Words used in quantifier proofs

Question

I'm creating a list of "gotcha words" that are often used in writing proofs (particularly quantifier proofs), but frequently in more than one possible way, and that beginners frequently misuse or misunderstand. But it seems likely that there are some I will forget to include, so I'm wondering whether anyone else has already created such a list and perhaps spent more time ensuring its completeness?

So far, here are the words I've got:

• "suppose" is used when proving a $\forall$ statement (suppose $x$ is an arbitrary real number) or using a known $\exists$ statement (we may suppose $x$ is a real number such that $x^2=2$).
• "assume" is used like "suppose", but more commonly restricted to statements rather than variables.
• "choose" is used when proving an $\exists$ statement (to show that there exists a $y$ with $xy=y+1$, choose $y = 1/(x-1)$) and using a known $\exists$ statement (we may choose an algebraic closure of the field $K$)
• "define" is used when proving an $\exists$ (define $y=1/(x-1)$)
• "arbitrary" is used when proving a $\forall$ (let $x$ be an arbitrary real number)
• "let" is used when proving a $\forall$ (let $x$ be an arbitrary real number), proving an $\exists$ (let $y=1/(x-1)$), or using an $\exists$ (let $L$ be an algebraic closure of $K$).

Edit: The current draft of my list is here.

Answer 1

(This is not exactly an answer to your question, but I hope it is useful.)

Fix, Set

(These are how I understand these terms. I'm not sure if my usage is popularly used.)

Fix means "choose an arbitrary value for this constant."

Fix an $m\in\mathbb{N}$. Note that $\frac{\mathsf{d}}{\mathsf{d}x}m=\frac{\mathsf{d}}{\mathsf{d}x}(\underbrace{1+1+\ldots+1}_{m\text{ times}})=0$

Set means "choose this specific value for this variable."

Set $r=1$ for the case of a unit circle.

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Because (since), Therefore (thus)

Because ($\because$) or since means "given the premise."

Therefore ($\therefore$) or thus means "the logical consequence is."

For example,

$\because$ All gods are immortals. $\because$ Zeus is a god. $\therefore$ Zeus is an immortal.

Also,

To denote logical implication or entailment, various signs are used in mathematical logic: $\rightarrow$, $\implies$, $\supset$, $\vdash$, $\models$. These symbols are then part of a mathematical formula, and are not considered to be punctuation. In contrast, the therefore sign is traditionally used as a punctuation mark, and does not form part of a formula.

More information can be found here.

Answer 2

Have you looked at Charles Wells's "Handbook of Mathematical Discourse"? It has quite a bit on this (and more generally on the use of language in mathematics teaching.)