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I am told that the “for some” and “for some arbitrary” are different.

For example, when proving the statement “if n is odd, then $n^2$ is odd”, one of the steps includes writing $$\text{$n = 2k+1,\:\:$ where $k$ is some integer}.$$ I am told that writing “$k$ is some arbitrary integer” here is wrong?

On the other hand, in mathematical induction, when performing the inductive step, before writing the inductive hypothesis for $k,$ we write “for some arbitrary $k\text”$. I am then told that the “arbitrary” here is compulsory to write.

To me, they sound the same. Am I wrong? Are they different in terms of notation?

PS I’m relatively new to this stuff, maybe explain less using notation and appeal more to logical understanding.

ryang
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bluesky
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    Essentially, "$P$ is true for arbitrary $k$" means that it's true for all $k$. "$P$ is true for some $k$" means that there exists a $k$ for which it's true. For example, for a given $n$, it's certainly not true that $n = 2k+1$ for all $k$; there simply exists a $k$ for which it is true. However, if we write "$k^2$ for arbitrary $k$ can be written as the sum of the first $k$ odd numbers," that's true for all (positive) $k$. – Brian Tung Jul 18 '23 at 03:25
  • Writing "for some arbitrary k" is fine both cases you described. – RyRy the Fly Guy Jul 18 '23 at 04:09

3 Answers3

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In your first example, I would just write “where $k$ is an integer”. But “some” is okay. The point is, since $n$ is already known, $k$ is completely determined. Writing “where $k$ is some arbitrary integer” implies, or at least suggests, that $k$ can be any integer.

In an inductive proof on the other hand, we do mean that $k$ can be any positive integer. Writing “for some arbitrary positive integer $k$, $P(k)$ implies $P(k+1)$” means the same as “for all positive integers $k$, $P(k)$ implies $P(k+1)$.”

Michael Weiss
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    This comment isn't a full answer, especially since OP prefers a notation-light response: but it might be worth noting that this is essentially the difference between the formal notations ∃ (the first case: this is true for “some” $k$) and ∀ (the second case: this is true for “all”, or for “arbitrary” $k$). – Aant Jul 18 '23 at 17:16
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    Agree. In the first example, I would say $n$ is an arbitrary odd integer. But $k$ depends on $n$ so it's not arbitrary. – Matthew Leingang Jul 18 '23 at 19:53
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In proofs like those you're describing, the purpose of singling out "some $k$" having "such and such property" is to prove something about all other numbers having the same property of $k$. For this reason, you can describe $k$ as arbitrary, although it is not strictly necessary. I have seen both covnentions used. What people typically mean by "for some $k$" or "for some arbitrary $k$" is "here is a particular number $k$ that is not uniquely different from any other number having the same properties."

RyRy the Fly Guy
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when proving the statement “if n is odd, then $n^2$ is odd”, I am told that writing $$n = 2k+1,\:\: \text{where }\boldsymbol k \textbf{ is some arbitrary integer}$$ is wrong?

The goal is to prove $\;\forall n{\in}\{\text{odd integers}\}\:\exists k{\in}\mathbb Z\:\:n = 2k+1.\;$ Since the value of $k$ that satisfies the given equation depends on the value of the arbitrary odd integer $n,$ the integer $k$ is certainly not arbitrary. Best to just write $$n = 2k+1\:\: \textbf{for some integer }\boldsymbol k.$$

I am told that the “for some” and “for some arbitrary” are different.

The former typically corresponds to existential quantification $(∃),$ whereas the latter is synonymous with “for an arbitrary”, which is related to universal quantification $(∀).$ To avert any potential for confusion, rewrite “for some arbitrary” as “for an arbitrary”.

in mathematical induction, when performing the inductive step, before writing the inductive hypothesis for $k,$ we write “for some arbitrary $k\text”$. I am then told that the “arbitrary” here is compulsory to write.

It's not compulsory (you could also write “take any natural number $k;$ suppose that $P(k)$ is true”), but certainly good practice.

ryang
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