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Working on the book: Daniel J. Velleman. "HOW TO PROVE IT: A Structured Approach, Second Edition" (p. 127)

Theorem. For every real number x, if $x > 0$ then there is a real number $y$ such that $y(y + 1) = x$.

I will use the proof of this theorem to give context to my question. The skeleton of the proof is:

Let $x$ be an arbitrary real number, and suppose $x > 0$. Let $$ y = \frac{-1+\sqrt{1+4x}}{2} $$ which is defined since $x > 0$.

I often see that, in the middle of an Existential statement proof, the word "let" appears. I understand the justification for the statement "Let $x$ be an arbitrary real number, and suppose x > 0...", since the statement he is trying to prove has a Universal Quantifier. Variable $x$ needs to be arbitrary; otherwise, it wouldn't be possible to use Universal Introduction inference rule.

However, I have two questions:

  • Is there some justification from a logic standpoint for the phrase "Let $ y = \frac{-1+\sqrt{1+4x}}{2} $...

  • If using a Natural Deduction system with identity, it is possible to use Identity Introduction at any point of a proof to introduce, for example, "3=3". But, in the former case, does it count as an abbreviation ? Is there some inference rule that allows the introduction of some statement like "Let $y = ...$" in the middle of a proof ?

I appreciate any insight into this matter.

F. Zer
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    You just want to prove that a $y$ exists. For each given $x$, there is no problem in putting your hand on that particular $y$ which is given by the formula. – Jeppe Stig Nielsen Sep 11 '20 at 20:50
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    Similarly as the equal signs has several meanings (definition, equality between two things, in-between computations...), the "let..." has two meanings here. The first one, "let x such that... is "choose any x such that...". The second one, "let y=..." is "define y to be ...". – Anthony Sep 11 '20 at 20:52
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    The easiest way to show that something exists is by finding it. Note that your thinking process here and the actual proof are different, you should reason as this: If I want to show that $y$ exists, let us find it. Solving the equation gives $y=...$, and we can see that this $y$ works. NOW, for the actual proof, it is irrelevant how you found the $y$, showing that this $y$ works is enough... – N. S. Sep 11 '20 at 21:34
  • I appreciated every comment. Thank you, everyone. @JeppeStigNielsen, could you clarify something? When you say "that particular $y$ which is given by the formula...", are you referring to this formula $(+1)=$ ? Would be possible to choose another $y$ ? It is a subtle details but I am not sure I get it. – F. Zer Sep 12 '20 at 01:15
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    I meant, logically there is nothing wrong with saying, hey, you got your $x$, now try looking at this $y$ given by $y = \frac{-1+\sqrt{1+4x}}{2}$. The formula actually yields a $y$. You do not even have to say where you got the formula from. If that $y$ can be shown to do the job, it does not matter if there are other $y$ as well, because the theorem did not say anything about it. You could prove a slightly stronger theorem saying there exist exactly two different such $y$, one positive and one negative, but that is another problem. – Jeppe Stig Nielsen Sep 12 '20 at 05:56
  • In general, "let" in regular mathematics means "if" in logic. Writing "let $a=1$" is the same as assuming that "$a=1$", or writing "if $a=1\dots$". Some of the time $a$ really does equal $1$ as well, but that is beside the point. – user400188 Sep 12 '20 at 06:03
  • Thank you, @JeppeStigNielsen. – F. Zer Sep 12 '20 at 16:19
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    Thanks for the input, @user400188. Thinking "let" as an "if" is useful and adds a different perspective. – F. Zer Sep 12 '20 at 16:21
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    I just came across this post. @user400188 is correct about the idiosyncratic and non-idiomatic use of the term "let" in mathematics, but it is a very bad idea to perpetuate this nonsense, as you can see from the huge amount of confusion to students. Instead, use "consider any" or "take any" for a ∀subcontext, and reserve "let" for ∃elim in line with its proper English meaning. (It seems that a troll downvoted my linked post recently but you can check with any professional logician if you don't believe it.) – user21820 Mar 10 '23 at 07:08
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    The word 'let' in mathematics basically assigns meaning. In comment #2 (@Anothony's) above, the "choosing any $x$" (i.e., existential quantification) is indicated *not* by the word 'let' per se, but by the word 'arbitrary' which is frequently tacit in the sentence; to wit: "let $x$ be an arbitrary real number blah blah" is commonly just written "let $x$ be a real number blah blah". – ryang Mar 30 '23 at 15:50

1 Answers1

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This kind if reasonning is known as "Analysis-Synthesis". I means that we begin by finding a necessay condition and we check that this condition is sufficient.

So, assuming that $ y $ exists, it will satisfy $$y^2+y-x=0$$

whose roots are $$y_1=\frac{-1+\sqrt{1+4x}}{2}$$ and $$y_2=\frac{-1-\sqrt{1+4x}}{2}$$

Now, you just need to check that $ y_1$ exists and works.

hamam_Abdallah
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