An excellent article, Alex.

First off, the natural numbers always contain 0. If you need evidence:

https://www.cantorsparadise.com/discovering-zero-among-the-prime-numbers-65a47cbf79ec

Here, the entire set of the natural numbers is the playing field. In it, a required use of zero is needed to explain one step. Hence, zero is from the same set, like the word ‘nothing’ is indeed part of the English language, and not treated in an exclusive manner.

That means that the other natural numbers may advance in steps, but that their reality is not as simple as you state it is. Said differently, the numbers exist in a larger reality and the overall situation is pretty much like Gödel's Incompleteness Theorems. No matter where we start with anything in the set of the natural numbers, we will come to an overarching conclusion that leads to an incompleteness because this set is ruled by all numbers including zero.

My conclusion is that one cannot distinguish between the real numbers and the natural numbers because the distinction is not real. It is the human brain that can make the distinction.

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The thing about infinity is that is the concept is very easy to understand and that the concept can continue well... ad infinitum. But the real contents petered out long before.

Half an apple and next half the pieces. Continue to half the pieces ad infinitum and we can do this forever, right?

We cannot. After thirty half-cuttings, we have apple sauce, and after one hundred half-cuttings not even the flavor sticks around.

So, what do we have on our hands?

We have a battle of abstractions. Let me immediately quote Spinoza who said 400 years ago already that God is real, as long as we view God as an abstract.

My beloved Gödel gave us the best example already to show what is going on. With matter in the universe we see:

* convergent behavior, found ubiquitously among all matter in the universe (desiring to become single masses where possible).

* at the single overall universal level, however, convergent behavior is missing. It does not exist. It is either divergent (all moving outwardly) or the behavior is one of simply no-connection among all.

Anything larger than a galaxy can be said to not be connected to any other galaxy. Full disconnect at the larger levels.

And there we have it. As soon as we agree that the universe is an abstract word in which we count all that exists but that collectively themselves are not connected, then we have found the end of the abstraction ladder.

At the bottom we can say 1 + 1 = 2, and already see that this is not meaningful because there is no contents to it.

1 apple + 1 apple = 2 apples, that is finally true, and I'd like to add:

1 apple + 1 orange = 2 pieces of fruit, to show how we can already play with abstracts at that simplest of levels and pick and choose however we want to eat our cake.

Move up with the abstract levels, and the top level is not based on itself but only on the lower levels.

Everything + the Solar System is for instance already incorrect because the Solar System is then mentioned twice.

Or, put your name on an envelope, street address, city, nation, Earth, Solar System, Milky Way, Universe, and the postal person can use all the information to direct your letter, except for the word Universe. It does not tell any story. It gives out no information of itself that directs anyone.

And that is how it must be. The very nature of abstracts is that we are pointing at something in a way that is not real, but captures something. At the largest of levels of doing that, we MUST end up with an empty position by itself and this then only pointing at that what exists for real.

Good article, Alex. I hope we can discuss further.