**
Q1.** *Is
it true that only first-order Predicate Logic with equality is
considered in this project.***
Answer.** Yes,
of course. Often, short terms such as "Logic", "Predicate Logic",
"Predicate Calculus", etc. are used instead of the full name
"first-order Predicate Logic with equality".* *

**Q2. ***Universal
construction represents the top result in the direction of expressive
possibilities of first order logic. But, this construction is too
complicated and therefore one hardly can understand both this
construction and the direction at a whole. Is it true ?*

**Answer.** No.
Currently, the most of results concerning expressive possibilities of
first-order predicate logic are obtained on the base of Canonical
construction as well as with using some other more weak constructions.
At the same time, Canonical construction is much more understandable
than Universal one. Thus, knowing the Canonical construction, give good
chance to view many interesting results in this direction. Moreover,
this will help to understand better the main moments of Universal
construction. So, current project is devoted to give real chance to
understand all main results in this direction.

**Q3. ***What
relation exist between Hahf's and Space constructions ?*

**Answer.** They
are identical. 'Hanfs'
construction' is just historical name, while 'Space construction' is new
name of the same construction, showing its logical significance.

**Q4.** *It
is well known that first-order Predicate Logic has very strong
expressive possibilities. As an example one can note finitely
axiomatizable set theory GB founded by Godel and Bernais. Why some extra
constructions are required to show expressive power of first-order
Predicate Logic ?*

**Answer.** The
matter is that here term "Power" is used in two meanings. First, "Power"
means maximum possible strength. For example, power of sound can be
measured in decibels. Second meaning of "Power" is diapason of
possibilities, that is, interval from very thin properties up to very
strong properties. As an analogy, a singer's "Power" means diapason of
its voice, not just maximum power of its voice in decibels.

In case of set theory GB, term
"expressive Power" of predicate logic is used to state its maximum
strength, while in the case of constructions of finitely axiomatizable
theories, "expressive Power" means value of diapason from very thin up
to very strong properties. This gives an explanation to the posed
question.

**Q5. ***In
well known construction of Kleene (1950) any recursively axiomatizable
theory of a finite signature is transformed to a finitely axiomatizable
theory in a signature with one additional binary predicate. This result
seems to be very general and significant, it shows deep expressive
possibilities of formulas of first-order logic. Why some extra
constructions are required to show expressive power of first-order
Predicate Logic ?*

**Answer.** Construction
of Kleene makes some transformation from any recursively axiomatizable
theory T to a finitely axiomatizable theory F. It is such that Peano
arithmetic P is some part of the theory F. As a result, Kleene's
construction, mainly, passes from T to F only 'negative' properties such
as undecidability, essential undecidability, etc. Particularly, if
theory T is taken decidable, complete, etc., then resulting theory F
will not have these properties. So, idea of Kleene's construction is
very general, but wishes to pass such thin properties as decidability,
completeness, etc., require some new constructions.