There's a category error that involves thinking that because we can't start at one and write down every subsequent natural number, they don't exist. — Banno

There's an ontology which presumes that numbers exist, it's called Platonism. It's been demonstrated to be a very problematic ontology, and many philosophers claim that it was successfully refuted by Aristotle, as inconsistent with reality.

It is also well-known that those issues do not arise in the same way at the macro scale. — Srap Tasmaner

That's the problem with this type of issue. The supposed universal principles work extremely well in the midrange of the physical domain. Since the midrange is our worldly presence, and that is the vast majority of applications, we tend to get the impression that the principles are infallible, and "true". However, application at the extremes evidently produces problems. Therefore we must take the skeptic's eye to address the real possibility of faults within the supposed ideals.

Logic and mathematics are mental tools or technologies, habits of mind, that we have developed for dealing with things at the macro scale. — Srap Tasmaner

What you call "the macro scale" is really the midrange, the realm of human dealings. Other than the micro scale and the macro scale, we need a third category which might be called the cosmological scale.

This is unsurprising since our mental lives consist, to a quite considerable degree, of making predictions. Logic and mathematics enable us to figure out ahead of time whether the bridge we're building can support six trucks at once or only four. — Srap Tasmaner

It is true, that this midrange scale, what you call the macro scale envelopes pretty much the entirety of our day to day lives. However, as philosophers with the desire to know, we want to extend our principles far beyond the extent of the macro scale. And this is where the issue of incorrectly representing infinity may become a problem.

For example, let's say that the macro scale is in the range of 45-55 in a scale of 0-100. So we might hypothesize and speculate about that part of reality beyond our mundane 45-55 range. If the application of mathematics, to the physical hypotheses leads to infinity in both directions at what is really only 35 and 65, then we have a problem because we place the majority of reality beyond infinity. And, if we close infinity by making it countable, then there is no way for us to know that there is even anything beyond 35 and 65. It appears from our physical hypotheses that we have reached infinity, therefore the extreme boundaries. And, if the mathematics has closed infinity, in the way that it does, then by that principle we actually have reached infinity. Therefore, by that faulty closure of infinity, 35 and 65 are conclude as the true ends of the universe, the true limits to reality, when reality actually extends much further on each side.

Which leads, at last, to my point, such as it is: there is something perverse, right out of the gate, about the insistence on "actually carrying it out". It misses an important point about the value of logic and mathematics, that we can check first, using our minds, before committing to an action, and we can calculate instead of risking a perhaps quite expensive or dangerous "experiment". ("If there is no handrail, people are more likely to fall and be injured or killed" -- and therefore handrail, without waiting for someone to fall.) — Srap Tasmaner

I don't see how this is relevant. The issue is not properly with "actually carrying it out", the problem is with the assumption that it is possible to carry it out. The defining feature of "infinite" renders it impossible to carry it out. So when we say that it is possible to carry out something which is defined as impossible to carry out, this is a problem regardless of "actually carrying it out".

This denigrates the status of "impossible". Now, "impossible" is a very important concept because it is the most reliable source of "necessity". When something is determined to be impossible, this produces a necessity which is much stronger and more reliable than the necessity of inducive reason. So the necessity of what is impossible forms the foundation for the most rigorous logic. For example, the law of noncontradiction, it is impossible for the same thing, at the same time, to both have and have not, a specified property. this impossibility is a very strong necessity. In mathematics, the impossible, and therefore the guiding necessity, is that we could have a count which could include all the natural numbers. if we stipulate that this is actually possible, then we lose that foundational necessity.

The natural numbers turn out to go on forever, and we can prove this without somehow conclusively failing to write them all down. — Srap Tasmaner

So this exposes the problem. We know that the natural numbers go on for ever. Therefore it is impossible to count them, or that there is a bijection of them. They could not have all come into existence therefore it is impossible that there is a bijection of them. This impossibility is a very useful necessity in mathematics. So if we stipulate axiomatically, that it is possible to count them, or have a bijection, then we compromise that very useful necessity, by rendering the impossible as possible.

To see the demonstration that the rational numbers are equinumerous with the natural numbers and complain that it is not conclusive because no one can "actually do them all" is worse than obtuse, it is an affront to human thought. — Srap Tasmaner

This is a misrepresentation of what I am arguing. My claim is that it is definitively impossible to count the numbers. Therefore to represent this as possible is a contradiction. This has nothing to do with whether a human being, computer, or even some sort of god, could "actually do them all". The system is designed so that they cannot be counted. Nothing can do them all, and this is definitional as a fundamental axiom. So, whether or not anything can actually do them all is irrelevant because we are talking about a definition. Therefore, to introduce another axiom which states that it is possible to do them all, is contradictory.

My claim is that it is definitively impossible to count the numbers. Therefore to represent this as possible is a contradiction. — Metaphysician Undercover

This is to spectacularly miss the point.

Because we can prove what the result would be, we do not have to actually carry out the pairing of every rational number with a natural number. Proof is a further refinement of prediction, beyond even calculation. Of course it's impossible to count the elements of an infinite set as you would the elements of a finite set. But for the results we're interested in here, you do not need to. That is the point. We already know what the result would be if it were in fact possible.

(In my old computability textbook, this was described by having Zeus count all the natural numbers: he could finish, by using half as much time to count each successor. But even Zeus could not count the real numbers, no matter how fast he went.)

I can put it another way: what you cannot calculate, you must deduce.

Infinite sets obviously present a barrier to calculation. So we deduce. Having deduced, we label our results, and then calculation becomes available again. We continually cycle between logic and mathematics, not just here but everywhere.

There's an ontology which presumes that numbers exist — Metaphysician Undercover

We don't need much ontology. Quantification will suffice.