A. S. Troelstra. Note on the fan theorem. The Journal of Symbolic Logic, 39:584--596, 1974. Home/Search Document Not in Database Summary Related Articles Check |
This paper is cited in the following contexts:
On the Uniform Weak König's Lemma - Kohlenbach (1999) (Correct)
....in WE PA (see [8] The schema of quanti er free choice is given by QF AC : 8x A 0 (x; y) 9Y ( A 0 (x; Y x) QF AC : 2T f QF AC g; where A 0 is a quanti er free formula. In the following we use the formal de nition of the binary ( weak ) K onig s lemma as given in [19] (here ; bx; lth(n) refer to the primitive recursive coding of nite sequences from [18] De nition 2.1 (Troelstra(74) WKL: 8f (T (f) 8x (lth(n) x fn = 0) 9b (f(bx) 0) where Tf : 8n ; m (f(n m) 0 0 fn = 0 0) 8n (f(n hxi) 0 0 x 0 1) i.e. T (f) ....
Troelstra, A.S., Note on the fan theorem. J. Symbolic Logic 39, pp. 584-596 (
The Computational Strength of Extensions of Weak König's Lemma - Kohlenbach (1998) (Correct)
.... keeping 2) would result in a principle called KL in the context of reverse mathematics which relative to the well known system RCA 0 is equivalent to arithmetical comprehension, whereas RCA 0 WKL is conservative over RCA 0 (see [15] The signi cance of the restriction 2) was pointed out rst in [17], where it is shown that the binary K onig s lemma applied to trees of arbitrary logical complexity implies comprehension of numbers for arbitrary complex predicates. In this paper we study an extension 1 WKL of (QF )WKL to a certain class of formulas 1 which are built up as follows: in front of ....
.... ( less or equal ) by induction on the type: x 1 0 x 2 : x 1 x 2 ; x 1 x 2 : 8y (x 1 y x 2 y) 2) min 0 (x 2 ) min(x 1 ; x 2 ) min (x 2 ) y : min (x 1 y; x 2 y) In the following we will need the de nition of the binary ( weak ) K onig s lemma as given in [17]: De nition 1.2 (Troelstra(74) WKL: 8f (T (f) 8x (lth n = 0 x fn = 0 0) 9b 1 k:18x (f(bx) 0 0) Tf : 8n ; m (f(n m) 0 0 fn = 0 0) 8n (f(n hxi) 0 0 x 0 1) i.e. T (f) asserts that f represents a 0,1 tree) Throughout this paper A 0 ; B 0 ; C 0 ; ....
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Troelstra, A.S., Note on the fan theorem. J. Symbolic Logic 39, pp. 584-596 (
Things that can and things that can't be done in PRA - Kohlenbach (1998) (1 citation) (Correct)
....functions can be obtained from the initial functions and it by substitution. The schema of quanti er free choice for numbers is given by AC qf : 8x A 0 (x; y) 9f8xA 0 (x; fx) where A 0 is a quanti er free formula. We also consider the binary K onig s lemma as formulated in [27]: WKL : 8f (T (f) 8x (lth(n) 0 x f(n) 0 0) 9b 1 18x (f(bx) 0 0) where b 1 1 : 8n(bn 1) and T (f) 8n (f(n m) 0 f(n) 0) 8n (f(n hxi) 0 x 1) here lth; bx; h i refer to a standard elementary recursive coding of nite sequences of ....
Troelstra, A.S., Note on the fan theorem. J. Symbolic Logic 39, pp. 584-596 (
Foundational and Mathematical Uses of Higher Types - Kohlenbach (1999) (2 citations) (Correct)
.... we de ne the relation : x 1 0 x 2 : x 1 x 2 ; x 1 x 2 : 8y (x 1 y x 2 y) 2) min 0 (x 2 ) min(x 1 ; x 2 ) min (x 2 ) y : min (x 1 y; x 2 y) In the following we will need the de nition of the binary ( weak ) K onig s lemma as given in [38]: De nition 2.2 (Troelstra(74) WKL: 8f (T (f) 8x (lth n = 0 x fn = 0 0) 9b 1 k:18x (f(bx) 0 0) where Tf : 8n ; m (f(n m) 0 0 fn = 0 0) 8n (f(n hxi) 0 0 x 0 1) i.e. T (f) asserts that f represents a 0,1 tree) Throughout this paper A 0 ....
....of n choice for numbers is given by n AC 9y 0 1 A(x; y) 9g 1 18x A(x; gx) where A(x; y) 2 n and may contain arbitrary parameters. Proposition 5. 11 Let T : E PA T n 1 WKL n CA (Likewise for n 1 WKL) Proof: We use the following tree predicate from [38]: A(k) k) lth(k) 1 1 ( k) lth(k) 1 = 0 A(lth(k) 1) k) lth(k) 1 = 1 :A(lth(k) 1) if lth(k) 0 true; otherwise: For A 2 n , A(k) can be written as a n 1 formula (using remark 5.3) By induction on n we can prove in E PA 9f 1 18 n n A(f ....
Troelstra, A.S., Note on the fan theorem. J. Symbolic Logic 39, pp. 584-596 (1974).
Mathematically Strong Subsystems of Analysis With Low Rate of.. - Kohlenbach (1995) (2 citations) (Correct)
....of infinite 0,1 trees there exists a sequence of infinite branches: Definition 4.24 WKL seq :# 8 : #f 1(0) #k 0 (T (fk) # #x 0 #n 0 (lth n = 0 x # fkn = 0 0) # #b # 1(0) #k 0 , i 0 . 1#k 0 , x 0 (fk( bk)x) 0 0) This formulation of WKL (which is used e.g. in [35] and [30] 31] 32] and in a similar way in the system RCA 0 considered in the context of reverse mathematics with set variables instead of function variables) and WKL seq uses the functional # ## bx = bx which is definable in GnA # i only for n # 3 and causes exponential growth. Since we ....
Troelstra, A.S., Note on the fan theorem. J. Symbolic Logic 39, pp. 584--596 (1974).
Things that can and things that can't be done in PRA - Kohlenbach (2000) (1 citation) (Correct)
....can be obtained from the initial functions and it by substitution. The schema of quanti er free choice for numbers is given by AC 0;0 qf : 8x 0 9y 0 A 0 (x; y) 9f8xA 0 (x; fx) where A 0 is a quanti er free formula. 4 We also consider the binary K onig s lemma as formulated in [28]: WKL : 8f 1 (T (f) 8x 0 9n 0 (lth(n) 0 x f(n) 0 0) 9b 1 18x 0 (f(bx) 0 0) 3 This means that we allow all the type 2 functionals n from [9] plus a bounded search operator and bounded recursion uniformly in function parameters on the ground type (see [9] 4 ....
Troelstra, A.S., Note on the fan theorem. J. Symbolic Logic 39, pp. 584-596 (1974). 26
Higher Order Reverse Mathematics - Kohlenbach (2000) (1 citation) (Correct)
....(f, g) where A# (f, g) expresses in terms of recursion equations that g = #(f ) Notation: For # = # 1 # . # # k # 0, we define 1 # : #x #1 1 . x #k k .1 0 , where 1 0 : S0. In the following we will need the definition of the binary ( weak ) Konig s lemma as given in [18]: Definition 2.1 (Troelstra(74) WKL:# #f 1 # T # (f) # #b # 1 #k.1#x 0 (f(bx) 0 0) # , where T # (f) # # # # #n 0 , m 0 (f(n # m) 0 0 # fn = 0 0) ##n 0 , x 0 (f(n # #x#) 0 0 # x # 0 1) ##x 0 #n 0 (lth n = 0 x # fn = 0 0) i.e. T # (f) asserts that f ....
Troelstra, A.S., Note on the fan theorem. J. Symbolic Logic 39, pp. 584--596 (1974).
Foundational and Mathematical Uses of Higher Types - Kohlenbach (2000) (2 citations) (Correct)
....the relation # # : 8 : x 1 # 0 x 2 :# x 1 # x 2 , x 1 # ## x 2 :# #y # (x 1 y # # x 2 y) 2) min ## (x ## 1 , x ## 2 ) #y # . min # (x 1 y, x 2 y) with min 0 from above. In the following we will need the definition of the binary ( weak ) Konig s lemma as given in [39]: Definition 2.3 (Troelstra(74) WKL:# 8 : #f 1 T (f) # #x 0 #n 0 (lth n = 0 x # fn = 0 0) # #b # 1 #k.1#x 0 (f(bx) 0 0) where Tf :# #n 0 , m 0 (f(n # m) 0 0 # fn = 0 0) # #n 0 , x 0 (f(n # #x#) 0 0 # x # 0 1) i.e. T (f) asserts that f ....
....by # 1,b n AC 0,0 : #x 0 #y # 0 1 A(x, y) # #g # 1 1#x A(x, gx) where A(x, y) # # 1,b n and may contain arbitrary parameters. Proposition 5.9. Let T : E PA # . Then T #n 1 WKL # # 1,b n CA (Likewise for #n 1 WKL) Proof: We use the following tree predicate from [39]: A(k) # 8 : k) lth(k) 1 # 1 # ( k) lth(k) 1 = 0 # A(lth(k) 1) # ( k) lth(k) 1 = 1 # A(lth(k) 1) if lth(k) 0 true, otherwise. For A # # 1,b n , A(k) can be written as a #n 1 formula. By induction on n we can prove in E PA # that ....
Troelstra, A.S., Note on the fan theorem. J. Symbolic Logic 39, pp. 584--596 (1974).
On uniform weak König's lemma - Kohlenbach (Correct)
....arbitrary types allowed) So the system RCA 0 from reverse mathematics (see [18] can be viewed as a subsystem of WE PRA # QF AC 0,0 by indentifying sets X # IN with their characteristic function. In the following we use the formal definition of the binary ( weak ) Konig s lemma as given in [21] (see also [22] here #, bx, lth(n) refer to the primitive recursive coding of finite sequences from [20] Definition 2.2 (Troelstra(74) Tf :# #n 0 , m 0 (f(n # m) 0 0 # fn = 0 0) # #n 0 , x 0 (f(n # #x#) 0 0 # x # 0 1) i.e. T (f) asserts that f represents a binary ....
Troelstra, A.S., Note on the fan theorem. J. Symbolic Logic 39, pp. 584--596 (1974).
Foundational and Mathematical Uses of Higher Types - Kohlenbach (2 citations) (Correct)
....define the relation # # : 8 : x 1 # 0 x 2 :# x 1 # x 2 , x 1 # ## x 2 :# #y # (x 1 y # # x 2 y) 2) min ## (x ## 1 , x ## 2 ) #y # . min # (x 1 y, x 2 y) with min 0 from above. In the following we will need the definition of the binary ( weak ) Konig s lemma as given in [38]: Definition 2.3 (Troelstra(74) WKL:# #f 1 T (f) # #x 0 #n 0 (lth n = 0 x # fn = 0 0) # #b # 1 #k.1#x 0 (f(bx) 0 0) where Tf :# #n 0 , m 0 (f(n # m) 0 0 # fn = 0 0) # #n 0 , x 0 (f(n # #x#) 0 0 # x # 0 1) i.e. T (f) asserts that f represents a ....
....by # 1,b n AC 0,0 : #x 0 #y # 0 1 A(x, y) # #g # 1 1#x A(x, gx) where A(x, y) # # 1,b n and may contain arbitrary parameters. Proposition 5.9 Let T : E PA # . Then T #n 1 WKL # # 1,b n CA (Likewise for #n 1 WKL) Proof: We use the following tree predicate from [38]: A(k) # 8 : k) lth(k) 1 # 1 # ( k) lth(k) 1 = 0 # A(lth(k) 1) # ( k) lth(k) 1 = 1 # A(lth(k) 1) if lth(k) 0 true, otherwise. For A # # 1,b n , A(k) can be written as a #n 1 formula. By induction on n we can prove in E PA # that ....
Troelstra, A.S., Note on the fan theorem. J. Symbolic Logic 39, pp. 584--596 (1974).
Intuitionistic Choice and Restricted Classical Logic - Kohlenbach (2000) (1 citation) (Correct)
....HA # AC 0,0 LNOS and d HA # AC 0,0 LNOS prove DWKL. Proof: We show the theorem for d HA # AC 0,0 LNOS. Analogously to the proof of the lemma above one verifies that d HA # AC 0,0 allows to reduce DWKL to the usual weak Konig s lemma WKL as defined in [12]: WKL:# #f 1 (T (f) # #x 0 #n 0 (lth(n) x # fn = 0) # #b 1 #x 0 (f(bx) 0) where Tf :# #n 0 , m 0 (f(n # m) 0 0 # fn = 0 0) # #n 0 , x 0 (f(n # #x#) 0 0 # x # 0 1) Consider the formula 3 ( 8 : #x 0 #n # 0 1#k 0(#m # 1k(lth(m) k # ....
Troelstra, A.S., Note on the fan theorem. J. Symbolic Logic 39, pp. 584--596 (1974).
Proof Mining in Subsystems of Analysis - Oliva (2003) (Correct)
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A. S. Troelstra. Note on the fan theorem. The Journal of Symbolic Logic, 39:584--596, 1974.
Relative Constructivity - Kohlenbach (1996) (1 citation) (Correct)
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Troelstra, A.S., Note on the fan theorem. J. Symbolic Logic 39, pp. 584--596 (1974).
Analyzing Proofs in Analysis - Kohlenbach (Correct)
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Troelstra, A.S., Note on the fan theorem. J. Symbolic Logic 39, pp. 584--596 (1974).
The Use of a Logical Principle of Uniform Boundedness in Analysis - Kohlenbach (1996) (1 citation) (Correct)
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Troelstra, A.S., Note on the fan theorem. J. Symbolic Logic 39, pp. 584--596 (1974).
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