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If we have a variable $x$ and we wish to express the assignment function taking the symbol '$x$' to a number $a$.

We might write that $s(x)=a$

I'm confused by this notation, as we use $s(x)$ which suggests a function that is applied to the real number $x$, as we usually use $x$ in an expression to refer to it's value not the variable itself.

This is acceptable if we consider $s(x)$ as strictly in the metalanguage, Is it more correct to use $s(x)$ to discuss the value of $x$? Or is $x$ a symbol in the metalanguage and a number in the object language?

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It's actually the other way round: It is the use of $x$ to refer to the value of the variable that's strictly speaking wrong. $x$ is a symbol and $s(x)$ is the real number. In common language use, when we say "$x$" we usually actually mean $s(x)$. The notion of an assignment function just describes more explicitly what is actually happening when we use a variable to refer to a value: We map symbols to real objects (such as numbers).

As for the language levels, $x$ is a symbol of the object language; $s$ and $a$ are part of in the meta language. The function $s$ talks about the formal symbol $x$ and says what it means in the real world, namely $a$.

Natalie Clarius
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  • If it is wrong strictly speaking, why do we use '$x$' in open formulas, is it because in the object language the symbol is used to refer to the object and it is incorrect to do so in the meta-language? We do so in quite formal contexts. –  Oct 27 '22 at 08:50
  • I see no issue in $x$ being an object in the same way $5$ is an object, but I might be confused between meta and object language. –  Oct 27 '22 at 08:56
  • If $s$ is an element of the meta language then I assume it is only used in the metalanguage? –  Oct 27 '22 at 09:10
  • Whether $x$ is bound by a quantifier or free in an open formula makes no difference. It is always evaluated by an assignment function. When we make a statement such as $x + 1 = 1+ x$, we iterate all possible assignment functions. – Natalie Clarius Oct 27 '22 at 11:42
  • $5$ is a symbol and strictly speaking not a real object in the same way $x$ is. In both cases, we use a symbol to refer to the real object because we can't put the real abstract number object on paper. For variables we use an assignment function to map symbols to objects, for constants we use an interpretation function. – Natalie Clarius Oct 27 '22 at 11:48
  • $x$ is strictly part of the object language, $s$ is strictly part of the meta language. What makes a formal context formal is that we bother even making the distinction. The notion of object and meta language and assignment functions describes more precisely what we actually do when we informally (i.e. outside of the branch of mathematics that specifically deals with symbolic logic, so even ordinary mathematical use would count as informal here) use $x$ to mean a number object. – Natalie Clarius Oct 27 '22 at 11:55
  • I agree it is a symbol that needs an interpretation, however if $x$ is a symbol what is the difference between representing the number as $s(x)$ which is itself just a set of symbols, how is it that we can interpret $s(x)$ as a number then? –  Oct 27 '22 at 13:01