Can we use symbols from the language model our universe is made of?

Question

In my question here, within my response to part 1, I used the symbol "$+$", which was not a part of the provided $\mathcal L$. Is this something that is allowed?

Thank you!

Answer 1

See the very first paragraphs of Leary's textbook [page 4]regarding structures and languages:

we have to be able to look at some symbols in a particular formal language and then take those symbols and relate them in some way to a mathematical structure. Different interpretations of the symbols will lead to different conclusions as regards the truth of the formal statement.

We have to take care of the difference between the structure, i.e. the domain of mathematical objects which is "talked about" and which is the subject matter of the discussion, and the language with which we "talk about" the objects and their properties.

We have a language $\mathcal L$ with the function symbol $S$ (for the successor) and the relation symbol $<$.

When we interpret the language into e.g. the structure $\mathbb N$ of natural numbers, we can say that $S(n)$ is interpreted with $n+1$, because "we know" what does it mean for a natural number $n$ to add one to it.

But you cannot use the symbol $+$ in the formulas of the language $\mathcal L$, because it is not parte of $\mathcal L$.