I am a little confused over the differences between phrases in proof writing. They are: 'Fix an abitrary $x \in S$', 'Fix $x \in S$', 'Let $x \in S$'. I would like to know what each of them means. In addition to this, I'd like further clarification on the two different interpretations of let $x \in S$ (in regards to particular and non-specific $x$'s, whether $x$ is an arbitrary or not constant). If I wanted $x$ to be a specific constant would I still write Let $x \in S$, or is that only for an arbitrary element $x$? When we write $x \in S$, what extra information do we have to give for $x$ to be non-arbitrary and specific?
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1The three phrases $$\text{Fix an abitrary $x\in S$.}$$ $$\text{Fix $x\in S$.}$$ $$\text{Let $x\in S$.}$$ mean the same. If authors don't not want $x$ to be arbitrary they better say so. – Kurt G. Aug 12 '23 at 06:55
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Alright thanks Kurt. Is there ever a case where $x \in S$ means $x$ is not arbitrary? Or is it always an arbitrary $x$. I find it hard to find a case where $x$ is a truly fixed value (not arbitrary). – Nav Bhatthal Aug 12 '23 at 06:58
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That's exactly what I am criticising about your question. Human language has room for interpretation. We are not robots that compile a program code. Again: if there is ever a case where $x$ means not arbitrary your authors better say so. If they don't tradition tells us $x$ is arbitrary. – Kurt G. Aug 12 '23 at 07:01
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'If they don't tradition tells' that is what I was looking for. Thanks. The 'default setting' is arbitrary and that's why most proof writing doesn't use the word, its so common usage it is almost assumed. – Nav Bhatthal Aug 12 '23 at 07:02
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It’s more like: if no further information is given, how could it not be arbitrary. – Malady Aug 12 '23 at 07:05
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1Q. "When we write $x\in S$ , what extra information do we have to give for $x$ to be non-arbitrary and specific?" A. Example: for a subset of $S$ you could give the extra information that $x$ is in that subset. Obviously that subset could be a singleton. In that case $x$ is totally not arbitrary. – Kurt G. Aug 12 '23 at 10:11
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Q: 'If I write $x \in \mathbb{R}$ and $y = 2x$, we would then start to call $x$ a variable. However earlier on we called $x$ an arbitrary number? Can we call a free variable like $x$ an arbitrary number? – Nav Bhatthal Aug 12 '23 at 11:07
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2Why not? $\phantom{.}$ Honestly: this discussion is happening in a vacuum. I would prefer to discuss concrete examples from the literature where you see mathematical (not linguistic!) ambiguities. – Kurt G. Aug 12 '23 at 11:20
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@Malady "if no further information is given, how could it not be arbitrary" Debatably, by using the word "choose" rather than "fix" or "let". As in, something like "By the fundamental theorem of algebra, choose $x\in\mathbb C$ satisfying $x^2-2x+1=0$. Then..." – Mark S. Aug 12 '23 at 11:41
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1All your logic-related questions are completely answered once you learn a proper deductive system for FOL, such as the one at this thread: Predicate logic: How do you self-check the logical structure of your own arguments?. – user21820 Aug 12 '23 at 13:22
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“Fixing” a value just means to not treat it as changing. An “arbitrary fixed value” may initially seem counterintuitive, but it just means that the value is one specific yet arbitrary value. Essentially, the value has no properties other than its membership in the set, and we are looking at only it. And “let $a$ equal…” simply means to treat the symbol $a$ as a stand-in for what proceeds. Does that clear it up?
Malady
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Under the context of induction proofs we may write 'Fix $k \in \mathbb{N}$' is that the same as 'Let $k \in \mathbb{N}$'? How do we know Fix $k \in \mathbb{N}$ doesn't refer to a specific $k$ but rather an arbitrary one? – Nav Bhatthal Aug 12 '23 at 06:56
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Usually, for the proof it doesn’t truly matter. I would interpret “fix $k$” to mean that we are now going to look at one specific value of $k$, but that value is still arbitrary. Imagine we have a matrix whose columns are indexed by $i$. If we fix $i$, all of a sudden we just have one column, but it is an arbitrary column. Nothing truly changed, just how we were thinking about it. – Malady Aug 12 '23 at 07:03
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That's the thing, if we use a letter like $k$, to me, it means $k$ is always arbitrary. If we wanted to write 'Fix $k=2$' then that is obviously TRULY fixed and not arbitrary. Is it not pretty ambiguous to use a letter such as $x$ or $k$ for a true fixed value, i.e the letter is always a stand in for an arbitrary value? – Nav Bhatthal Aug 12 '23 at 07:05
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Fixed and arbitrary are not exclusive. For my matrix example, fixing $i$ did not require us to give a value to $i$. All it did was shift our focus to the columns of the matrix. – Malady Aug 12 '23 at 07:07
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Can you give an example of using a letter like $x$ or $i$, fixing the value but it not being arbitrary? – Nav Bhatthal Aug 12 '23 at 07:13
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1“Let $i$ and $-i$ be the roots of the polynomial $x^2+1$.” i is fixed, but certainly not arbitrary. – Malady Aug 12 '23 at 07:16
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I was referring to using $i$ as a variable, not the imaginary unit. – Nav Bhatthal Aug 12 '23 at 07:21
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You asked me for an example of fixing a symbol to a non-arbitrary value. I gave you such an example. If $i$ is taken to be a variable then it cannot be fixed. Those are simply opposites. – Malady Aug 12 '23 at 07:24
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I'll mark your answer as accepted but do you have any more example? – Nav Bhatthal Aug 12 '23 at 07:27
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$e$ and $\pi$ and all sorts of other things. We use symbols to mean things, sometimes we use them to mean non-arbitrary things. – Malady Aug 12 '23 at 07:30
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Finally, I'm struggling to find an example where we might use notation like $x \in S$ for specific $x$. If that's all that's given, isn't it assumed $x$ is arbitrary? – Nav Bhatthal Aug 12 '23 at 07:42