Here is a silly question, but I am a silly person.  Consider the: Natural Numbers. Natural Numbers X Natural Numbers. Natural Numbers X Natural Numbers X Natural Numbers ...  Now take the union of all of these sets. In other words, this is the set of all sequences of natural numbers/tuples of any size, of natural numbers.  The cardinality is certainly at least that of the power set of the naturals, or the cardinality of the reals, because this is essentially like the power set except now, order matters/repetition is allowed.  But does having order matter, and allowing repetitions, increase cardinality of an infinite set? Will this set have a cardinality aleph-one, or will it be higher? I suspect it is the same, but maybe this does push the cardinality up in the same way that taking a power set would?  Thank you.
While I was reading Enderton's "A mathematical introduction to Logic", I came across the proof of the following sentence: "The set of all finite sequences of members of the countable set A is also countable".   Proof: The set S of all such finite sequences can be characterized by the equation  $$S=\bigcup_{n \in N} A^{n+1}$$ Since A is countable, we have a function f mapping A one-to-one into N. The basic idea is to map S one-to-one into N by assigning to $(a_0,a_1,...,a_m)$ the number $2^{f(a_0)+1}3^{f(a_1)+1}\cdot ... \cdot p_m^{f(a_m)+1}$, where $p_m$ is the $(m+1)$st prime. This suffers from the defect that this assignment might not be well-defined. For conceivably there could be $(a_0,a_1,...,a_m)=(b_0,b_1,...,b_n)$, with $a_i$ and $b_j$ in A but with $m\neq n$. But this is not serious; just assign to each member of S the smallest number obtainable in the above fashion. This gives us a well-defined map; it is easy to see that it is one-to-one.  Note: P is a finite sequence of members of A iff for some positive integer $n$, we have $P=(x_1,...,x_n)$, where each $x_i \in A$.  First of all, I cannot understand why the former assignment might not be well-defined and the latter assignment is well-defined. Secondly, I cannot understand what Enderton means by "just assign to each member of S the smallest number obtainable in the above fashion". By the way, is $(a,b,c,d) = ((a,b),(c,d))$ true? Also, in which cases can I omit/add parentheses in a tuple so as to have an equal tuple?
What Constitutes Spam? Over the course of the past several months, there have been one or more users who have been spamming about an XML parsing alternative named &quot;VTD-XML&quot;. I have occasionally assisted the moderators by pointing these answers out to them, and they have responded by deleting the answers, and the offending user (which hasn't stopped this person from immediately creating another user account and continuing). Now, there is a user , who claims to be the actual author of VTD-XML. In my mind, several of his answers, like his  to  looks a great deal like the original spam.  has asked me why I consider his answer to be spam. I decided to do so in public, in an attempt to get him to engage with this community, assuming he's not another sock puppet of the original spammer.  My reason is simple: he's been answering with one or two lines of text, followed by the exact same link to the exact same site that the spammer was using:  VTD-XML may well be worth a look as it is 1/3~1/5 the memory usage of DOM   Now, maybe I'm wrong. Maybe this does not constitute spam. Maybe Mr. Zhang can explain to us why we should not treat these posts as spam. Let's see.  Somehow, I missed the fact that in the same question, on July 31 2009:  how about vtd-xml  http://vtd-xml.sf.net   This one was from a user named &quot;Jimmy Zhang&quot;, which has since been deleted. Should I not be suspicious? Additionally, as pointed out by , he's answering a question about Delphi by posting a link to a site that doesn't even mention Delphi! Throughout the nonsense with the sock puppets, I kept asking for some real engagement from the sock puppet. It all just made VTD-XML look bad that instead of truly answering questions, we just kept getting incessant spam against old XML questions. 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