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Until now I thought that in logic and theoretical computer science, if we consider an alphabet consisting of symbols and strings over that alphabets, the symbols can be any objects (for example natural numbers, sets, ...), we just imagine them to be things one can "write down" (but that's not important from a mathematical point of view). But recently I read phrases like:

For each real number $x$ we add a constant symbol $c_x$ to the language/signature.

But if symbols can be any object, we don't we add $x$ itself but instead first turn $x$ into $c_x$ which looks more like a "symbol", and then add? How would one precisely define $c_x$?

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  • $\begingroup$ We first turn $x$ into $c_x$ to differentiate the real number from the symbol supposed to represent it. Some authors write $\underline{x}$, and some don't even bother to make a distinction (because technically, you could use the reals themselves as symbols). I don't think it's essential (but maybe someone will correct me ?) but it helps for comprehension. For instance if you used the reals for the symbols denoting them, $x=y$ could either mean that $x$ and $y$ are reals that are equal, or it could simply be the formula "$x=y$" and so it may be confusing $\endgroup$ – Max Mar 14 '17 at 20:25
  • $\begingroup$ @Max: I think this would be a reasonable answer. Feel free to post it as such. Thanks! $\endgroup$ – user7280899 Mar 14 '17 at 20:30
  • $\begingroup$ You should also consider that symbols are pure syntax, meaningless characters. We then give them a meaning, an interpretation, a reference. The symbol $c_x$ is not an alias or the number $x$ itself but a pure meaningless syntactic construction representing $x$. The same idea can be found in the proof theory of first-order logic as in Skolemization of formulas and Herbrand structures. $\endgroup$ – Boris E. Mar 14 '17 at 20:45
  • $\begingroup$ @BorisEng: "You should also consider that symbols are pure syntax, meaningless characters." Yes, that's the idea. But we do logic in a meta theory and in this theory, we consider symbols to be themself objects. And we can compare them for equality, put them into sets, ... In this sense, they are not "meaningless". $\endgroup$ – user7280899 Mar 14 '17 at 20:51
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You're right that it doesn't matter what we use as symbols, as long as they are mathematical objects that we can compare for equality, put into sets, etc.

There are at least two reasons for writing $c_x$ instead of just $x$ in situations such as the one you quote:

  • It reminds the human reader that the new symbols are just symbols: the proof systems and rules of logic don't care about their properties as numbers, except to the extent we define new axioms to force them to.

  • It allows us to rename the symbols if we need to -- for example it might be that some of the symbols that are already in the language happen to be numbers. (Usually we haven't said what exactly they are either, which means that we cannot be sure they are not numbers). In that case we need to choose something different for our new symbols.

So the content of what you quote is just

Choose some set $S$ of the right cardinality that is disjoint from the set of symbols we're already working with, and fix a bijection $c:\mathbb R\to S$. We will write $c(x)$ as $c_x$. Consider a new logical language whose symbols are our old vocabulary together with $S$ ...

The phrasing in your quote is intended to be the "minimally invasive" way to imply something like this, which is still explicit enough to avoid being sidetracked by clever students protesting, "but what if $2$ is already a symbol? Where did we assume it is not?"

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