An infinity which depends on the axiom of choice
Applied Mathematics and Computation 218 (2012) 8196–8202
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
An infinity which depends on the axiom of choice Vladimir Kanovei ⇑,1, Vassily Lyubetsky 2 Institute for the information transmission problems, Bolshoy Karetny Per. 19, Moscow 127994, Russia
a r t i c l e
i n f o
Keywords: Pantachy Axiom of choice The Solovay model
a b s t r a c t In the early years of set theory, Du Bois Reymond introduced a vague notion of infinitary pantachie meant to symbolize an infinity bigger than the infinity of real numbers. Hausdorff reformulated this concept rigorously as a maximal chain (a linearly ordered subset) in a partially ordered set of certain type, for instance, the set NN under eventual domination. Hausdorff proved the existence of a pantachy in any partially ordered set, using the axiom of choice AC. We show in this note that the pantachy existence theorem fails in the absense of AC, and moreover, even if AC is assumed, hence pantachies do exist, one may not be able to come up with an individual, effectively defined example of a pantachy. 2011 Elsevier Inc. All rights reserved.
1. Introduction Linear order relations, which typically appear in conventional mathematics, are countably cofinal, that is, they admit countable strictly increasing cofinal subsequences. In fact every Borel (as a set of pairs) linear order on a subset of a Polish space is countably cofinal: see, e.g. [11]. Uncountably cofinal orders were introduced in mathematics, in the form of partial rather than linear orders, by Du Bois Reymond. The rate of growth partial order 6RG is defined on positive real functions so that f 6RG g iff the limit lim x!þ1 gðxÞ exists and fðxÞ is >0. This ordering of functions was known long before Du Bois Reymond, but he was the first who considered 6RG in [1] as a relation on the whole totality of positive real functions. He also proved in [1] that the ordering 6RG is great deal nonseparable: in particular, for any countable collection ffn gn2N of positive real functions there is a function f satisfying fn 0. This was the first application of the diagonal method in mathematics.) Somewhat later, Du Bois Reymond published a monograph [2], with a mixed mathematical and philosophical content, where he stipulated that the totality of all real functions ordered by 6RG, which he called the infinitary pantachy, might serve as an extension of the continuum of real numbers, where infinitesimal and infinitely large quantities coexist with usual reals (corresponding to constant functions), thus manifesting a sort of infinity which exceeds the infinity of the real continuum. This concept was met with mixed reception among contemporary mathematicians. In particular, Hausdorff [7,8] noted that obvious existence of 6RG-incomparable functions makes the infinitary pantachy rather useless in the role of an extended analytic domain (see more on controversies around Du Bois Reymond’s approach in [5]). Instead, Hausdorff suggested to consider maximal linearly ordered sets of functions (or infinite real sequences, that can be ordered the same way), in the sense of 6RG or any other similar order based on the comparison of behaviour of functions or sequences at infinity. He called such maximal linearly ordered sets pantachies. ⇑ Corresponding author. 1 2
E-mail addresses: [email protected] (V. Kanovei), [email protected] (V. Lyubetsky). Partial financial support of RFBR and the University of Bonn acknowledged. Partial financial support of RFBR acknowledged.
0096-3003/$ - see front matter 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2011.05.003
V. Kanovei, V. Lyubetsky / Applied Mathematics and Computation 218 (2012) 8196–8202
8197
Hausdorff [7,8] proved the existence of a pantachy in any partially ordered set. This result was one of the earliest explicit applications of the axiom of choice AC. The method of one of two Hausdorff’s pantachy existence proofs is known nowadays as the maximality principle. Typically for the AC-based existence proofs, Hausdorff’s pantachy existence proof did not produce anything near a concrete, individual, effectively defined example of a pantachy in the 6RG ordered set of real functions or in any partial order of the same kind. Haudorff writes in [7,p. 110]: Since the attempt to actually legitimately construct a pantachy seems completely hopeless, it would now be a matter of gathering information . . . about the order type of any pantachy . . .3 Working in this direction, Hausdorff proved, in particular, that any pantachy is uncountably cofinal, uncountably coinitial, and has no (x, x⁄)-gaps – hence, is extremely nonseparable, a type of infinity rather uncommon for mathematics of the early 1900s. Yet those studies left open the major problem of effective existence of pantachies. One may ask: (1) whether the pantachy existence can be established not assuming the axiom of choice AC, (2) whether, even assuming AC, one can actually define an individual example of a pantachy. Advances in modern set theory allow us to answer both questions in the negative, both for the 6RG-ordering of positive functions and for a variety of similar partial orderings. This is the main result of this paper, and it supports Haudorff’s observation cited above. The result is not unexpected. The unexpected feature is that we will have to apply two difficult special results in set theory related to Solovay’s models (Propositions 12 and 13), since the basic technique of Solovay’s models does not seem to be sufficient in this case. The negative answer we obtain is a motivation for the title of the paper: pantachies in the 6RG-ordering of positive real functions is the type of infinity which depends on the axiom of choice!
2. Preliminaries We precede the formulation of our main results with several definitions and notational comments. First of all, we adjust to modern terminology related to partial and linear orderings. Definition 1. A partial quasi-order, PQO for brevity, is a binary relation 6 satisfying x 6 y ^ y 6 z ) x 6 z (transitivity) and x 6 x (reflexivity) on its domain. In this case, an associated equivalence relation and an associated strict partial order < are defined, on the same domain, so that
x y iff x 6 y ^ y 6 x and x < y iff x 6 y ^ y i x: If a PQO 6 also satisfies the antisymmetry condition x 6 y ^ y 6 x ) x = y (which is not assumed, generally speaking) then it is called a partial order, PO for brevity. Thus, a PQO is a PO iff the associated equivalence relation is the equality. A PQO is linear, LQO for brevity, if we have x 6 y _ y 6 x for all x, y in its domain. A linear order, or LO, is any LQO which satisfies the same antisymmetry condition x 6 y ^ y 6 x ) x = y. An LQO hX; 6i (meaning: X is the domain of 6) is of countable cofinality iff there is a set Y # X, at most countable and cofinal in X, that is, if x belong to X then there exists an element y 2 Y such that x 6 y. In this case, we also say that X is countably 6-cofinal. h For instance, if X has a 6-largest element x then X is countably cofinal: indeed, take Y = {x}. S The set 2<x1 ¼ n<x1 2n consists of all transfinite binary sequences of length <x1, and if n < x1 then 2n is the set of all binary sequences of length exactly n. By
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
An infinity which depends on the axiom of choice Vladimir Kanovei ⇑,1, Vassily Lyubetsky 2 Institute for the information transmission problems, Bolshoy Karetny Per. 19, Moscow 127994, Russia
a r t i c l e
i n f o
Keywords: Pantachy Axiom of choice The Solovay model
a b s t r a c t In the early years of set theory, Du Bois Reymond introduced a vague notion of infinitary pantachie meant to symbolize an infinity bigger than the infinity of real numbers. Hausdorff reformulated this concept rigorously as a maximal chain (a linearly ordered subset) in a partially ordered set of certain type, for instance, the set NN under eventual domination. Hausdorff proved the existence of a pantachy in any partially ordered set, using the axiom of choice AC. We show in this note that the pantachy existence theorem fails in the absense of AC, and moreover, even if AC is assumed, hence pantachies do exist, one may not be able to come up with an individual, effectively defined example of a pantachy. 2011 Elsevier Inc. All rights reserved.
1. Introduction Linear order relations, which typically appear in conventional mathematics, are countably cofinal, that is, they admit countable strictly increasing cofinal subsequences. In fact every Borel (as a set of pairs) linear order on a subset of a Polish space is countably cofinal: see, e.g. [11]. Uncountably cofinal orders were introduced in mathematics, in the form of partial rather than linear orders, by Du Bois Reymond. The rate of growth partial order 6RG is defined on positive real functions so that f 6RG g iff the limit lim x!þ1 gðxÞ exists and fðxÞ is >0. This ordering of functions was known long before Du Bois Reymond, but he was the first who considered 6RG in [1] as a relation on the whole totality of positive real functions. He also proved in [1] that the ordering 6RG is great deal nonseparable: in particular, for any countable collection ffn gn2N of positive real functions there is a function f satisfying fn 0. This was the first application of the diagonal method in mathematics.) Somewhat later, Du Bois Reymond published a monograph [2], with a mixed mathematical and philosophical content, where he stipulated that the totality of all real functions ordered by 6RG, which he called the infinitary pantachy, might serve as an extension of the continuum of real numbers, where infinitesimal and infinitely large quantities coexist with usual reals (corresponding to constant functions), thus manifesting a sort of infinity which exceeds the infinity of the real continuum. This concept was met with mixed reception among contemporary mathematicians. In particular, Hausdorff [7,8] noted that obvious existence of 6RG-incomparable functions makes the infinitary pantachy rather useless in the role of an extended analytic domain (see more on controversies around Du Bois Reymond’s approach in [5]). Instead, Hausdorff suggested to consider maximal linearly ordered sets of functions (or infinite real sequences, that can be ordered the same way), in the sense of 6RG or any other similar order based on the comparison of behaviour of functions or sequences at infinity. He called such maximal linearly ordered sets pantachies. ⇑ Corresponding author. 1 2
E-mail addresses: [email protected] (V. Kanovei), [email protected] (V. Lyubetsky). Partial financial support of RFBR and the University of Bonn acknowledged. Partial financial support of RFBR acknowledged.
0096-3003/$ - see front matter 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2011.05.003
V. Kanovei, V. Lyubetsky / Applied Mathematics and Computation 218 (2012) 8196–8202
8197
Hausdorff [7,8] proved the existence of a pantachy in any partially ordered set. This result was one of the earliest explicit applications of the axiom of choice AC. The method of one of two Hausdorff’s pantachy existence proofs is known nowadays as the maximality principle. Typically for the AC-based existence proofs, Hausdorff’s pantachy existence proof did not produce anything near a concrete, individual, effectively defined example of a pantachy in the 6RG ordered set of real functions or in any partial order of the same kind. Haudorff writes in [7,p. 110]: Since the attempt to actually legitimately construct a pantachy seems completely hopeless, it would now be a matter of gathering information . . . about the order type of any pantachy . . .3 Working in this direction, Hausdorff proved, in particular, that any pantachy is uncountably cofinal, uncountably coinitial, and has no (x, x⁄)-gaps – hence, is extremely nonseparable, a type of infinity rather uncommon for mathematics of the early 1900s. Yet those studies left open the major problem of effective existence of pantachies. One may ask: (1) whether the pantachy existence can be established not assuming the axiom of choice AC, (2) whether, even assuming AC, one can actually define an individual example of a pantachy. Advances in modern set theory allow us to answer both questions in the negative, both for the 6RG-ordering of positive functions and for a variety of similar partial orderings. This is the main result of this paper, and it supports Haudorff’s observation cited above. The result is not unexpected. The unexpected feature is that we will have to apply two difficult special results in set theory related to Solovay’s models (Propositions 12 and 13), since the basic technique of Solovay’s models does not seem to be sufficient in this case. The negative answer we obtain is a motivation for the title of the paper: pantachies in the 6RG-ordering of positive real functions is the type of infinity which depends on the axiom of choice!
2. Preliminaries We precede the formulation of our main results with several definitions and notational comments. First of all, we adjust to modern terminology related to partial and linear orderings. Definition 1. A partial quasi-order, PQO for brevity, is a binary relation 6 satisfying x 6 y ^ y 6 z ) x 6 z (transitivity) and x 6 x (reflexivity) on its domain. In this case, an associated equivalence relation and an associated strict partial order < are defined, on the same domain, so that
x y iff x 6 y ^ y 6 x and x < y iff x 6 y ^ y i x: If a PQO 6 also satisfies the antisymmetry condition x 6 y ^ y 6 x ) x = y (which is not assumed, generally speaking) then it is called a partial order, PO for brevity. Thus, a PQO is a PO iff the associated equivalence relation is the equality. A PQO is linear, LQO for brevity, if we have x 6 y _ y 6 x for all x, y in its domain. A linear order, or LO, is any LQO which satisfies the same antisymmetry condition x 6 y ^ y 6 x ) x = y. An LQO hX; 6i (meaning: X is the domain of 6) is of countable cofinality iff there is a set Y # X, at most countable and cofinal in X, that is, if x belong to X then there exists an element y 2 Y such that x 6 y. In this case, we also say that X is countably 6-cofinal. h For instance, if X has a 6-largest element x then X is countably cofinal: indeed, take Y = {x}. S The set 2<x1 ¼ n<x1 2n consists of all transfinite binary sequences of length <x1, and if n < x1 then 2n is the set of all binary sequences of length exactly n. By
