The role of 'arbitrary' in proofs

Generally, when one is going to prove a result regarding a set of elements, they begin their proof with those first few pleasing words: "Suppose...is an arbitrary element in..."

My question is, why does considering an arbitrary element in a proof imply that the proven result applies to every element in the set? Although I think I have an vague about this, I am still interested in seeing what others have to say in relation to this idea. Perhaps, if possible, answer the question in the manner that you would if a student of yours posed this question to you.

Thank you!

Solution 1:

You could instead say "let $x$ be any element of our set" that is the same as saying "let $x$ be an arbitrary element of our set". This just means we are not assuming anything about $x$ other than it is in our set. So, anything we prove about $x$ holds for any element in our set.

Solution 2:

Recall that a statement of the form $\forall x\varphi(x)$ is true in a structure $M$ if and only if for every $m\in M$ it holds that $\varphi(m)$ is true.

In the proof we usually show that there is a schema which can prove for each $m\in M$ that $\varphi$ holds. And so we can show that $\forall x\varphi(x)$ holds in $M$.

Solution 3:

This will be clarified if you study formal logic. For a nice introduction see the entry on Classical Logic in the Stanford Encyclopedia of Philosophy. The article addresses your question explicitly in a comment following the $\,\forall$-introduction rule, $ $ namely

This rule (∀I) corresponds to a common inference in mathematics. Suppose that a mathematician says “let n be a natural number” and goes on to show that n has a certain property P, without assuming anything about n (except that it is a natural number). She then reminds the reader that n is “arbitrary”, and concludes that P holds for all natural numbers. The condition that the variable v not occur in any premise is what guarantees that it is indeed “arbitrary”. It could be any object, and so anything we conclude about it holds for all objects.

See also the following section for the semantics of universal quantificaton, and the final section on Meta Theory for relationships between deductive systems and their models (e.g. soundness).

Solution 4:

Let's say we have an argument. You say "for all x in the set S, A (x) is true". I say "I don't believe it. I'm sure there is an x in the S where A (x) is not true". You say "well, tell me an x and I'll show you A (x) is true". I say "I can't think of any x right now, so just assume any x". You say "Ok then, we will assume that x is any element of the set S, and I'll prove that A (x) is true. And since we don't make any assumption about that x, the same proof will work for any x in the set, so S (x) is true for all x".