Why is the set $w={0,1,2,\ldots}$ ill-defined?

In the comment section of this question, Thomas Andrews say that the set $w=\{0,1,2,\cdots\}$ is ill-defined. For convenience, I copy parts of the question here:

For a set $A$, we define $A^+:=A\cup\{A\}$. When we define, $$0=\emptyset,\ 1=0^+,\ 2=1^+,\ \cdots$$ set of natural number $w$ is defined as $$w=\{0,1,2,\cdots\}=\{0,0^+,(0^{+})^+,\cdots\}$$ (for clarity $\omega$ is changed to $w$)

I agree that $w$ is ill-defined because the "$\ldots$" does not specify how many steps we will go. If we want $w=\omega_0$ then we have to specify that there can only be finitely many $+$ above $0$. If we use infinite or even uncountable many $+$ then $w\neq \omega_0=\omega$. Is this the true reason why $w$ is ill-defined?

It appears to me that if we limit the number of $+$ to be finite, then $w=\omega_0$. However, I don't know how to say this in a rigorous way.

Questions:

1. Does $w=\text{Ord}$, the proper class of all ordinals if we are allowed to add $+$ on and on without stopping? (It seems we have not even defined the meaning of infinitely many $+$, so what does "infinitely many $+$" mean?)
2. How can I say the phrase "only finitely many $+$" in the language of set theory? I do know what "finite set" mean in set theory($A\cup\{A\}$ does not have the same cardinality as $A$), but $+$ is not quite the same thing.

If "dots" are not really something we can use to define something, then what notation should we use instead?

PS: I know the usual definition of $\omega_0$ as the minimal infinite ordinal.

EDIT At the very beginning, I have pointed out that "$\ldots$" is not something we can use to define, but "$\ldots$" is used so often in Analysis that I feel I can make it a valid definition somehow. Now I realize that "dots" is just a matter of practice, not something formal, at least in this context.

The real reason it is ill-defined is that it is ill-defined ! Or better, if you like, the reason is : it is not well-defined.

Asking why it is ill-defined is akin to asking why the set $\{2, 26, 43, 17, 57380, ...\}$ is ill-defined : who knows what I meant by these $...$ ? (hint : not even I know)

The thing is mathematics is a formal, rigourous thing, and we try to make everything as precise as we can. To do this, we base what we do on axioms : a mathematical argument must use the axioms clearly (with of course the caveat that people with more training are used to various things and so don't need to state the axioms they use, and don't need to go back to very basic levels when they explain their arguments - but that is a question of practice, not principle). In particular, the definitions we make must be "validated" from the axioms (by this I mean : if we define an object and assert its existence/uniqueness - you don't need axioms to say "a set is called a bird if it satisfies such and such things", but doing so will not give you the fact that birds exist, or that there is a unique bird)

Now in ZF ( which is the commonly accepted/used foundation for mathematics - with again, some caveats) there is no axiom that says "if OP is pretty certain of what they mean by $...$, then it's ok to define a set using $...$" - you can understand why.

Mathematicians often do this, however : they define a set with $...$ or a sequence by giving the first few terms and saying that "the pattern is obvious" : again, this is a matter of practice, not principle. In principle, they should give the precise definition, and the reason they don't is simply that they know that they could, if asked to do so, give a precise definition. In your case, when we're very clearly at the beginning of learning formal mathematics, it is not clear that you could give a precise formulation of what's hidden in those "$...$".

And in fact, as it was hinted at in the comments, the precise formulation of these "$...$" lies in the axiom of infinity : it is with this axiom that we can make things like "$0$, then $1$, then $2$, and for all $n$, $n+1$" precise.

The existence of the set $w$ you mention is essentially what is stated by the axiom of infinity : it is a set that contains $0$ and is closed under $(-)^+$.

Once we have this set, and proved its properties, we can allow ourselves to write things such as $\{u_0, u_1,u_2,...\}$, but that's just a matter of convenience, and in principle this should be defined precisely, referring to specific axioms/theorems.

This is the way the set of natural numbers was introduced to me the first time I ever received a course in set theory:

First, we introduce the:

Axiom of Infinity (AI): There exists a set that has the empty set as one of its elements, and it is such that if $x$ is one of its elements, then $x\cup\{x\}$ is also one of its elements.

In formal language, this can be translated as:

$$\exists y(\varnothing\in y\;\wedge\;\forall x(x\in y\rightarrow x\cup\{x\}\in y)),$$

or, without any defined terms:

$$\exists y(\exists z(z\in y\wedge\forall t\neg(t\in z))\;\wedge\;\forall x(x\in y\rightarrow\exists u(u\in y\wedge\forall v(v\in u \leftrightarrow v=x\vee v\in x))).$$

In other words, we will say that a set $A$ is inductive if:

1. $\varnothing\in A$,

2. For each $a\in A,\;a\cup\{a\}$ is also an element of $A$.

Then, AI can be expressed simply by:

$\qquad\qquad\qquad\qquad\qquad\qquad\quad\quad$There exists an inductive set.

If $A$ is an inductive set, then the sets $\varnothing,\;\{\varnothing\},\;\&\;\{\varnothing,\{\varnothing\}\}$ are all elements of $A$. A natural number is a set that is an element of all inductive sets. As we stated before, $\varnothing,\;\{\varnothing\},\;\&\;\{\varnothing,\{\varnothing\}\}$ are natural numbers.

Theorem: There exists a set whose elements are all the natural numbers

Dem Let $A$ be an inductive set, that exists by the axiom of infinity (AI). The axiom of subsets corresponding to the property $P(x)$:

$\qquad\qquad\qquad\qquad\qquad\qquad\quad$''$x$ belongs to every inductive set''

that can be expressed in the formal language of the theory by the formula:

$$\forall y(y\text{ is inductive}\rightarrow x\in y)$$

that is:

$$\forall y(\varnothing\in y\wedge\forall z(z\in y\rightarrow z\cup\{z\}\in y)\rightarrow x\in y)$$

ensures that for the inductive set $A$, there exists a set whose elements are those elements $x$ of $A$ that have the property $P(x)$, or in other words, $\{x\in A|\;P(x)\}$ is a set. But if a set $x$ has the property $P(x)$, then we have that it is an element of every inductive set, and, in particular, is an element of the inductive set $A$, so every natural number belongs to $A$ and:

$$\{x\in A|\; P(x)\}=\{x| x\text{ is an element of every inductive set}\}=\{x| x\text{ is a natural number}\}$$

is a set.

$\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\square$

This set is unique, by the Axiom of Extensionality, and is the set of the natural numbers, which we represent by $\mathbb{N}$.

Now, I will pose the following questions:

Was it necessary at all to use any dots, at any point, in the construction of the natural numbers? Can these dots be implemented in the formal language of the theory of ZF? The answer to both questions is no; the usage of dots is simply for notational purposes; that is, you cannot use dots to define the set of natural numbers, but rather to represent that set after you have proved it exists, and it is clear to the reader what are the elements omitted by the dots.