 
I just went back to school a year ago, I have the crazy dream of trying to get to a Math Ph.D from no previous knowledge and the more I learn the more I'm scared by the magnitude of my ignorance.
And you're right, it is quite frustrating to get people to re-evaluate anything
 
@Zophikel: As I always say, people who do not have a complete grasp of logic are never capable of even vaguely understanding or appreciating its absolute necessity for correct reasoning, in every field of mathematics as well as in real life.
 
@user21820 intersting
 
4:59 PM
@Zophikel I think it makes you able to learn pretty much anything on your own
 
I was first exposed to logic when I was doing programming
 
@Phyllotactic <- Yes this is absolutely true.
@Zophikel: If you're capable of writing recursive programs easily, you should have no trouble studying logic.
 
@user21820 I did learn a little bit of logic while doing intro to proofs
 
For example, if you can write a sudoku solver from scratch by yourself in any common programming language without any libraries, that shows a more or less complete grasp of the fundamental capability of programs, which include conditional branches and iteration. Classical first-order logic is similarly fundamental for mathematics.
 
@user21820 i'm interested in logic from the viewpoint of quantum mechanics
In quantum mechanics, quantum logic is a set of rules for reasoning about propositions that takes the principles of quantum theory into account. This research area and its name originated in a 1936 paper by Garrett Birkhoff and John von Neumann, who were attempting to reconcile the apparent inconsistency of classical logic with the facts concerning the measurement of complementary variables in quantum mechanics, such as position and momentum. Quantum logic can be formulated either as a modified version of propositional logic or as a noncommutative and non-associative many-valued (MV) logic. Quantum...
 
5:03 PM
In my personal opinion, one needs a firm foundation in classical logic first before it is meaningful or useful to look at other logics.
 
@user21820 good point
@user21820 you know a good book on mathematical logic i'll explore it when i'm done with analysis
 
The reason is basically that the ultimate underlying logic in mathematics is classical. We study all sorts of logics in the meta-system, but the meta-system is itself a classical system.
 
The Meta-System ?
 
I have recommendations linked under "Introduction" from my profile; you should skip Suppes' book and go straight to Stephen's notes.
 
all right thanks @user21820
 
5:05 PM
The meta-system is the system we use to study other formal systems, including itself.
 
interesting
I should pick up logic as early as possible since it has a role in QC
 
Indeed it does, but I don't know enough about QC to comment much on that.
However, certainly current foundations for mathematics (ZFC set theory) can reason about and simulate all other practical formal systems ever conceived or ever to be conceived.
Actually much weaker than ZFC is enough, but ZFC is current, so...
@Deedlit: Hello!
@Deedlit @amWhy @SimplyBeautifulArt: I now have 256-char programs for the small and large Veblen ordinals. First I'll post the bare programs. Both are in the same style as my super-tree programs, though surprisingly the one for the large Veblen ordinal was shorter before diagonalizing. Their growth rate in terms of the initial value of c should be at least f[v*ω] in the FGH, where k is the small or large Veblen ordinal respectively.
Small:
c=99
o=[]
i=[o]
def r(x):
	v=o
	y,*d=x
	if y!=i and y[-1]==o: *y,_=y
	z,*p=y
	t=z!=o
	if len(y)<2:
		if t: v=[[r(z)]]*c
	else:
		v=[i]
		if t: v+=[[r(z)]+p]
		m=0
		while p[0]==o:
			m+=1
			_,*p=p
		a,*b=p
		for k in x*c: v=[i*m+[v,r(a)]+b]
	return v+d
def f(n):
	if n:
		v=[i*c+[[i]]]
		while v!=o:
			c*=c
			f(n-1)
			v=r(v)
			print(c)
f(c)
f(c)
Large:
c=99
o=[]
i=[o,o]
j=[i]
def r(s):
	v=o
	y,*z=s
	_,x,*l=y
	t=x!=o
	if t: u=[o,r(x)]
	if l==o:
		if t: v=[u]*c
	else:
		v=j[:]
		if t: v+=[u+l]
		p,q,*l=l
		for k in s*c:
			v=[i*(p!=j)+[r(p),v]+[p,r(q)]*(q!=j)+l]
	return v+z
def f(n):
	if n:
		v=j
		for k in i*c:
			v=[i+[v,j]]
		while v!=o:
			c*=c
			f(n-1)
			v=r(v)
			print(c)
for k in i*c: f(c)
I also made friendly versions with test cases and pretty outputs.
Small friendly:
c=2
o=[]
i=[o]
def dbg(t,x,k):
	if x==o: return "0"
	if x!=i and x[-1]==o: *x,_=x
	if x==i: return "1"
	if x==[[i]]: return "w"
	if t==1 and len(x)==1:
		v=dbg(1-t,x[0],k)
		if len(x[0])>1 and x[0][-1]!=i: v="("+v+")"
		return "w^"+v
	v=""
	if t==1: v+="f("
	l=len(x)
	j=0
	for y in x[::-1]:
		if t==0 and y==i:
			v+=str(l-j)+"+"
			break
		j+=1
		if k==0:
			v+="..."
			break
		k-=1
		v+=dbg(1-t,y,k)+"+,"[t]
	v=v[:-1]
	if t==1: v+=")"
	return v
def r(x):
	v=o
	y=x[0]
	if y!=i and y[-1]==o: *y,_=y
Large friendly:
o=[]
i=[o,o]
j=[i]
def dbg(t,x,k):
	if x==o: return "0"
	if x==j: return "1"
	if t==0:
		v=""
		while x!=[]:
			*x,y=x
			if y==i:
				v+=str(len(x)+1)+"+"
				break
			if k==0:
				v+="..."
				break
			k-=1
			v+=dbg(1,y,k)+"+"
		v=v[:-1]
	else:
		if len(x)==2:
			_,x=x
			v=dbg(0,x,k)
			if len(x)>1 and x[-1]!=i: v="("+v+")"
			if v=="1": return "w"
			return "w^"+v
		v="f("
		while x!=[]:
			*x,p,q=x
			if k==0:
				v+="..."
				break
			k-=1
			if p!=o: v+=dbg(0,p,k)+":"
			v+=dbg(0,q,k)+","
These programs more or less follow my description here, for anyone interested in analyzing them. =)
@Deedlit: By the way, I think I know the correct comparison function. Say for the small Veblen ordinal first. If we want to two terms x = f(l+[a]+m) and y = f(l+[b]+m) where l,m are lists of ordinals and a < b, there are 3 cases. If b > every parameter in m, then x < y. If b = some parameter in m and all subsequent parameters are 0, then x = y. Otherwise x > y. This comparison can be done recursively because the total length of the compared terms will strictly decrease.
I think it shouldn't be hard to prove it is a total-ordering and that every reduction always decreases the ordinal, but proof of well-ordering is not obvious to me.
 
6:24 PM
Hello @user21820
Congratz on getting to the large Veblen ordinal!
 
 
2 hours later…
8:09 PM
@user21820 et al. If anyone has a teeny weeny bit of spare time, this question should be deleted.
 
 
3 hours later…
10:44 PM
@SimplyBeautifulArt I think I am not smart enough to answer your question. Or, I still don't understand your question.
 
11:05 PM
@Zophikel a pole is a point outside a function's domain where an infinitely differentiable function (at all points in its domain) has a limit diverging to that point. That's how I think of it.
@Simple lol, don't say that. The question is supposed to incite thought and be a learning experience
@amWhy I was hurling my guts for 9 hours is all
 
@SimplyBeautifulArt Ouch!! Sorry to hear that! Are you feeling any better? In any case, make sure you get some extra sleep!
 
well, I still don't know how to find a number the fill on a piece of paper, just don't know how
 
@amWhy well, I've been periodically napping for the past hours, so I hope I can sleep at the right time
 
As you said, the number is finite, I think it must be an integer, but you say, we can use exponential or something
 
@Simple well, you needn't actually fill the whole paper. Rather, reach stepping points such that the next stepping point is larger than filling the entire paper of the previous step.
For example, if your first number was 10^100, then find a number so large that it wouldn't fit on the paper using basic exponentiation and multiplication of 10^100
 
11:12 PM
ok
 
If you can't think of a number so large that it would be 'unreachable' in terms of the previous, then we'll call it 'significantly larger', which is what we want
@user21820 lol, I need to go understand whatever language you used better
 
What you mean is, if I find a sufficient large number $n$ that might fill the paper, $n$ is a desire number if $e^n$ and multiplication of $n$ exceed the paper
For multiplication, do you mean $n\times n$ or $n\times k$ where $k$ is any constant?
 
Yeah. The hidden question is if you can come up with, say, some notation 'larger' than filling the paper with exponentiation.
Either (though one could argue that n*n = n^2 is better/simpler notation)
Ugh, while I haven't thrown up in the past two hours, I feel dead internally, though better than before. :-(
 
@SimplyBeautifulArt Take it easy, @SimplyBeautifulArt. Wait until your stomach cooperates with your mind, and drink whatever fluids you can keep down!
 
11:28 PM
I can drink fluids right now without regurgitating it, but only so much. Much dehydrated still
@Simple want a push in some direction?
 
i kind of see how to find the number
 

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