I am looking for an easy explaination about Metalanguage in mathematics

Question

We define that statements are objects that fullfill a certain syntax. But this definition itself is a statement. It is a variant of saying: If the object fulfils a certain syntax then it is a statement because it also fulfils a certain syntax. This is again a statement about the statement and so on...

I know this is very informal and I have started to read a book about logic but the author said he will talk about this subject first at chapter 7 and I hoped that I maybe could get an informal but still comprehensible "peek" of the problem: That we are defining an object in a lower language that still is applicable to higher languages. To me it seems like proving something for fields and then saying that it also holds for rings.

Answer 1

See An Elementary Latin Grammar :

The statement "Cesar scribit" [means] Caesar is writing.

We have here the Latin language, which is the object of the study; call it : object language.

And we have the English language, used to perform the study; call it : meta-language.

The statement "Cesar scribit" is a statement in the object language.

The statement "The statement "Cesar scribit" [means] Caesar is writing." is a statement of the meta-language that expresses a fact about the object language statement "Cesar scribit".

And now compare with : D.van Dalen, Logic and Structure, page 7 :

Definition 1.1.2 The set $\text {PROP}$ of propositions is the smallest set $X$ with the properties

(i) $p_i ∈ X(i ∈ \mathbb N), \bot ∈ X$,

(ii) $ϕ, ψ ∈ X ⇒ (ϕ ∧ ψ), (ϕ ∨ ψ), (ϕ → ψ), (ϕ ↔ ψ) ∈ X$,

(iii) $ϕ ∈ X ⇒ (¬ϕ) ∈ X$.

It is a statement in the meta-language : the usual mathematical argot, made of natural language plus symbols used as abbreviations, regarding the syntax of the object language : the language of propositional calculus.

"this definition itself is a statement" ?

Yes; it is a statement in the meta-language defining the formal syntax of the objcet language.