Separating Variants of LEM, LPO, and MP

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1 Matt Hendtlass, U Canterbury Robert S. Lubarsky, FAU Proof Theory, Modal Logic and Reflection Principles Mexico City, Mexico September 29 - October 2, 2014

2 Implications Definition The Law of Excluded Middle (LEM): For any proposition A, either A is true or A is false (A or A). The Weak LEM (WLEM): For any proposition A, either A or A. The Limited Principle of Omniscience (LPO): For any binary sequence α, either α(n) = 0 for all n or there exists n such that α(n) = 1. Markov s Principle (MP): If it is impossible for all terms of α to be zero, then there exists an n such that α(n) = 1.

3 Implications Definition The Law of Excluded Middle (LEM): For any proposition A, either A is true or A is false (A or A). The Weak LEM (WLEM): For any proposition A, either A or A. The Limited Principle of Omniscience (LPO): For any binary sequence α, either α(n) = 0 for all n or there exists n such that α(n) = 1. Markov s Principle (MP): If it is impossible for all terms of α to be zero, then there exists an n such that α(n) = 1. The Lesser Limited Principle of Omniscience (LLPO): For any binary sequence α with at most one non-zero term, either α(n) = 0 for all even n or α(n) = 0 for all odd n.

4 Implications Implications LEM LPO MP WMP WLEM WLPO WKL Π 0 1 -ACω, LLPO MP... WLEM n LLPO n MP n WLEM n+1 LLPO n+1 MP n+1... WLEM ω LLPO ω MP ω

5 More Principles Implications Definition LLPO n : Let (P i ) i<n be a decidable partition of ω into blocks of size ω, and let α be a binary sequence with at most one non-zero term. Then there exists k < n such that α(m) = 0 for all m P k.

6 Implications More Principles Definition LLPO n : Let (P i ) i<n be a decidable partition of ω into blocks of size ω, and let α be a binary sequence with at most one non-zero term. Then there exists k < n such that α(m) = 0 for all m P k. MP n : If α has at most one non-zero term and it is impossible for all terms of α to be zero, then there exists k < n such that α(m) = 0 for all m P k.

7 Implications More Principles Definition LLPO n : Let (P i ) i<n be a decidable partition of ω into blocks of size ω, and let α be a binary sequence with at most one non-zero term. Then there exists k < n such that α(m) = 0 for all m P k. MP n : If α has at most one non-zero term and it is impossible for all terms of α to be zero, then there exists k < n such that α(m) = 0 for all m P k. WLEM n : i,j<n,i j A i A j i<n A i.

8 Implications Definition (Mandelkern) A real r is positive, r > 0, if r is within 1/n of some rational q > 1/n.

9 Implications Definition (Mandelkern) A real r is positive, r > 0, if r is within 1/n of some rational q > 1/n. r is almost positive, r 0, if r is not negative and r is not 0. (Equivalently, r is not not positive.)

10 Implications Definition (Mandelkern) A real r is positive, r > 0, if r is within 1/n of some rational q > 1/n. r is almost positive, r 0, if r is not negative and r is not 0. (Equivalently, r is not not positive.) r is pseudo-positive if, for all x, either x 0 or r x.

11 Implications Definition (Mandelkern) A real r is positive, r > 0, if r is within 1/n of some rational q > 1/n. r is almost positive, r 0, if r is not negative and r is not 0. (Equivalently, r is not not positive.) r is pseudo-positive if, for all x, either x 0 or r x. Easily, r > 0 r is pseudo-positive r 0. Mandelkern was interested in the converses.

12 Implications, Binary Version Definition A binary sequence α is positive, α > 0, if α(n) = 1 for some n.

13 Implications, Binary Version Definition A binary sequence α is positive, α > 0, if α(n) = 1 for some n. α is almost positive, α 0, if α is not 0. (Equivalently, α is not not positive.)

14 Implications, Binary Version Definition A binary sequence α is positive, α > 0, if α(n) = 1 for some n. α is almost positive, α 0, if α is not 0. (Equivalently, α is not not positive.) α is pseudo-positive if, for all β, either i) β 0 (i.e. n (β(n) = 1)) or ii) α β (i.e. n (α(n) = 1 k n β(k) = 0)).

15 Implications, Binary Version Definition A binary sequence α is positive, α > 0, if α(n) = 1 for some n. α is almost positive, α 0, if α is not 0. (Equivalently, α is not not positive.) α is pseudo-positive if, for all β, either i) β 0 (i.e. n (β(n) = 1)) or ii) α β (i.e. n (α(n) = 1 k n β(k) = 0)). Easily, positive pseudo-positive almost positive.

16 Implications, Binary Version Definition A binary sequence α is positive, α > 0, if α(n) = 1 for some n. α is almost positive, α 0, if α is not 0. (Equivalently, α is not not positive.) α is pseudo-positive if, for all β, either i) β 0 (i.e. n (β(n) = 1)) or ii) α β (i.e. n (α(n) = 1 k n β(k) = 0)). Easily, positive pseudo-positive almost positive. It is not hard to see that the assertion if α is almost positive then α is pseudo-positive is MP.

17 Implications, Binary Version Definition A binary sequence α is positive, α > 0, if α(n) = 1 for some n. α is almost positive, α 0, if α is not 0. (Equivalently, α is not not positive.) α is pseudo-positive if, for all β, either i) β 0 (i.e. n (β(n) = 1)) or ii) α β (i.e. n (α(n) = 1 k n β(k) = 0)). Easily, positive pseudo-positive almost positive. It is not hard to see that the assertion if α is almost positive then α is pseudo-positive is MP. WMP: If α is pseudo-positive then α is positive: β ( n β(n) = 1 n(α(n) = 1 β(n) = 0)) n α(n) = 1

18 LEM LPO MP WMP WLEM.. Th. 3. [1,8,10] WLPO WKL Th. 13, see also [10] [3] LLPO. Th. 5 MP. [15] or Th. 12 WLEM n LLPO n MP n WLEM n+1 LLPO n+1 MP n n MP n Th. 11 Th. 4, see also [9].. WLEM ω Th. 10 Th. 9 LLPO ω MP ω

19 Proof 1 Theorem Over IZF, WLEM does not imply WMP. Proof. Let f : V M be an elementary embedding of the universe of sets V into a model M with non-standard integers.

20 Proof 1 Theorem Over IZF, WLEM does not imply WMP. Proof. Let f : V M be an elementary embedding of the universe of sets V into a model M with non-standard integers. Consider the Kripke model: M V.

21 Proof 2 Theorem Over IZF, WLEM does not imply WMP. Proof. To get WMP to be false, iterate:. M 2 M 1 V.

22 Proof 3 Theorem Over IZF, WLEM does not imply WMP. Proof. Same picture:. M 2 M 1 V. Only this time, sets are allowed to change once. They have to settle down by the node after the one where they first appear.

23 Proof 4 Theorem Over IZF, WLEM does not imply WMP. Proof. To get DC to hold, use a topological model: Let X be ω { } and U be a non-principal ultrafilter on ω. Take the discrete topology on ω, and a neighborhood of to be { } u where u U.

24 Proof 4 Theorem Over IZF, WLEM does not imply WMP. Proof. To get DC to hold, use a topological model: Let X be ω { } and U be a non-principal ultrafilter on ω. Take the discrete topology on ω, and a neighborhood of to be { } u where u U. WLEM: Consider X 0 := {k ω : {k} A} and X 1 := {k ω : {k} A}.

25 Proof 4 Theorem Over IZF, WLEM does not imply WMP. Proof. To get DC to hold, use a topological model: Let X be ω { } and U be a non-principal ultrafilter on ω. Take the discrete topology on ω, and a neighborhood of to be { } u where u U. WLEM: Consider X 0 := {k ω : {k} A} and X 1 := {k ω : {k} A}. WMP: Consider α such that {k} α(n) = 1 iff k = n.

26 Proof 4 Theorem Over IZF, WLEM does not imply WMP. Proof. To get DC to hold, use a topological model: Let X be ω { } and U be a non-principal ultrafilter on ω. Take the discrete topology on ω, and a neighborhood of to be { } u where u U. WLEM: Consider X 0 := {k ω : {k} A} and X 1 := {k ω : {k} A}. WMP: Consider α such that {k} α(n) = 1 iff k = n. No neighborhood of forces α to be positive.

27 Proof 4 Theorem Over IZF, WLEM does not imply WMP. Proof. To get DC to hold, use a topological model: Let X be ω { } and U be a non-principal ultrafilter on ω. Take the discrete topology on ω, and a neighborhood of to be { } u where u U. WLEM: Consider X 0 := {k ω : {k} A} and X 1 := {k ω : {k} A}. WMP: Consider α such that {k} α(n) = 1 iff k = n. No neighborhood of forces α to be positive. To see that α is pseudo-positive, given β, consider X 0 := {k ω : {k} n β(n) = 1} and X 1 := {k ω : {k} n β(n) = 0}.

28 Proof 5 Theorem Over IZF, WLEM does not imply WMP. Proof. To get DC and WMP, iterate. Let T be ω <ω, the nodes of the countably branching tree.

29 Proof 5 Theorem Over IZF, WLEM does not imply WMP. Proof. To get DC and WMP, iterate. Let T be ω <ω, the nodes of the countably branching tree. A basic open set O contains a unique shortest node σ O, and, for all σ O, {n σ n O} U.

30 Theorem Over IZF, WMP does not imply MP. Proof. Take the topological model over the rationals Q. For MP, let O n be a nested sequence of irrational intervals with intersection {r}. Let α(n) = 0 = O n.

31 Theorem Over IZF, WMP does not imply MP. Proof. Take the topological model over the rationals Q. For MP, let O n be a nested sequence of irrational intervals with intersection {r}. Let α(n) = 0 = O n. For WMP, suppose O β( n (β(n) = 1) n (α(n) = 1 β(n) = 0)).

32 Theorem Over IZF, WMP does not imply MP. Proof. Take the topological model over the rationals Q. For MP, let O n be a nested sequence of irrational intervals with intersection {r}. Let α(n) = 0 = O n. For WMP, suppose O β( n (β(n) = 1) n (α(n) = 1 β(n) = 0)). For r O, let O n decide α(n) (r O). Worst case: O n α(n) = 0.

33 Theorem Over IZF, WMP does not imply MP. Proof. Take the topological model over the rationals Q. For MP, let O n be a nested sequence of irrational intervals with intersection {r}. Let α(n) = 0 = O n. For WMP, suppose O β( n (β(n) = 1) n (α(n) = 1 β(n) = 0)). For r O, let O n decide α(n) (r O). Worst case: O n α(n) = 0. WLOG r is the left endpoint of O n. Let β(n) = 0 = O n (r, ). Then β contradicts the hypothesis on O.

34 Theorem Over IZF, WKL does not imply WLPO. Proof. 0 1 V 0 V 1 f,0 f,1 V V 0 the standard V, V 1 ω-non-standard.

35 Theorem Over IZF, WKL does not imply WLPO. Proof. 0 1 V 0 V 1 f,0 f,1 V V 0 the standard V, V 1 ω-non-standard. For WKL, if at Tr is an infinite tree, consider a non-standard branch.

36 Theorem Over IZF, WKL does not imply WLPO. Proof. 0 1 V 0 V 1 f,0 f,1 V V 0 the standard V, V 1 ω-non-standard. For WKL, if at Tr is an infinite tree, consider a non-standard branch. For WLPO, let α(n) be 0 for all standard n and 1 for some non-standard n.

37 Matt Hendtlass and Robert Lubarsky, Separating Fragments of WLEM, LPO, and MP, to appear (hopefully), available at Hajime Ishihara, Constructive reverse mathematics: compactness properties, in L. Crosilla and P. Schuster, eds., From Sets and Types to Topology and Analysis: Towards Practicable Foundations for Constructive Mathematics, Oxford University Press, p , Ulrich Kohlenbach, Applied Proof Theory: Proof Interpretations and their Use in Mathematics, Springer, Mark Mandelkern, Constructively complete finite sets, Zeitschr. f. math. Logic und Grundlagen d. Math. 34, p , Fred Richman, Polynomials and linear transformations, Linear Algebra Appl. 131, p , 1990.

Constructive (functional) analysis

Constructive (functional) analysis Hajime Ishihara School of Information Science Japan Advanced Institute of Science and Technology (JAIST) Nomi, Ishikawa 923-1292, Japan Proof and Computation, Fischbachau,