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.