also is there any way to express the floor function in a structured way in tex? the only way i know is \lfloor whatever \rfloor whatever but i would prefer to say \floor{whatever} which doesn't work
@Sean That is correct, we are given a specific function $f$. The case I showed you was to give a feel for how $A$ is constructed, not to do it in the hypotheses of the proof (that would be impossible for many reasons).
@BenjaLim Is the proof of a limit of a sequence as follows: Start with the definition of the limit of a sequence via the inequality, manipulate the inequality to get an inequality of the form $n>N(\epsilon)$, and let $N=\lceil N(\epsilon)\rceil$ to complete the proof?
@Sean For any function $f$ whose domain is $S$ and codomain is $P(S)$ we can define a set $A\subseteq S$ by allowing only those elements $x$ of $S$ which satisfy $x\not\in f(x)$ to be in $A$.
@Sean We now assume the existence of a surjective function $f:S\to P(S)$. So we can construct $A$ as we did in my first note. Notice that $A\in P(S)$ the codomain. By the second note there exists $s\in S$ satisfying $f(s)=A$.
@BenjaLim, i don't remember the theory, but I guess I roughly know what it is - so I try to write down what the going-up/going-down operators are etc. Perhaps this is the wrong idea.
@Sean Finally, we ask whether $s\in A$. Firstly, if $s\in A$, then $s\not\in f(s)$ by definition of $A$, but $f(s)=A$, a contradiction. Otherwise, $s\not\in A$; hence $s\in f(s)$, but again $f(s)=A$, so $s\in A$, a contradiction.
Thinking about it a bit more, is there any reason you cant just use maple/mathmatica to calculate it to a decimal place or two? — remusDec 19 '10 at 19:18
i'm imagining generalizing away from $e$ to some extent while preserving the idea of iterated powers. e.g. if we have some finite set of somehow unrelated (?) reals, can we show that any finite power-tower comprised of them is not an integer?
@peoplepower since we can't answer the quesiton of e^e^e^79 we can't answer any generalization either, but i wonder what would make a plausible conjecture?
@DanBrumleve Actually that is not entirely good. Anecdotal evidence suggest that it is better to speak in a language that the the listener can understand rather than the language the speaker can speak in, as the "guessing" is much more improved. For example, I would rather have someone say to me that a function is going up from 4 to 5 in reals rather than saying that the function is monotonic on the interval [4,5] in french.
which google translate says in portuguese is "A função é monotónico durante o intervalo [4,5]"
The listener and the speaker have the same context, and it is the listener who wants to connect the dots. It is better for him to connect the dots in the language he is a master in.
@Orange that is a good counterpoint. but we shouldn't discount the value of bilingual users who might edit english translations into questions... i'm imagining in that situation the gap might be bridged more easily than with the current monoculture
"However, one another factor that I missed is the fact that someone really good in both languages can "translate" the post into English. "
But then, instead of separate sites, we should have separate "language tags" and like close votes, it should be ensured that all posts with language tags other than english are translated into english. It should be responsibility of the "language editors".
