You don't seem to quite grasp why I reject "closer". — Metaphysician Undercover

Perhaps I don't grasp it, or perhaps I just don't agree with it. Let's assume it's the former. Please tell me whether the following points aligns with your view:

1) One can travel along y=1/x in the positive-x direction, without bound.
2) The limit of the journey corresponds to the final destination, which if anything would be (∞,0).
3) The point (∞,0) does not exist (since ∞ is not a number) therefore there is no limit.

2) The limit of the journey corresponds to the final destination, which if anything would be (∞,0). — Ryan O'Connor

I surely disagree. There is no "final destination." That's @MU's error, why are you amplifying it?

I surely disagree. There is no "final destination." That's MU's error, why are you amplifying it? — fishfry

I'm trying to condense his argument into a few points in hopes that it brings to focus where the misunderstanding lies.

Perhaps I don't grasp it, or perhaps I just don't agree with it. Let's assume it's the former. Please tell me whether the following points aligns with your view:

1) One can travel along y=1/x in the positive-x direction, without bound.
2) The limit of the journey corresponds to the final destination, which if anything would be (∞,0).
3) The point (∞,0) does not exist (since ∞ is not a number) therefore there is no limit. — Ryan O'Connor

I surely disagree. There is no "final destination." That's MU's error, why are you amplifying it? — fishfry

Yes that's what I'm saying, there is no final destination, so to even produce any representation (such as ∞,0), as if it is a final destination, is a misrepresentation amounting to contradiction.

There is a particular line we are talking about, and #1 ought to state that this line is extended "without bound", which means "there is no limit". So #3 ought to read "the point (∞,0) does not exist because there is no limit". Now we could add #4: "∞" is a description of the entirety of the line (not a point on the line), and 0 is completely unrelated to the line, therefore irrelevant.

Here's another way to look at the position of zero. It is an ideal, like infinite is an ideal. The two are opposing ideals, like hot and cold are opposing ideals. The line takes the characteristic of the one ideal, the infinite, therefore the opposing ideal, zero, is excluded. It's just like if we were talking about the absolute, ideal hot, cold would be completely excluded.

Yes that's what I'm saying, there is no final destination — Metaphysician Undercover

This is where the misunderstanding is. Nobody is saying that there is a final destination or that (∞,0) exists. The standard definition of a limit does not require a final destination, it only requires one to approach a number as they advance along the journey. A limit is the process of approaching not the act of arriving. And you must admit that any workable definition of 'approach' will have one approach y=0 as they travel along y=1/x. [As mentioned before, your definition of approach involving looking at the number of intermediate numbers is not workable for number systems which are dense in the reals, e.g. the rational numbers].

In short, I believe our disagreement is simply the result of us having a different definition of limit.

Did you read the rest of my post? What I'm saying is that 0 is not even relevant. That's what I've been trying to explain, that to describe the value of y as approaching 0 is a false representation. Y is always infinitely far away from zero, because zero is impossible on that line. The value for y never "approaches 0". It is correct to say that the value gets lower and lower, but it is incorrect to say that it approaches 0, because no matter how low it gets it never approaches 0. 0 is not at all relevant to this line.

To say that y approaches zero is an inaccurate simplification, nothing but a rounding off in your description. The real description is that the value of y gets lower and lower without ever approaching zero. Of course the true description is "not workable", that's the nature of any infinity. The appearance of infinity is the result of something being not workable. To make an infinity into something workable is to provide a false representation.

In short, I believe our disagreement is simply the result of us having a different definition of limit. — Ryan O'Connor

I think where we disagree is in the role that zero can play in this measurement. You think 0 can play a role, as the value which y approaches. I think that since the line has no start nor end, 0 is not an applicable number.

It is correct to say that the value gets lower and lower, but it is incorrect to say that it approaches 0, because no matter how low it gets it never approaches 0. — Metaphysician Undercover

The real description is that the value of y gets lower and lower without ever approaching zero. — Metaphysician Undercover

Our disagreement is also due to us having different definitions of approach. We both agree that y gets lower and lower (and perhaps you would even agree that the greatest value which y never reaches is 0) but I call that approach and you call that not approaching. Let us agree to disagree on definitions!

(and perhaps you would even agree that the greatest value which y never reaches is 0) — Ryan O'Connor

There is no such value. Might it be 100? Or 1000?. y=1/x, x>1 has a greatest lower bound, 0, which it never reaches. Your attempts at the philosophy of mathematics may never bear fruit if you consider this a cogent statement.

There is no such value. Might it be 100? Or 1000?. y=1/x, x>1 has a greatest lower bound, 0, which it never reaches — jgill

You're right. You said what I was intending to say. Thanks for the correction!

We both agree that y gets lower and lower (and perhaps you would even agree that the greatest value which y never reaches is 0) but I call that approach and you call that not approaching. Let us agree to disagree on definitions! — Ryan O'Connor

As I explained, by way of example, to assume such a "greatest value", or "lowest value" is contradiction. When we say that the natural numbers are infinite, and therefore have no highest value, it's contradiction to say that 20 is closer to the highest value than 10. Likewise, when there is no lowest value, it's contradiction to say that .01 is closer to the lowest value than .02.

What is misleading in the example of Gabriel's horn, is that the x and y axes are set to converge at 0, at a right angle in relation to each other. This proposed point of convergence creates the illusion that 0 is a valid value, where x and y are 'the same". However, as I described earlier, the spatial representation of two dimensions at right angles to each other is actually a false representation, making the two dimensions incommensurable, as demonstrated by the irrationality of the square root of two. This incommensurability indicates that the two proposed lines, x and y, cannot actually be modeled as intersecting, and sharing a common point at 0.

So this false idea that x and y actually meet each other at that point, 0, is what misleads you into thinking that 0 is a valid measurement.

To put it simply, non-dimensional existence, which is represented by the point, 0, is incompatible with our representations of dimensional existence. So 0 cannot enter into our scales for measuring dimensional existence, as a valid measurement point until we establish commensurability between non-dimensional and dimensional existence. This problem with zero becomes very relevant when we start to consider motions, and acceleration from rest. An infinite acceleration is required to go from rest to moving. The problem is somewhat avoided with relativity theory which denies the reality of rest, making acceleration simply a change in direction. But what that does is make the mathematics extremely complex, still working with points and vectors, rather than resolving the problem of how the non-dimensional truly relates to the dimensional.