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A map $s : \mathbb{N} \to X$ is a computable sequence in $(X,\nu_X)$ when there exists a computable map $f : \mathbb{N} \to \mathbb{N}$ such that $s(n) = \nu_X(f(n))$ for all $n \in \mathrm{dom}(\nu_X)$.

My best guess would be, "A map s taking N onto X is a computable sequence in the ??? (X, nu??) when there exists a computable map f taking N onto N such that s at n equals ??? of f ??? for all n elements of the domain of nu ???."

I am searching for a way to read it aloud that encodes all the elements of the sentence into speech.

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What book is this from? Otherwise, I verbalize $\nu_X$ as "nu sub x" –  J. M. Oct 9 '10 at 13:49
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Why the downvote? This is a reasonable question methinks. –  J. M. Oct 9 '10 at 14:11
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As an addition to the answers below, here is a remark that may be useful: when describing maps, such as $f:\mathbb N \to X$, one should say "into" rather than "onto" unless you specifically mean for the map $f$ to be surjective. (Probably many audience members won't pay attention to this distinction, but some will, and they will be confused if you say "onto" when you don't really mean it.) –  Matt E Oct 10 '10 at 0:07
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4 Answers 4

up vote 14 down vote accepted

"A map ess from en to ex is a computable sequence in ex nu-ex when there exists a computable map eff from en to en such that ess of en equals nu-ex of eff of en for all en in the domain of nu-ex."

The hyphens are meant to indicate that the pause between these syllables is shorter than between two separated words.

Yes, I am not verbalizing that the X is subscripted, the ordered pair, or the difference between $n$ and $\mathbb{N}$. I assume you are talking about a situation where you are writing this statement on the board as you speak. Communicating any significant amount of math in a purely oral manner is incredibly hard because mathematical notation is much denser than ordinary spoken language; I find that I have to leave out some detail in order for the sentence to fit in my listener's head.

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"Communicating any significant amount of math in a purely oral manner is incredibly hard because mathematical notation is much denser than ordinary spoken language" - indeed. +1 for this. –  J. M. Oct 9 '10 at 15:09
    
I would use 'the naturals' or 'the natural numbers' where needed, and maybe omit the subscript on 'nu' since that is more or less implicit. That last bit will stop helping when you start considering maps between (X,nu_X) and (Y,nu_Y) but that's why the Greek alphabet has more letters :-) –  yatima2975 Oct 9 '10 at 15:59
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@yatima: nope, the Greek alphabet has only twenty-four; the Latin alphabet has twenty-six. –  J. M. Oct 9 '10 at 16:25
    
I didn't say the Greek alphabet had 26; only that there's more than just 'mu' :-) –  yatima2975 Oct 11 '10 at 11:50
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I would say "A map s is a computable sequence when there exists a computable map f satisfying certain properties" while writing down the property.

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Agreed. Translating every mathematical statement to words is just too cumbersome; it's best to do it this way. –  Noldorin Oct 9 '10 at 19:02
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@Noldorin Personally, I absorb it much better when I can read it out loud to myself. I employ the audio channel heavily when I study things. I'd only use the visual channel exclusively for learning how to draw or paint. –  ixtmixilix Oct 14 '10 at 15:24
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David Speyer wrote how I would say it in practise, in a context where I was writing it on a black/whiteboard. Here's how I would say it in a pub or walking down the street:

"Let's define a 'representation map for X' [or your own preferred jargon] to be just some partial function nu, from the natural numbers to X. Then we can define a computable sequence for that representation map nu to be any function s, from the natural numbers to X, which [is consistent with / agrees with / extends] the composition of nu with a computable function f on the natural numbers."

When using natural language, choose your nouns wisely and characterize them. Do you care about the ordered pair $(X,\nu_X)$, or really just the map $\nu_X$ (for which $X$ is just the background against which the idea is presented)? What is the role of the partial map $\nu_X$ in the idea you are communicating? Do you care about the integers $f(n) \in \mathop{\mathrm{dom}}(\nu_X)$ over which you quantify, or really just the domain of the composite function $\nu_X \circ f$?

Identify the main characters in the synopsis of your play, and their roles: you will have a better chance of transporting the objects and morphisms of your idea faithfully to your interlocutors.

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If you'd have incorporated David Speyer's answer into your own, I'd have chosen yours. I think it's good to have two ways of saying this stuff. Personally, I absorb things much better if I can say them aloud to myself. –  ixtmixilix Oct 14 '10 at 15:22
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@ixmixilix: thanks for the compliment, but I don't think it would have been graceful of me to do so. –  Niel de Beaudrap Oct 14 '10 at 15:26
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I would try to keep it really simple. Rather than verbalizing all the function and set symbols and subscripts, I would just say, "A computable sequence is the result of composing a notation after a computable function." I assume you're talking Computable Analysis theory and by $\nu_X$ you intend a notation $\nu_X:\subseteq \Sigma^* \to X$. Of course, this reading assumes your audience already knows what you mean by a notation and a computable function.

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