Cfrgtkky

I have this definition: "Let $Rinmathbb{N}$ be called arbitrarily large if and only if

$$
forallquad xinmathbb{N}quadexistsquad Rinmathbb{N}quad:quad x<R~~text{''}
$$

When I wrote it, I was thinking $forall$ meant "for any," but now I see that it means "for all." (Probably why LaTeX symbol is called forall.) I think this definition makes sense and is well posed because $mathbb{N}$ is an open set whose supremum is not in $mathbb{N}$.

Is this definition ok or is it problematic? When I use, "for all" in my head I say, "Choose $x=R$, then is $R$ greater to itself?" The answer must be no, but my intention in constructing the definition was that $R$ cannot be chosen and instead only exists as a concept.

What I mean by that is if you take $x=R$ then I will say, "Which natural number is that?" The counterpart will say its ten million and five," and then I would say, "That number does not have the requisite property to be called $R$ because it is less that ten million and six."

Is this definition ok or inherently stupid?

It's not really meaningful to speak about whether a single number is "arbitrarily large" or not.

What one does often speak of is things like "$P(x)$ holds for arbitrarily large $x$". That can be formalized as

"$P(x)$ holds for arbitarily large $x$" means
$$ forall ninmathbb N : exists x>n : P(x) $$

-- in other words, no matter how large an $n$ you're thinking of, you can find an even larger $x$ with the property $P$.

A related notion is

"$P(x)$ holds for all sufficiently large $x$" means
$$ exists ninmathbb N : forall x>n : P(x) $$

Note that both of these are claims about the property $P$. They don't say something about any particular number $n$ or $x$.

An issue nobody pointed out is that you define a property about a number $R$, but in your definition you quantify over $R$. This doesn't make sense.

Such $R$ doesn't exists, we can refer to a sequence $a_n$ (that is a function $mathbb{N}to mathbb{R}$) arbitrarly large for example when we deal with limits stating that

$$forall Rinmathbb{N} quad exists xinmathbb{N}quad forall nge xquad a_nge R$$

and in that case we say that $a_n to infty$.

According to your definition no natural number is arbitrary large so why are you trying to define something which you know it does not exist?

You may want to define the concept of a sequence diverging to infinity in which case you have a sequence $R_n$ which gets larger than any arbitrary number.