We have two theorems about Cartesian product:   A finite Cartesian product of countable sets is countable; A countable infinite Cartesian product of countable sets is not necessarily countable.   To prove the first theorem, we can show   Firstly, we can prove $A\times B$ is countable,  Secondly, suppose it is true for $n-1$, i.e. $X=A_1\times A_2\times \ldots \times A_{n-1}$ is countable, we prove the case for $n$ by the fact $A_1\times A_2\times \ldots \times A_{n-1}\times A_{n}=X\times A_{n}$ is in fact the product of two countable sets, in above we know it is still countable.   The above method to prove the first theorem is exactly the mathematical induction, which should show the theorem is true for all $n\in\mathbb{Z}_{+}$, so it should works for countable infinite unions.  However, we may use Cantor diagonal method to prove at least $\{0,1\}^w$ is not countable, thus the second theorem is also right. Does this means that the mathematical induction is wrong?    EDIT: After the following discussions, I have the answer:  The key point is that any element $n\in\mathbb{Z}_+$ is always a finite number, thus induction principle only guarantee the theorem is true for any $n\in\mathbb{Z}_+$, i.e. for any finite number.  When discuss a countable infinite union, it is infinite number of unions, this infinite number no more belong to $\mathbb{Z}_+$, thus can not be guaranteed by induction principle.
This is a follow-up question to the discussion about the finite axiom of choice .  Suppose we have a countable collection of non-empty sets $\{A_1, A_2, A_3,\cdots\}$ Reasoning as indicated in that discussion, we can prove the existence of a choice function $c_n$ such that $c_n(i)$ belongs to $A_i$ for $i = 1,\cdots,n$  Again reasoning as suggested and using induction, we can prove the existence of a function $C$ defined over $\{1, 2, \cdots\}$ such that $C(n)$ is a choice function defined for $i=1,\cdots,n$ with $C(n)(i)$ an element of A_i. We can require in addition that $C(n)(i)$ and $C(n-1)(i)$ are equal for $i=1,\cdots,n-1$.  Now it would it seem that we can prove the countable axiom of choice by taking the union of $C(1)$, $C(2)$, $\cdots$  The question is as follows: why is that proof wrong? I suspect the answer may have something to do with the axiom of union.
I created a interior scene and  i don't know how to bake a texture using cycles bake. I'm trying to show that the group $U(Z_{54})$ is cyclic.  To start, I found the divisors of 54 = {1, 2, 3, 6, 9, 18, 27, 54}  Then I started to find the elements using the powers of a.  Where $U(Z_{54})= &lt;a&gt;$  Therefore, $&lt;a^{54/1}&gt;$={ e }  $ &lt;a^{54/2}&gt;$= { e, $ a^{36}$ }  and such, where the last would be &lt; $a^{54/54}$> = { e }  The things I would like to know is, how does this process show that it's cyclic? Is it because the first divisor (1) and the last divisor (54) is equal to e? Therefore creating a cycle?  Also, what would the &lt; a > to the powers finding the elements process be called?  Thank you! .NET Process.Start default directory? I'm firing off a Java application from inside of a C#  console application. It works fine for the case where the Java application doesn't care what the "default" directory is, but fails for a Java application that only searches the current directory for support files.  Is there a process parameter that can be set to specify the default directory that a process is started in? I have a hazy notion of some stuff in differential geometry and a better, but still not quite rigorous understanding of basics of differential topology.  I have decided to fix this lacuna once for all. Unfortunately I cannot attend a course right now. I must teach myself all the stuff by reading books.  Towards this purpose I want to know what are the most important basic theorems in differential geometry and differential topology. For a start, for differential topology, I think I must read Stokes' theorem and de Rham theorem with complete proofs.   Differential geometry is a bit more difficult. What is a connection? Which notion should I use? I want to know about parallel transport and holonomy. What are the most important and basic theorems here? Are there concise books which can teach me the stuff faster than the voluminous Spivak books?  Also finally I want to read into some algebraic geometry and Hodge/Kähler stuff.  Suggestions about important theorems and concepts to learn, and book references, will be most helpful. How is canonicity of derivative works determined for Star Wars? 3/4 circle with plus in it? After I recently updated my phone about a day ago, this strange sign showed up on the top left that I have never seen before. Does anyone know what it means? Or how to get rid of it? How to prove a general matrix invertible given by as below?  How to prove that $A^TA+I$ is always invertible for $\forall A \in \mathbb{R}^{n\times n}$? Try to do something vs. Try doing something Is there any differences between      Try to do   And     Try doing   Thank you in advence. Prime spectrums bijection but not homeomorphism Accessing the index in 'for' loops? Rounding a number to the nearest 5 or 10 or X Given numbers like 499, 73433, 2348 what VBA can I use to round to the nearest 5 or 10? or an arbitrary number?  By 5:   499 -&gt;  500 2348 -&gt; 2350 7343 -&gt; 7345   By 10:   499 -&gt;  500 2348 -&gt; 2350 7343 -&gt; 7340   etc. What is a virtual terminal for? Time dependent Schrödinger equation with time independent potential and separation of variables suppose we have a potential that's independent of time $V(x,t) = V(x)$ so in Schrödinger equation we get: $$i\hbar \frac{\partial \Psi (x,t)}{\partial t}=-\frac{\hbar^2}{2m}\frac{\partial^2 \Psi (x,t)}{\partial x^2}+V(x)\Psi(x,t)$$ since the LHS involves a variation of $\Psi$ with $x$ and the LHS involves a variation of $\Psi$ with $t$ can we say that a basis for the solutions will be with the form $\Psi(x,t)=\psi(x)T(t)$? Is there a convenient way to import a schema into Oracle 11gR2 using a single new or different tablespace than where the data originated?  As an example, I have exported BLOG_DATA from OLDDB, where all user data is stored in the USERS tablespace.   On NEWDB, I would like to import the BLOG_DATA schema, but store the user objects in the BLOG_DATA tablespace, created specifically for this user.   I have created the BLOG_DATA user, created the BLOG_DATA tablespace and set it as the default tablespace for that user and added an appropriate unlimited quota.  CREATE TABLESPACE blog_data DATAFILE SIZE 1G;  CREATE USER blog_data IDENTIFIED BY secretpassword DEFAULT TABLESPACE blog_data QUOTA UNLIMITED ON blog_data;  GRANT connect,resource TO blog_data   The schema was exported from OLDDB with something like  exp blog_data/secretpassword@OLDDB file=blog_data.dmp    After reading Phil's excellent answer below, I found myself wondering:   Since data has no other place to go than the default tablespace - the only tablespace the user has a quota on - will this effectively force imp to put all user objects in that default tablespace?  imp blog_data/secretpassword@NEWDB file=blog_data.dmp   Would this then place the entire blog_data schema in the blog_data tablespace on NEWDB? Is there any reason why this wouldn't work or that I would run into problems with certain objects etc?  update:  I did a quick test and found this to be the case. Imp places objects in the default tablespace for that user, provided it can't place it in the original tablespace (eg. the tablespace does not exist). Full explanation:   Still, I suppose that using Data Pump like Phil suggests might be the preferred option. Is it possible to upgrade a BGA 1224 CPU? Most efficient way to convert an HTMLCollection to an Array Is there a more efficient way to convert an HTMLCollection to an Array, other than iterating through the contents of said collection and manually pushing each item into an array? Looking for a single word for 'blind worship' Is there a single word for 'blind worship' as in worshiping an actor blindly notwithstanding the bad performance? Why comment reply auto-completion feature (@) sometimes doesn't work? How do I backup data (SMS/contacts) from a device with a broken screen? I just broke the screen of my (unrooted) Galaxy S.  How do I get back my SMS and contacts?   PS: I tried Kies but it won't detect my phone for some reason. (Kies works fine with another Galaxy I possess but not this one) A word to describe "work that is ready to be done"