Confused about Enderton's mathematical logic notation I am currently reading Enderton's math logic text. I am on section 1.7 where he is proving the compactness theorem. I came across this weird notation that I have never seen before; then I referred to his chapter 0.
He states: A useful operation is that of adjoining one extra object to a set. For a set $A$, let $A$; $t$ be the set whose members are (i) the members of $A$, plus (ii) the (possibly new) member $t$. Here $t$ may or may not already belong to $A$, and we have
$A; t = A \cup \{t\}$
He uses this notation to prove the compactness theorem. So does this mean that we are redefining $t$ like how they do it in programming languages?
For example, $t = 2$ and $A = \{1,3\}$. Then, $A; t = A \cup \{t\}$ redefines $t = \{1,2,3\}$. Sorry if this a stupid question, but I don't really get it.
 A: I’m not sure I see any redefining going on here. In the example you gave,
$A$ stands for $\{1,3\}$,
$t$ stands for $2$, and
$A;t$ stands for $\{1,2,3\}$.
Maybe you can clarify your confusion a bit more.
A: In reading a mathematical definition, you need to be careful about bringing in ideas from other places.  In particular, this is not a text on programming, and one should not assume that notation from any particular programming language is related to the notation being defined in the text.
When an author writes something like "Let ___ be...," they are introducing a new symbol, and then telling you exactly what that symbol means.  In this case,

For a set $A$, let $A\mathbin{;}t$ be the set whose members are (i) the members of $A$, plus (ii) the (possibly new) member $t$.

Here, the new symbol "$A\mathop{;} t$" is being introduced.  You should, for the moment, regard this as one symbol.  The author then explains what this new symbol means:  it is a set whose elements are all the elements of $A$, plus the possibly new element $t$.  That is, the set which is denoted by the single symbol "$A\mathop{;} t$" is the union of $A$ and $\{t\}$, i.e.
$$ A\mathop{;} t = A \cup \{t\}. $$
Again, the entire symbol to the left of the equals is being defined.  Hence
$$ \{1,3\} \mathop{;} 2 = \{1,2,3\}. $$
In this case, it might also be helpful to regard the symbol "$;$" as a binary operator or function, similar to "$+$", "$\cap$", or "$\times$".  The symbol "$;$" sits between a set (on the left) and an object (on the right), and "combines" them in some way.  Specifically, it includes the object into the set.
