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I'm reading Halmos' Naive Set Theory.

According to the usual and natural convention "for some $y\,(x\, \epsilon \, A)$" just means "$x\, \epsilon \, A"$. It's equally harmless if the letter used has already been used with "for some-" or "for all-." Recall that "for some $x \, (x\, \epsilon \, A)$ means the same as "for some $y\, (y\, \epsilon \, A) $; it follows that a judicious change of notation will always avert alphabetic collisions.

It seems they change the letter to avoid these alphabetic collisions, but what's the problem with it? For me, it's clearer when it's stated without the letter chaging, just as in the bold text. Where are these alphabetic collisions going to be harmful? In the past, I felt confused when I saw "for some $y\,(x\, \epsilon \, A)$", I thought it was a statament about two objects.

I've also asked something similar before.

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    This doesn't make sense. Can you give more context?2012-11-21
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    The $>$ is a typo.2012-11-21
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    do you mean to use the symbol $\in$?2012-11-21
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    @Holdsworth88: The $\in$ symbol is known as $\epsilon$-relation. In fact the original notation used $\epsilon$ and you can find old papers which used it that way. Halmos' book is quite old nowadays.2012-11-21
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    I believe I have corrected two typos in the quoted text.2012-11-21
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    @Ted I don't know why this thing of changing the letters exist. Is this helping?2012-11-21
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    I think that when Halmos writes "alphabetic collisions" he is referring to "for some $y(x\in A)$", and he suggests avoiding these by writing "for some $x(x\in A)$" instead. That is, Halmos wants to do what *you* want to do, and wants to avoid the sort of statement you find confusing. He is agreeing with you.2012-11-22
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    Part of the problem is that Halmos is discussing logic at a somewhat informal level. One needs to pay attention to these issues when treating first order logic in a more formal setting, after all, typically we present axiom schemas where we start with a formula $\varphi(y)$ and end with $\varphi(x)$. Informally, it is clear what this means. Formally, we need to talk about free and bound appearances of a variable in a formula, and things are simpler to define if $x$ is just not present in $\varphi$ to begin with.2012-11-22
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    @GerryMyerson So do people write like "For some $x$ $($$y$ $\in$ $A$$)$"?2012-11-23
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    People write all kinds of things. I'm marking an advanced undergraduate exam and seeing $3-2=-1$.2012-11-23
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    @GerryMyerson It's strange because I've seen this on books written by professional mathematicians.2012-11-24
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    Cite?${}{}{}{}$2012-11-24
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    @GerryMyerson Look in the link I provided in the end of the question, there's an example of letter switching for no known reason.2012-11-24
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    I don't see any similarity between C & R using $n$ in one formula and later using $r$ instead, and someone writing "For some $x$ ($y\in A$)."2012-11-24
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    @GerryMyerson Why not?2012-11-24
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    Why not? Either because I'm too stupid to see the similarity, or because there isn't any. What similarity do you see between a formula that has no existential quantifier and no set inclusion statement, and a formula that has both?2012-11-25
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    @GerryMyerson I'm not seeing similarity between the formulas, I'm seeing similarity only in the change of letters.2012-11-26
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    Sorry, Gustavo, you've lost me. You ask why people write "For some $x$ ($y\in A$)", you claim to have seen this written by mathematicians, but the only example you produce is an example of something else entirely. I don't know what you want, but I'm beginning to think it has nothing to do with "For some $x$ ($y\in A$)". Maybe you should think about what you really want, and edit the body of your question to reflect it, or post a new question entirely (but with a link back to this one).2012-11-26
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    @GerryMyerson My entire point is similar to the post I linked, why do people change the letters in these cases.2012-11-27
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    @GerryMyerson: I'm pretty sure Halmos is *not* telling the reader to write “for some $x$, $x\in A$” instead of “for some $y$, $y\in A$,” since those are completely different sentences.2014-03-15
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    @Steve, I didn't write that Halmos was saying that, so why address such a comment to me?2014-03-16
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    @GerryMyerson: Sorry: I mistyped one $x$ as a $y$. You wrote (I quote you) that Halmos suggests writing **"'for some $x(x\in A)$' instead [of] 'for some $y(x\in A)$'"**." Which I'm sure he was not saying, since that rewriting would change the mathematical meaning. (What I accidentally said you said was actually what I think Halmos was saying. What you did say he couldn't have been saying.)2014-03-16

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