Why, when we impose restrictions such as the ones of our physical world, new properties arise?

Question

Unfortunately I am not and expert mathematician nor a philosopher so I don't have the right words to phrase the concept, but the following example should be able to make the question clear:

Take the complex numbers. We can't impose an order relationship between them (>, <). But as we impose a new constrain by saying that $\sqrt{-1}$ doesn't exist, we find that we can now tell which number is bigger than another.

How can we rigorously explain what's going on?

EDIT: Second example concerning Achilles and the tortoise removed since multiple users pointed out that it contained numerous errors. Minor corrections.

Answer 1

Turning my comment above into an answer:

As Don Thousand commented above, narrower classes of objects have more guaranteed properties a priori. You're focusing on tameness properties, but it's also worth noting that we can get guaranteed pathologies this way too. For example, here are a couple ways in which adding a natural requirement to a starting class of structures can limit our ability to do algebra:

• When we shift from the class of all fields (which includes both $\mathbb{R}$ and $\mathbb{C}$) to the class of ordered fields (which includes $\mathbb{R}$ but not $\mathbb{C}$), we are guaranteed to lose the ability to solve the equation $x^2+1=0$.

• When we shift from the class of all rings (which includes both $\mathbb{Z}$ and $\mathbb{Q}$) to the class of discrete rings (which includes $\mathbb{Z}$ but not $\mathbb{Q}$), we are guaranteed to lose the ability to solve the equation $2x=1$.

More restrictions make things more decisive, but even very "nice" restrictions do not necessarily make things better.

Answer 2

The "paradox" of Achilles and the Tortoise has nothing to do with uncountable infinities. It can be easily be phrased within the (countable) rationals, since it is about $$\tag1 \sum_{n=1}^\infty\frac1{2^n}=1. $$ And in fact it doesn't require infinity to be phrased, as it could be written as $$\tag2 \forall \varepsilon>0,\ \exists N\in\mathbb N:\ \ \ \ 1-\varepsilon<\sum_{n=1}^N\frac1{2^n}<1 $$ and proven within the field of rational numbers. That said, the only paradox in the "paradox" lies in the mistake of not noticing that in the setup of the paradox time is also subject to $(2)$.

As for your example with the complex numbers, I fail where you want to go, but that kind of thinking can be made in scores of situations:

• In going from $\mathbb R$ to $\mathbb Q$ you "gain" that now every number is a fraction.

• In going from $\mathbb Q$ to $\mathbb Z$ you "gain" that every element has a predecessor and a successor.

• In going from $\mathbb Z$ to $\mathbb N$ we "gain" that every subset has a first element.

It is easy to argue that there lots of "gains" going the other way, too.

Answer 3

First, regarding your first point, I agree with comments and answers pointing out that the Paradox of Achilles and the Tortoise has virtually nothing to do with uncountable infinities.

Regarding your second point, I would like to add something which I think is not stressed enough in other answers: Namely that inclusion of structures $A \subset B$ says virtually nothing about how "complicated" or "interesting" the structures $A$ and $B$ are relative to each other. That is because one could be interested in different structural properties of them.

As a graphical example, the Menger sponge (approximation on the right) is contained in the cube (on the left):

(Image from Wikipedia https://en.wikipedia.org/wiki/Menger_sponge#/media/File:Menger_sponge_(Level_0-3).jpg)

Now which of these objects do you find more interesting, which has "more" or "less" properties? It very much depends on what you want. With the cube, you can compute a volume, study its symmetry group, do many non-trivial things ... But still the sponge on the right seems to have other, and more intricate, properties.

I like to compare that with the inclusion $\mathbb R\supset \mathbb Z$ (the real numbers being a bit like the cube).

• If you have, say, a series $\sum a_n$ with the $a_n \in \mathbb R$, there's quite an intricate theory about whether it absolutely converges or conditionally converges or does not converge at all etc.

• But if that series has terms $a_n$ taken just from $\mathbb Z$, this is absolutely boring: It converges if and anly if all but finitely many $a_n$ are $=0$, and then it'S just a finite sum.

So in that sense (namely, calculus / topology), the structure $\mathbb R$ is far more interesting than $\mathbb Z$.

But:

• For $n \ge 3$, does the equation $x^{n} + y^{n} =z^{n}$ have solutions with all $x,y,z \neq 0$ in $\mathbb R$? Well, trivially yes, infinitely many, you can graph them nicely and compute them to great accuracy with a pocket calculator.

• Does the equation $x^{n} + y^{n} =z^{n}$ have solutions with all $x,y,z \neq 0$ in $\mathbb Z$? No. But it took mathematicians hundreds of years and very intricate theories to figure that out.

So in that sense (namely, solving equations / arithmetic), the structure $\mathbb Z$ is much more interesting than $\mathbb R$.