Something about Apologetics.

[Abracadabra]

Zeno's claim is that you can't perform an infinite number of finite steps. And the calculus limit doesn't challenge that, nor does it show that Zeno was wrong.

How does a methodology for describing infinite sequences/sums/etc. not challenge the claim that you can't perform an infinite number of finite steps?

It's right there in the methodology itself.

What do you need to show in order to prove that a calculus "limit" exists?

Do you need to show that it's possible to complete an infinite number of steps?

No. Not at all. In fact, if that's what the calculus limit was saying then all the calculus limit would be proclaiming is that infinity is finite.

All you need to do to satisfy the formal calculus definition for a limit is to show the existence of trends and/or boundaries. And whether or not those trends and/or boundaries exist or not.

If you can show that certain trends exist, and they are unbounded, then you have satisfied the definition of the limit and therefore you can say that the limit "exists".

Moreover, when you have satisfied this definition and have proven the existence of the "limit" does that prove that this actual point must exist?

No it doesn't even prove that. You can easily have a function that is not include a given point. Yet, you can still prove that this undefined (and therefore nonexistent point) can still satisfy the calculus limit, and be said to 'exist' as a 'limit'. But that in no way asserts or proves that the actual point must exist. That's simply not what the calculus concept of limit is saying.

Given an infinite sum or other process, the calculus limit simply states that if you can show specific properties of trends and boundary conditions, then you are premitted to say that "If this process could be completed it would arrive at this particular target called the Limit".

But the neither the methodology used to arrive at that conclusion, nor the formal definition of the calculus limit itself make any statements requiring that this process can actually be completed, or that it should be able to be completed.

In short, the calculus limit does not assert that any infinite process is completable or even should be completable.

In fact, if it were actually saying that (which it clearly isn't) then it would be proclaiming that a finite process can indeed be completed and therefore it would have proved that infinite is finite.

The calculus limit does not do that, nor does it even imply that such a notion should be possible.

Just because it allows you to say what the sum of an infinite addition would be IF you could supposedly complete the sum, doesn't mean that it's saying that such a summation process could actually be completed.

This is a gross misunderstanding of the calculus limit.

The formal epsilon-delta definition of the calculus limit cannot be used to support any such conclusions.

That's not what the calculus limit is saying.

So the calculus limit does not having anything at all to do with Zeno.

Zeno holds that you could never complete an infinite number of tasks.

The calculus limit does not challenge this.

The calculus limit has nothing to do with Zeno's objections concerning the idea of being able to actually complete an infinite number of tasks. They simply have nothing to do with each other at all.

Yet, unfortunately many mathematicians have been erroneously taught that the calculus limit solves Zeno's concerns. I'm certain that it doesn't and if Zeno were alive today I'm sure he would object as well.

The calculus limit is addressing a totally different concept.