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Intuitionistic Formal Spaces
, 1989
"... This paper is exactly the same as Intuitionistic formal spaces  a first communication, in: Mathematical Logic and its Applications, D. Skordev ed., Plenum 1987, pp. 187204 by the same author, except for: (i.) the conditions on the positivity predicate (part 3. of definition 1.1 and end of section ..."
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This paper is exactly the same as Intuitionistic formal spaces  a first communication, in: Mathematical Logic and its Applications, D. Skordev ed., Plenum 1987, pp. 187204 by the same author, except for: (i.) the conditions on the positivity predicate (part 3. of definition 1.1 and end of section 1) and the treatment of Scott domains (section 8), which have been modified as explained in the addendum Intuitionistic formal spaces vs. Scott domains, in: Atti del Congresso Temi e prospettive della logica e della filosofia della scienza contemporanee, vol. 1, CLUEB, Bologna 1988, pp. 159163; (ii.) the correction of some of the misprints; (iii.) one change in notation (now \Delta is used for covering relations, rather than ) and one in terminology (now `weak transitivity' replaces `weakening'). For an update on the development of formal topology, see the survey Formal topology  twelve years of development, in preparation, by the same author.
A construction of Type:Type in MartinLöf's partial type theory with one universe
"... ing on w and pairing with oe(p(c); (x)Ap(q(c); x) ! p(c)) in the first coordinate yields hoe(p(c);(x)Ap(q(c); x) ! p(c)); (w)(Ap(q(c); p(w)); (x)Ap(q(c); Ap(q(w); x)))i 2 PAR; i.e. s (c) 2 PAR. We define the operator that builds the universe (U 1 ; T 1 ) by putting f(c) := s (c) +hn 1 ; (x ..."
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ing on w and pairing with oe(p(c); (x)Ap(q(c); x) ! p(c)) in the first coordinate yields hoe(p(c);(x)Ap(q(c); x) ! p(c)); (w)(Ap(q(c); p(w)); (x)Ap(q(c); Ap(q(w); x)))i 2 PAR; i.e. s (c) 2 PAR. We define the operator that builds the universe (U 1 ; T 1 ) by putting f(c) := s (c) +hn 1 ; (x)R 1 (x; p(c))i; for c 2 PAR, and let e := fix((c)f(c)). Hence e 2 PAR is a fixed point of f , e = f(e). The right summand of f corresponds to the rules (2). We now interpret Type:Type. The universe (U 1 ; T 1 ) is defined by letting U 1 := T (p(e)) and T 1 (a) := T (Ap(q(e); a)); for a 2 U 1 . Thus the rules (1) are verified. Using the equality e = f(e) and the commutation of T with \Sigma, \Pi and + we get U 1 = T (p(e)) = T (p(f(e))) (4) = T (oe(p(e); (x)Ap(q(e); x) ! p(e))) + T (n 1 ) = (\Sigmax 2 T (p(e)))[T (Ap(q(e); x)) \Gamma! T (p(e))] +N 1 = (\Sigmax 2 U 1 )[T 1 (x) \Gamma! U 1 ] +N 1 and hence j(0 1 ) 2 U 1 . Furthermore we have T 1 (j(0 1 )) = T (Ap(q(...