real numbers, constant symbols in first order logic

Question

I have this exercise in my book on first order logic:

Let's work out a language for elementary trigonometry. To get you started, let us suggest that you start off with lots of constant symbols- one for each real number. It is tempting to use the symbol 7 to stand for the number seven, but this runs into problems. (Do you see why this is illegal? 7, 77, 7/3, ...) Now, what functions would you like to discuss? Think of symbols for them. What are the arities of you function symbols? Do not forget that you need symbols for addition and multiplication! What relation symbols would you like to use?

What symbols should I use for the real numbers? I can't use 7 because 7 is contained in 77 and constant symbols can't be properly contained in other symbols. I also have the problem that irrational number have an infinite non-repeating sequence of decimals, and constant symbols have to be finite? How do I solve this?

Is there a way to choose the constant symbols for the real numbers?

Answer 1

Consider a circular clock face with one hand. Let $[0,2\pi)$ be the positions of the hand where $0$ is the point where the hand points to $12:00$ and the hand moves clockwise with increasing angle.

We know that $[0,2\pi)$ is uncountable so let $\phi:\mathbb R\to [0,2\pi)$ be a bijection. For $r\in\mathbb R$, let the symbol be the clock with its hand pointed at $\phi(r)$.

Answer 2

You can choose an arbitrary set as symbols (the simplest way will be to take $\mathbb R$ itself), and you need a notation to write these symbols on paper - you can just say that symbol for number $123$ is written as $\overline{123}$ (and it is separate symbol, not a string $\bar{1} \bar 2 \bar 3$).

At this point word "symbol" has two meaning: the actual thing you write on paper / show on screen, and part of signature of theory, where it is element from arbitrary set, and we only need to distinguish it from other parts. You may need a notation to write (some) formulas on paper, and no notation will allow you to write arbitrary real number, but this is usually not a problem.