What does it mean that $V$ is transitive? It means that a subset is also an element.
Another way to say that is to say that if $y\in V$ and $x\in y$ then $x\in V$.
However if $V$ "sees" $x\in y$ then it thinks that $x\in V$. It is possible to have a model with some $y$ and some element of $y$ that $V$ doesn't know about, but we do (externally).
Consider the following case, which may be easier to understand.
$V=(V,E)$ is a model of ZFC (which is an element of the universe $M$), take a countable ultraproduct of $V$.
The result is a model $V^*$ of ZFC, but this model is ill-founded in $M$.
However $V^*$ does not know about the decreasing $x_{n+1}\in x_n$ which lives in $V^*$.
$M$, however knows about this sequence so $M$ can tell us that $V^*$ is ill-founded.
I'm going to see if my advisor is free. Be back later.
Well, it's far from enough to be staying. This morning at around 6 there was a little mousse covering the green in front of my house, but it was gone quickly when dawn came.
Realistically, there will be no trace of snow by Wednesday or so. But oh well. It's not like I go skiing or anything, I just like it because it makes things go quiet.
An undergraduate was telling me about a puzzle he'd found: the idea was to make $2011$ out of the numbers $1, 2, 3, 4, \ldots, n$ with the following rules/constraints: the numbers must stay in order, and you can only use $+$, $-$, $*$, $/$, ^ and $!$. In words, "plus minus times divide, exponenti...
@Matt Oh, didn't know about that, looks good! I like Steven Moffat's work quite a bit, so I should check it out when it's available. Thanks for the link
Does anyone know a good book that gives exercises to improve concentration, reading speed and memory? Those are for me at least above average but improvement or slowing down decay is always great :-). The books I find are full of scary buzz words.
@tb Can you try and help me locate a decent reference for Cohen's second model of ZF? Jech's presentation is cumbersome and I can't find it anywhere else.
I only read a few bits from that book, but I like his style. I found several of the proofs very elegantly written. I also have to say that professor Herlich is very nice as a person. In this case the name describes the man perfectly: de.wiktionary.org/wiki/herrlich
@MartinSleziak Just a note on usage. Herrlich is almost never (if ever) applied to persons. You would describe an excellent meal, the weather or a landscape as herrlich, but not a person.
Aces. I'm forced to do a seminar next term and I get to choose between numerical analysis, probability or statistics. Or "Representation theory of associative algebras" but given the lecturers it'd rather commit seppuku.
@AsafKaragila Okay, I give up, I can't find anything of relevance.
The only thing I found that you might want to look at is the references in the Howard-Rubin list for model M7. But Google doesn't let me look there, so...
@Matt You can join any of the seminars here. We have about many different topics. I will give a lecture about that question I recently answered regarding second countability and Lindolef-ness without the axiom of choice.
Question. Would it be much of breaching copyright, if I gave you the pages I downloaded from google books (hypothetically, let's say I have cca 370 out of 430)? Anyway, they're available on google books, so they're publicly available I would say. (Not sure if storing and sharing them would be ok.)
@AsafKaragila I'm about to wander off the right path again: I wanted to get this over with asap and then leave. If I do a term abroad it'll take even longer.
@robjohn the algebra people do in homotopy theory using spectra. It's a thing that became quite fashionable a while ago when Manin lauched a campaign for it.
So given the FPO $$ M_\mathcal{U} = \{ (s,x) \mid s \in fin ( \omega ) \land x \in \mathcal{U} \land \max (s) < \min (x) \} $$ where $$ (s,x) \leq (t,y) \iff s \subseteq t \land y \subseteq x \land t \setminus s \subseteq x$$ and $$ m_G := \bigcup \{ s \in fin( \omega ) \mid \exists x \in \mathcal{U} : (s,x) \in G \} $$ ($\mathcal{U} $ is an ultrafilter) I'm asked to show that for any $x \in \mathcal{U}$: $m_G \subseteq^\ast x$ which means that $| m_G \cap x^c | < \omega$. I've been thinking about a proof by contradiction but I think it's a dead end and I'm also worried that I'm making the …
I went to the library with another lib card I picked up and told her I was having trouble getting on the computers with it. She scanned it and told me the PIN, asking if I typed it correctly. I simply said "whoops" and now my max time on Sunday library computers is doubled forevermore. :D
$(s,x)$ and $(t,y)$ are compatible if there is an $(r,z)$ such that $$ s \subseteq r \land z \subseteq x \land r \setminus s \subseteq x \text{ and } t \subseteq r \land z \subseteq y \land r \setminus t \subseteq y$$
I'm somewhat preoccupied, so I don't see it quite clearly yet. However the main point is that the finite sets grow and given $x\in\mathcal U$ we can find a dense set which has smaller infinite sets (e.g. $x\cap (n,\omega)$) and larger and larger finite sets (e.g. $x\setminus (0,n]$) and therefore this dense subset of the forcing poset ensures that we meet $x$ almost everywhere.
so, $D=(-2,2), f(x)=\sqrt{|x|}sin(1/x), x \neq 0$ and $f(0)=0$. I have to show that if it's continuous (or not). I find this trivial to say that it's continuous, but of course I have to proove it. I wonder if I have to use the delta-epsilon proof thing or if there is another way I can prove it? Can I say that both functions $sqrt{|x|}$ and $sin(1/x)$ are continuous in the domain (except for 0)? So, the product of these functions is also continuous?
@robjohn yes, as matt mentioned, I ment pm as personal message
yeah, I think read somewhere that PM function was not implemented to avoid knowledge not being public, i.e discussions happening through PMs instead of being public
@Clash I guess that's a better way as opposed to delta-epsilon, it's be long for a function like that. The product of continuous (real-valued) functions is continuous.
@Gigili I see. How can I show that $f(x)$ is continuous on $f(0)$? I mean, we know it's 0, but we don't know about $f(x)=\sqrt{|x|}sin(1/x)$. Maybe I'm confused. Can I write that because $\sqrt{|x|}=0$ for $x=0$ and $f(x)=0$, then the function is continuous?
Because $\sqrt(|x|) $ is zero either $x$ tends to $0^+$ or $0^-$, doesn't matter what the amount of other function is - at that point, I guess. $\lim_{x \to 0} ... =0$
I need a studying tip for Schaum's Outline for Topology...the answers are there. And I don't feel like doing the hard work when the answers are already there. What if I just read through?
@AsafKaragila Okay I will give an example. There was a puzzle I was trying to solve called Mahavira's Arrow about number of arrows in a hexagonal quiver. After 5 mins, I looked at the solution. My question is since now I know the answer now what will be the next step so to speak? :)
Concerning your linguistic question: I'd say $x$ and $y$ are solutions and you count the number of them. You can have an amount of money or speak of the amount of chocolate in the cake, but the amount of solutions seems somewhat odd to me.
@Gigili Instead of $$"mx = 2m-1$$ So $x$ and $y$ are $(2m-1)/m$ and $1-m$, respectively." I'd really write: If $m \neq 0$ then we may divide by $m$ and get $x = (2m-1)/m$ and $y = 1-m$. If $m = 0$...
Then conclude as you did and say what happens if $m = 1$ or $m=2$.
@robjohn "we name something in algebra simply by how it looks, such as a + b is called the sum of a and b" is a common phrase thrown around in pre-algebra that creates a lot of confusion, would you care to elaborate on its intended meaning?
In an abelian group you can call the group operation "+" and then things like "a+b=b+a" and "a+(b+c)=(a+b)+c" carry over to the new setting as well, even though these aren't even numbers. In algebra when the formal structure of a general setting (like abelian groups) "looks" like something familiar and usual - addition - then we can abuse our words and start calling things like a+b "sums" in the higher setting as well as the lower.
@robjohn "we name something in algebra simply by how it looks, such as a + b is called the sum of a and b" is a common phrase thrown around in pre-algebra that creates a lot of confusion, would you care to elaborate on its intended meaning?
@Skullpatrol I saw that before, but I am trying to figure out what you are trying to communicate. I wouldn't use that phrasing since it sounds too randomly motivated.
We name something in algebra so that it corresponds to something with which we are already familiar so that we might better understand its function in the new setting.
@robjohn What I'm trying to communicate is that a + b is called the sum of a and b, but what we are referring to is the resulting number c which is the sum of a and b.
You can often look at things that way. When people write expressions out for a group the symbols get interpreted one way, and when they write out expressions for arithmetic they have a different interpretation.
This doesn't always work since often people write strange things that have no compositional meaning.
"we name something in algebra simply by how it looks, such as a + b is called the sum of a and b" is a common phrase thrown around in pre-algebra that creates a lot of confusion" In pre algebra? Really? What does 'something' here refer to if not actual addition for a prealgebra audience?
@robjohn The progression is from a phrase to a variable expression to the result of that variable expression with values substituted in. But sometimes, for example we talk about just the sum of a and b as a + b having certain properties without ever mentioning that it is the result of the sum that we are referring to.
Ah, the distinction between the result of an operation and the operation itself expressed in the same way with notation. We've officially hit Spoonwood territory here.
