Difference between “for some $k$” and “for some arbitrary $k$”

Question

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.

Answer 1

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 $P(k+1)$.”

Answer 2

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."

Answer 3

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 “$\boldsymbol k$ is some arbitrary integer” here is wrong?

The full statement, written without ambiguity, is:

• Given an arbitrary odd integer $\boldsymbol n,$ there is some integer $\boldsymbol k$ such that $n = 2k+1.$

  (That is, $\;\forall n{\in}\{\text{odd integers}\}\:\:\exists k{\in}\mathbb Z\:\:n = 2k+1.)$

Once we have arbitrarily taken an odd integer $n$ (in other words, once we have taken any odd integer $n$), only a particular integer $k$ satisfies the given equation. Since $k$ depends on the choice of $n,\;k$ is certainly not arbitrary.

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

The former means existential quantification $(∃),$ whereas the latter means universal quantification $(∀).$ To avert any potential for confusion, it is better to rewrite the latter as “for an arbitrary” or “given an arbitrary” or “for each”.

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.

When indicating universal quantification, the word “arbitrary” is actually frequently tacit; “take an integer” is typically understood to mean “take an arbitrary integer” (“for each integer”), which of course does not mean “for some integer”.