Class object of subclass in super-class method
How to determine an object's class?
LIM is cofinal in ON I'd like to show that the class of all limit ordinals, LIM, is cofinal in the class of all ordinals ON.  I thought that I could do this by contradiction and assume that it wasn't cofinal in ON. Then there is an $\alpha \in$ ON such that for all $\beta \in $ LIM: $\beta &lt; \alpha$, i.e. $\beta \in \alpha$ and hence LIM $\subset \alpha$.  This means that LIM is a set so we can consider $\bigcup$ LIM. Then we have $\beta &lt; \bigcup$ LIM for all $\beta \in$ LIM by definition of $\bigcup$ LIM. So LIM $\subseteq \bigcup$ LIM.   $\bigcup$ LIM is either a limit or a successor ordinal.  If it is a limit ordinal then $\bigcup$ LIM $\in$ LIM and by transitivity of ON, $\bigcup$ LIM $\subseteq$ LIM, so $LIM = \bigcup$ LIM in this case which would mean that $\bigcup$ LIM $\in$ LIM $= \bigcup$ LIM which would be a contradiction because sets cannot contain themselves. Hence $\bigcup$ LIM cannot be a limit ordinal.  If $\bigcup$ LIM is a successor ordinal then $\bigcup$ LIM = $\beta \cup \{&#160;\beta \}$ for some $\beta \in $ ON.  Now I'm not sure how to proceed from here. I'm also not sure about whether I'm on the right track with this proof because I think there should be a shorter way  to prove this.  Thanks for your help. How to make a water vortex I have a spreadsheet in Google Docs with two data rows and lots of columns like so:    Is it possible to easily convert these rows into two columns and lots of rows (obviously retaining all information and formulas)? In other words, to something like this:   19.9.2009   967,15 €  28.9.2009   950,78 € 12.11.2009   942,03 € ..           ..   There's probably a neat mathematical term for such a conversion, but I don't remember which (matrix something?). Feel free to suggest a better name for the question.  Edit: To clarify, ultimately I want to get rid of the original (two row) version and continue working on the converted one. 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Please help me  to evaluate combinatorially the following sum: $$\sum_{k=0}^n \binom{n}{k}$$   Thank you. Dictionaries on iOS 8 not working Change default CSS of Google Chrome I would like to be able to change the default CSS used by chrome, such as removing the underlines for links, etc. However, I cannot find the default CSS file used by Chrome. Does anyone know where it is? Why is "improve this question / answer" shown to anonymous users instead of "edit"? If I browse a site as an anonymous user, instead of "edit" I see: "improve this question / answer".        I don't get it, are anonymous users considered to be kinda dumb and don't understand "edit"?  Why to reword this action? Inverse of function is continuous or not? If $f:I\subseteq \Bbb R \rightarrow \Bbb R$ is injective and everywhere continuous in its domain then $f^{-1}$ is also continuous everywhere. Here  $I$ cannot be discrete set and $f$ is single variable function The above statement is always correct or not? I thought it is always correct. Am I correct or not? DBRefs (Mongo Document references) not fetched eagerly