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Zapiski Nauchnykh Seminarov LOMI, 1974, Volume 40, Pages 101–109 (Mi znsl2685)  

Heytiag predicate calculus with $\varepsilon$-symbol

G. E. Mints
Abstract: It is known that the introduction of $\varepsilon$-symbol with $\varepsilon$-axioms $A[t]\to A[\varepsilon x A]$ leads to non-conservative extension. For example $\exists x(\rceil P_x\to\rceil Pb\&\rceil Pa)$ becomes derivable. A conservative extension is obtained by treating $\varepsilon$-symbol like $\iota$-symbol: for every occurence $\varepsilon x A[x,\alpha_1\dots,\alpha_n]$ in a sequent from a deduction formula $\forall\alpha_1\dots\forall\alpha_n\exists x A$ should occur in the antecedent of this sequent. Cut-elimination is proved for the resulting system $HPC^{\varepsilon}$. It is pointed out that the proof could be extended to $HPC$ with decidable equality and to Heyting arithmetic with free function variables and the principle of choice:
$$ \Gamma\to\forall x\exists y A;\quad\forall x A_y[f(x)],\quad\Gamma\to C\vdash\Gamma\to C. $$

The extension to Heyting arithmetic with bound variables of higher types and corresponding choice principle requires new ideas.
Bibliographic databases:
UDC: 51.01
Language: Russian
Citation: G. E. Mints, “Heytiag predicate calculus with $\varepsilon$-symbol”, Studies in constructive mathematics and mathematical logic. Part VI, Zap. Nauchn. Sem. LOMI, 40, "Nauka", Leningrad. Otdel., Leningrad, 1974, 101–109
Citation in format AMSBIB
\Bibitem{Min74}
\by G.~E.~Mints
\paper Heytiag predicate calculus with $\varepsilon$-symbol
\inbook Studies in constructive mathematics and mathematical logic. Part~VI
\serial Zap. Nauchn. Sem. LOMI
\yr 1974
\vol 40
\pages 101--109
\publ "Nauka", Leningrad. Otdel.
\publaddr Leningrad
\mathnet{http://mi.mathnet.ru/znsl2685}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=398794}
\zmath{https://zbmath.org/?q=an:0358.02023}
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