Words 
Explained in 
Comment 
Mathematics ; mathematical logic 
1.1 

theory 
1.1, 1.3, 1.A 

system 
1.1, 1.3, 1.4, 3.1, 3.2 
= a set with structures, interpreting given
types and symbols. Could not call this its officially name of structure which I left to its other standard meaning which sounds better. 
model 
1.1, 1.A 
= interpretation of a theory. See also (Mod, Tru) in 4.8 
content / components of a theory 
1.1, 1.3 

foundation, primitive 
1.1 
Ref : wikipedia Primitive notion 
development 
1.1, 1.3, 4.8, 4.9 
Did not find this concept made explicit
elsewhere. 
axiom 
1.1, 1.3, 1.5, 1.10, 1.A 

statement 
1.1, 1.9  The syntactic difference (1.9) making them
not particular formulas seems original, the special syntax
of binders in set theory (1.8) being absent from traditional
presentations (ZF). 
set theory 
1.11.4, 1.7, 1.8, 1.10... 

model theory 
1.1, 1.31.7, 1.9... 

theorem, proof, provable, proof theory, (in)consistent 
1.1, 1.6, 1.9, 1.F, 4.6, 5.7 
see also "refutation" 
constant (symbol) 
1.2, 1.4  
value (of a variable) 
1.2  
variable (symbol) 
1.2, 1.5, 4.1  
fixed variable 
1.2, 1.3 

free variable bound variable 
1.2, 1.8 
Ref: wikipedia 
range ; range over 
1.2  
element 
1.2, 1.7 

pure element  1.2, 1.7  elsewhere called urelement or atom, while
the latter word has another
standard meaning 
set  1.2, 1.7, 1.E 

function, argument, domain  1.2, 1.10 

value of a function 
1.2  
functor 
1.2, 1.4, 1.D, 1.E 
Not found this named elsewhere. Not the concept of functor in category theory (particular case, not named in this work but implicitly used such as in actions of categories). 
operation 
1.2, 1.10, 2.3 

relation 
1.4, 1.10, 2.3 

arity ; nullary ; unary; binary... 
1.2 
See also: tuple, operation, relation 
notion 
1.3, 1.7 
here precisely defined in a way not seen
elsewhere 
object of a theory 
1.3  to not confuse with objects of a category 
onemodel theory 
1.3  Did not see it named nor conceived elsewhere. 
logical framework 
1.1, 1.3, 1.A  This concept is of course crucial but not
named elsewhere this way, maybe not any other way, but
specific ones are named by the word "logic" with adjectives. 
firstorder logic generic theory 
1.3  see also: model theory 
[T,M]  1.3 
only used in 1.B and 1.7; invented notation
(how else would you write that?) 
meta(notions of onemodel theory) metaobject 
1.3  
meta(notions of set theory)  1.7 

type (abstract type, interpreted type) 
1.3  Hardly ever mentioned in the math literature,
where it is also called sort,
but that latter word does not require them to be disjoint
(see "ordersorted logic") 
language 
1.3, 3.1 
= list of structure symbols 
structures 
1.3, 1.4 
here only firstorder structures : operators
and predicates 
secondorder structures 
(1.3) 5.1  
interpretation (of a theory into set theory)  1.3 
designates a model in the universe 
generic interpretation 
1.3  I coined that name 
standard 
1.3, 1.D, 4.7 

universe 
1.3, 1.D, 1.F, 5.3 

operator, predicate 
1.4 

Boolean 
1.4 
Logicians use
this name in some ways but elsewhere call it propositional,
while in ordinary math "proposition" is synonymous with
theorem, which is different. 
paraoperator 
1.4 
I coined that name 
(logical) connective 
1.4, 1.6 

∈  1.4 

f(x) evaluator Dom 
1.2, 1.4 
I coined the name "evaluator" as I am not
aware of any standard name for this. 
ZF 
1.4 
Only commented here, not described nor used.
See also 1.D (specification) and 5.B 
ground, expression, formula 
1.5 
This definition of "expression" as "term or
formula" just seems to be an implicit
standard. 
term, occurrence, root  1.5, 4.1 

binder  1.5, 1.8  I coined it, for what seems to be elsewhere
called "Variablebinding operator" as I give "operator"
another meaning and cannot see how else to do. 
equality = 
1.5, 1.6 

variable structure 
1.5 

definition, abbreviation 
1.5, 4.8 

parameter 
1.5 
see also : constructions 
invariant  1.5, 3.1, 3.3, 3.6, 4.8, 5.1 

tautology logically valid formula 
1.6 

⇔ , ¬ , ≠ , ∉, ⇎ 
1.6 
equivalent, not (negation), different, not in, inequivalent (exclusive or) 
∧ , ∨  1.6 
conjunction, disjunction 
implication ⇒ necessary condition sufficient condition contrapositive, converse 
1.6 

∴  1.6 
Therefore. This notation seems standard but I
only found it in lists of html symbols, not in math texts. Wikipedia says it only serves as natural language abbreviation, while I use it in formulas as synonym of ∧ (but not of ⇒). 
chain of conjunctions chain of disjunctions chain of implications 
1.6 
Did not see these officially declared
elsewhere though they are so much needed. 
⊢ refutation, refutable, decidable, lemma, proposition, corollary 
1.6 
see also "theorem" 
axioms of equality 
1.9 
References : wikibooks
 Encyclopedia
of math. Used in 4.6 (proof of the completeness
theorem) 