Basic Topological Concepts and a Construction of Real Numbers in Alternative Set Theory
aa r X i v : . [ m a t h . L O ] M a y Basic Topological Concepts and a Construction ofReal Numbers in Alternative Set Theory
Kiri Sakahara ∗ Takashi Sato † Abstract
Alternative set theory (AST) may be suitable for the ones who tryto capture objects or phenomenons with some kind of indefiniteness ofa border. While AST provides various notions for advanced mathemati-cal studies, correspondence of them to that of conventional ones are notfully developed. This paper presents basic topological concepts in AST,and shows their correspondence with those of conventional ones, and iso-morphicity of a system of real numbers in AST to that of conventionalones.
To represent a collection of human beings who are Charles Darwin’s ancestors,what kind of mathematical concepts are appropriate? Not a set, wrote Vopˇenkain his
Mathematics in the Alternative Set Theory [5], since it has no clear bound-ary between human beings and apes, both of which are Mr. Darwin’s predeces-sors and in blood relationship within certain steps. But the former is included inthe collection while the latter is not, the border must remain indefinite. To cap-ture such nebulous collections properly, one has to develop alternative concepts,other than sets, admitting this kind of indefiniteness.This example is too peculiar. However, the type of indefiniteness is preva-lent when one tries to put together a huge amount of objects and to repre-sent it by a single mathematical entity. When Vopˇenka set out to rebuild aset-theoretical framework from a different perspective, putting in place this in-definiteness within its framework consistently was one of his main objectives.The resulting system, the alternative set theory(AST for short), enables thiskind of representation possible, especially when one tries to capture objects orphenomenons in the real world.Vopˇenka [5] displays various notions of AST, each of which is fundamental toadvance mathematical studies. While formal aspects of the system are relativelydetailed, correspondence of them to that of conventional ones are not fullydeveloped. For example, basic ideas of topological spaces, such as open classor continuity, are not mentioned. Lack of them may, unfortunately, keep oneswho need them most away from these attractive frameworks only because thoseconcepts they are accustomed to are absent. ∗ Yokohama National University and Kanagawa University, Kanagawa, Japan. † Toyo University, Tokyo, Japan. continua and morphisms are due to Tsujishita [4]. The readers can find rel-atively compact explanation of AST, such as an axiomatic system of AST, inVopˇenka and Trlifajov´a [6] and Sakahara and Sato [3].
Let us start with constructing a number system in accordance with Vopˇenka [5].The class of natural numbers N is defined as: N = (cid:26) x ; ( ∀ y ∈ x ) ( y ⊆ x ) ∧ ( ∀ y, z ∈ x ) ( y ∈ z ∨ y = z ∨ z ∈ y ) (cid:27) , while the class of finite natural numbers FN consists of the numbers representedby finite sets FN = { x ∈ N ; Fin( x ) } in which Fin( x ) means that each subclass of x is a set. The class is said tobe countable iff it is infinite class and any of its initial segment is finite. Oneexample of countable classes is FN since all of its initial segments are finite,while N is uncountable .The class of all integers Z and that of all rational numbers are definedrespectively as: Z = N ∪ {h , a i ; a = 0 } and Q = (cid:26) xy ; x, y ∈ Z ∧ y = 0 (cid:27) . BQ ⊆ Q denotes the class of bounded rational numbers and FQ ⊆ BQ the classof finite rational numbers , i.e., BQ = { x ; ( ∃ i ∈ FN ) ( | x | ≤ i ) } and FQ = (cid:26) xy ; x, y ∈ FN ∧ y = 0 (cid:27) . The arithmetic and orders on Q are defined as same as usual. Real numberswill be defined later.Next, let us proceed to the concepts that are necessary to build topologicalstructures. A class X is a σ -class (a π -class) iff X is the union (the intersection)of a countable sequence of set-theoretically definable classes .The basic properties of these classes are listed in the next theorem. Theorem 1 (The 6th theorem at the Section 5 of Chapter 2 of Vopˇenka [5]) . (1) X is a σ -class iff X is the union of a countable ascending sequence of set-theoretically definable classes. X is a π -class iff X is the intersection of acountable descending sequence of set-theoretically definable classes. A set-theoretically definable class is the class defined of the form { x : ψ ( x ) } in which ψ ( · ) is a set formula, the one which includes set variables only. A universal class V , classes ofnatural numbers N and rationals Q are all set-theoretically definable, while classes of finitenatural numbers FN , finite rationals FQ and bounded rationals BQ are not since Fin is nota set-formula.
2) The union of two π -classes is a π -class. The intersection of two σ -class isa σ -class.(3) The union of a countable sequence of σ -classes is a σ -class. The intersectionof a countable sequence of π -classes is a π -class.(4) If X , Y are σ -classes ( π -classes) then X × Y is a σ -class (a π -class).(5) If X is a σ -class (a π -class) then dom( X ) ≡ { u ; h u, v i ∈ X } is a σ -class (a π -class). A class . = is a π -equivalence iff . = is a π -class and an equivalence relation.A sequence { R n ; n ∈ FN } is a generating sequence of an equivalence . = iff thefollowing conditions hold:(1) For each n , R n is a set-theoretically definable, reflexive, and symmetricrelation.(2) For each n and each x , y , z , h x, y i ∈ R n +1 and h y, z i ∈ R n +1 implies h x, z i ∈ R n ; R = V .(3) . = is the intersection of all the classes R n .If { R n ; n ∈ FN } is a generating sequence of . = then (i) . = is a π -equivalence,(ii) R n +1 ⊆ R n for each n , and (iii) x . = y holds iff h x, y i ∈ R n for each n . It isalso true that every π -equivalence has its generating sequence. Theorem 2 (The first theorem at the Section 1 of Chapter 3 of Vopˇenka [5]) . Each π -equivalence has a generating sequence. An equivalence . = is said to be compact iff for each infinite set u there are x, y ∈ u such that x = y and x . = y . A relation . = is called an indiscernibilityequivalence iff . = is a compact π -equivalence.In the environment of alternative set theory, each indiscernibility equivalencerepresents a topological structure. Let us denote this structure as continuumafter Tsujishita [4]. A continuum is a pair C = h C, . = C i of a set-theoreticallydefinable class C and an indiscernibility equivalence defined on C . C is said tobe the support of the continuum C .It will be investigated how the structures of continuum correspond to thestandard topological concepts in the following sections. Let us fix one continuum C = h C, . = C i . A class X is a figure of C iff X containswith each x all y which satisfies x . = C y .A monad of x is defined asmon C ( x ) = { y ∈ C : y . = C x } . Evidently, mon C ( x ) is a figure for each x . A class X is a figure iff the monadof each element of X is a subclass of X .The figure of X is defined for each X as follows:fig C ( X ) = { y ∈ C : ( ∃ x ∈ X )( x . = C y ) } . Basic properties of figures are given as the following theorem.3 heorem 3 (the second theorem at the Section 2 of Chapter 3 of Vopˇenka [5]) . For each X , Y and x of C we have the following:(1) fig C ( X ) is a figure.(2) X ⊆ Y implies fig C ( X ) ⊆ fig C ( Y ) .(3) If X ⊆ Y and if Y is a figure then fig C ( X ) ⊆ Y .(4) fig C ( X ∪ Y ) = fig C ( X ) ∪ fig C ( Y ) .(5) fig C ( X ) ∩ fig C ( Y ) = ∅ iff fig C ( X ) ∩ Y = ∅ .(6) mon C ( x ) = fig C ( { x } ) . X , Y of C are separable , noted as Sep C ( X, Y ), iff there is a set-theoreticallydefinable class Z of C such that fig C ( X ) ⊆ Z and fig C ( Y ) ∩ Z = ∅ . It is evidentthat Sep C ( X, Y ) is symmetric.The closure of a class X , denoted as cl C ( X ), is defined as followcl C ( X ) = { x ∈ C : ¬ Sep C ( { x } , X ) } . The interior of a class X is given dually as int C ( X ) = C \ cl C ( C \ X ).It is evident that the following equation holds:cl C ( C \ M ) = C \ int C ( M ) , since C \ int C ( M ) = C \ ( C \ cl C ( C \ M )) = cl C ( C \ M ). It is also easy toverify that the following equation holds:cl C ( M ) = C \ int C ( C \ M ) . Each closure cl C ( X ) has the properties illustrated in the next theorem. Theorem 4 (the 5th theorem at the Section 2 of Chapter 3 of Vopˇenka [5]) . For each X , Y of C we have the following:(1) cl C ( X ) is a figure.(2) fig C ( X ) = fig C ( Y ) implies cl C ( X ) = cl C ( Y ) .(3) X ⊆ cl C ( X ) .(4) X ⊆ Y implies cl C ( X ) ⊆ cl C ( Y ) .(5) cl C ( X ∪ Y ) = cl C ( X ) ∪ cl C ( Y ) . It is also helpful to remind the next theorem.
Theorem 5 (the 10th theorem at the Section 2 of Chapter 3 of Vopˇenka [5]) . Let X be a figure of C . Then the following statements are equivalent:(1) X is the figure of a set u .(2) X is a π -class.(3) C \ X is a σ -class. X = cl C ( X ) .(5) C \ X = int C ( C \ X ) . A figure X is closed in C iff it has one (and therefore all) of the properties ofTheorem 5. Contrastively, a class X is open in C iff C \ X is a closed figure in C . It is worth mentioning that not only closed classes but also open classes arefigures. In general, if X is a figure, so too C \ X is. Suppose that if it is not,then for some element y ∈ C \ X , there exists x ∈ mon C ( y ) ∩ X . Since X isa figure, y must be an element of X just because y ∈ mon C ( x ) ⊆ X . It isa contradiction. Open classes are also figures by the same token, since closedclasses are figures.A figure X is said to be clopen in C iff it has one (thus, both) of the propertiesof Theorem 6. Theorem 6 (the 6th Theorem at the Section 3 of Chapter 3 of Vopˇenka [5]) . For each figure X in C , the following are equivalent(1) X is set-theoretically definable(2) X and C \ X are closed figures. The equivalence . = C is said to be totally disconnected iff it has a generatingsequence ( S i ) i ∈ FN such that S i is an equivalence for each i . Theorem 7 (The 10th theorem at the Section 3 of Chapter 3 of Vopˇenka [5]) . The following are equivalent:(1) . = C is set-theoretically definable.(2) mon C ( x ) is a clopen figure for each x ∈ C .(3) The class V / . = C is finite. Now, it is possible to construct a topological space out of a continuum C .Let us check the claim with the next theorem. Theorem 8.
Given a continuum C = h C, . = C i in which C is clopen, the space h C, O i is topological where O is given as O = { int C ( X ) : X ⊆ C } Proof.
It is evident that C ∈ O and ∅ ∈ O .Let X, Y ∈ O . Then, by Theorem 5. there exist x, y ⊆ C which satisfy X = C \ fig C ( x ) and Y = C \ fig C ( y )Then, by (4) of Theorem 3, X ∩ Y = C \ (fig C ( x ) ∪ fig C ( y )) = C \ fig C ( x ∪ y ) . Since fig C ( x ∪ y ) is a π -class, C \ fig C ( x ∪ y ) is σ -class, so that C \ fig C ( x ∪ y ) = X ∩ Y = int C ( X ∩ Y ) by (3) and (5) of Theorem 5, and thus, X ∩ Y ∈ O .5astly, let ( C i ) i ∈ FN be a sequence of elements of O . Then, for each i thereexists a set c i ⊆ C which satisfies C i = C \ fig C ( c i ) . Then, the following equation holds. [ i ∈ FN C i = C \ \ i ∈ FN fig C ( c i )Since T i ∈ FN fig C ( c i ) is a π -class by (3) of Theorem 1, C \ T i ∈ FN fig C ( c i ) is σ -class, so that C \ T i ∈ FN fig C ( c i ) = S i ∈ FN C i ∈ O by exactly the same reasoningas the previous case. A monad mon C ( x ) is said to be a point of C and x its position . A position isnot uniquely determined since each point contains multiple different elementsin general, and each elements are equally qualified to be its position.For any subclass A of C , a point mon C ( x ) is said to be an accumulationpoint of A iff it satisfies that x ∈ cl C ( A \ mon C ( x )) or mon C ( x ) ⊆ cl C ( A \ mon C ( x )) . When it is not but the point of A , it is said to be an isolation point of A .Let (mon C ( a i )) i ∈ τ and ( a i ) i ∈ τ be sequences of points and their positionsof C respectively, and ( R i ) i ∈ FN be a generating sequence of . = C . A sequence(mon C ( a i )) i ∈ FN of C is said to converge to a point mon C ( x ) iff( ∀ k ∈ FN ) ( ∃ i ∈ FN ) ( ∀ j > i ) ( h a j , x i ∈ R k ) . It is simply denoted as lim i ∈ FN mon C ( a i ) = mon C ( x ) if it converges tomon C ( x ).Let Z n ( a ) denote an image of a ∈ C with R n , defined as Z n ( a ) ≡ { x ∈ C : h a, x i ∈ R n } . Then, the next theorem holds.
Theorem 9.
Let . = C be a non totally disconnected indiscernibility equivalenceand A ⊆ C . A point mon C ( a ) is an accumulation point of A iff it is a limit ofsome converging sequence (mon C ( a i )) i ∈ FN consisting only of mutually differentpoints of A .Proof. Let ( R i ) i ∈ FN be a generating sequence of . = C and Z n ( a ) be an imageof a . Since . = C is not totally disconnected, every element R n of its generatingsequence is strictly bigger than . = C , so that Z n ( a ) is also strictly bigger thanmon C ( a ) for all n ∈ FN .Suppose there exists a set-theoretically definable class Z which separatesmon C ( a ) and A \ mon C ( a ), that is, which satisfies mon C ( a ) ⊆ Z and Z ∩ A \ mon C ( a ) = ∅ . Since mon C ( a ) = T i ∈ FN Z i ( a ), there exists j ∈ FN such that Z k ( a ) ⊆ Z for all k > j . Since (mon C ( a i )) i ∈ FN converges to mon C ( a ), there6xists a i ∈ Z k ( a ) \ mon C ( a ) for every k > j . It means Z ∩ A \ mon C ( a ) = ∅ ,since a i ∈ A \ mon C ( a ). It is a contradiction.Conversely, suppose mon C ( a ) is an accumulation point of A . Then, it holdsthat Z n ( a ) ∩ A \ mon C ( a ) = ∅ for all n ∈ FN . Since . = C is not totally dis-connected, there exists a ∈ Z n ( a ) which satisfies a . = a . It is also evidentthat there exists n ∈ FN which satisfies n > n and Z n ( a ) ∩ mon C ( a ) = ∅ ,otherwise a must be indiscernible with a by definition of . = C , and that thereexists a ∈ Z n ( a ) which satisfies a . = a . By repeating it countably many times,we can get a sequence of points of C as (mon C ( a i )) i ∈ FN which converges tomon C ( a ) and its elements are mutually different since each pair of positions a i and a i +1 is separated by set-theoretically definable class Z n i +1 ( a ).This theorem guarantees the well-known property of closed classes as shownin the next theorem. Theorem 10.
Every converging sequence of A has its limits in A iff A is closed.Proof. Let (mon C ( a i )) i ∈ FN be a converging sequence of A . When it has asubsequence consisting only of mutually different points, the limit is an accu-mulation point of A by Theorem 9. Otherwise, the limit is an isolation point.Both limits are included in A since A is closed. Converse is also true.Points of topological structures of AST satisfy the following separabilityproperty. Theorem 11.
Every pair of different points mon C ( x ) = mon C ( y ) is separable.Proof. Since x . = y , there exists n ∈ FN which satisfy h x, y i / ∈ R n . It impliesthat mon C ( x ) ⊆ Z n ( x ) and Z n ( x ) ∩ mon C ( y ) = ∅ . A class X ⊆ C is a neighborhood of a point mon C ( x ) iff x ∈ int C ( X ) . The open class which contains x is called an open neighborhood of x . A com-plete system of neighborhoods V ( x ) for a point mon C ( x ) is the collection of allneighborhoods for the point x . Theorem 12.
Let . = C be a non totally disconnected indiscernibility equivalence, (mon C ( a i )) i ∈ FN be a converging sequence, and V ( x ) be a complete system ofneighborhoods. A point mon C ( a ) is a limit of (mon C ( a i )) i ∈ FN iff for each V ∈ V ( a ) there exists i ∈ FN , { a j : j > i } ⊆ V. Proof.
Let ( R n ) n ∈ FN be a generating sequence of . = C and Z n ( a ) be an imageof a with R n . Then, for every neighborhood V ∈ V ( a ), there exists ℓ ∈ FN which satisfies Z n ( a ) ⊆ V for all n > ℓ . Since (mon C ( a i )) i ∈ FN converges tomon C ( a ), there exists a i n ∈ Z n ( a ) for each Z n ( a ).Conversely, since each Z n ( a ) is a neighborhood of a , or Z n ( a ) ∈ V ( a ), thereexists n i ∈ FN which satisfies { a j : j > n i } ⊆ Z n ( a ) for each n ∈ FN . Itimplies (mon C ( a i )) i ∈ FN converges to mon C ( a ).7 Compactness
A family A of classes is a cover of a class X iff X ⊆ ∪ A . A continuum C is compact iff every open cover of C has its finite subcover. A class X ⊆ C is compact iff every open cover of X has its finite subcover. Theorem 13.
A continuum C is compact iff every countable sequence has aconverging subsequence and its limit in C .Proof. Suppose C is compact and (mon C ( a i )) i ∈ FN has no converging subse-quences. Then, F = { mon C ( a i ) : i ∈ FN } is a closed class by Theorem 10 sinceit has no accumulation points. Let O = C \ F , then, O is open and a i / ∈ O for all i ∈ FN . Since each point is discernible, there exists a sequence of naturals ( n i ) i ∈ FN andset-theoretically definable classes Z n i ( a i ) which are pairwise disjoint. Then, C is covered by the union of the following classes C = O ∪ [ i ∈ FN O i where O i = int C ( Z n i ( a i )) . However, no finite subcover exists. It contradicts with the compactness of C .Conversely, suppose C is not compact and every countable sequence hasconverging subsequences. Let C = S i ∈ FN O i be a countable cover of C but hasno finite subcover. Choose for each n ∈ FN a n / ∈ [ i ∈ n +1 O i . Then, { a i } i ∈ FN has an accumulation point. It implies there exist a strictlyincreasing function F : FN → FN and subsequence (cid:0) a F ( j ) (cid:1) j ∈ FN which satisfieslim j →∞ mon C (cid:0) a F ( j ) (cid:1) = mon C ( a ). Since C = S i ∈ FN O i , there must be k ∈ FN which satisfies mon C ( a ) ⊆ O k . Since (cid:0) a F ( j ) (cid:1) j ∈ FN converges to mon C ( a ), theremust be ℓ ∈ FN which satisfies ℓ > k and a m ∈ O k ⊆ S i ∈ m +1 O i for all m > ℓ .But, a m / ∈ S i ∈ m +1 O i by definition. It is a contradiction.Let us, next, introduce an R -net . A class X ⊆ C is an R -net iff there areno distinct element x, y ∈ X such that h x, y i ∈ R . X is maximal R -net on C iff X ⊆ C and for each y ∈ C there is an x ∈ X such that h x, y i ∈ R .A relation R is an upper bound of an equivalence . = iff R is symmetrical, set-theoretically definable and . = is a subclass of R , i.e., x . = y implies h x, y i ∈ R for each x, y ∈ C . Theorem 14 (the third theorem at the Section 1 of Chapter 3 of Vopˇenka [5]) . Let . = be a compact equivalence and let R be its upper bound. Then there is afinite number n such that, for each R -net X , X is subvalent to n , i.e., there isa one-one mapping of X onto a subclass of n . X - n denotes that X is subvalent to n hereafter.The following theorem implies the equivalence between compactness andbeing closed in an AST environment. 8 heorem 15. A figure X of C is compact iff it is closed.Proof. Suppose that X is compact. Let us choose x ∈ X and y ∈ C \ X arbitrarily. Since X is a figure and every points are separated by Theorem 11,there exists n ∈ FN and set-theoretically definable classes Z n ( x ) and Z n ( y )which are mutually disjoint and separate mon C ( x ) and mon C ( y ). Then, openclasses O n ( x ) = int C ( Z n ( x )) and O n ( y ) = int C ( Z n ( y )) are mutually disjoint.Now, let us fix y ∈ C \ X and choose large enough n ∈ FN so that the followinginclusion holds: X ⊂ [ x ∈ X { O n ( x ) : O n ( y ) ∩ O n ( x ) = ∅} . Since X is compact, there exists a finite subset m of X which covers XX ⊂ [ x ∈ m { O n ( x ) : O n ( y ) ∩ O n ( x ) = ∅} . For each Z n ( x ) is set-theoretically definable, S x ∈ m { Z n ( x ) : Z n ( y ) ∩ Z n ( x ) = ∅} too is set-theoretically definable, so that it separates { y } and X . Since y ischosen arbitrarily, it implies that X is closed.Suppose, conversely, that X is closed but not compact. Then, it has acountable sequence with no converging subsequences. Let us choose one suchsequence of points (mon C ( a i )) i ∈ FN arbitrarily. Now, let X be a maximal R -net which satisfies X - n for some n ∈ FN . Then, there must be at least oneelement x i ∈ X whose image Z i ( x i ) include infinite points for each i ∈ FN , thatis, S j ∈ J i mon C ( a j ) ⊆ Z i ( x i ) where J i is a countable subclass of FN . Let usdefine a function F ( i ) = j which satisfies a j . = x i . Then, (mon C (cid:0) a F ( j ) (cid:1) j ∈ FN isa subsequence of (mon C ( a i )) i ∈ FN and converges to mon C ( a ) ∈ T i ∈ FN Z i ( x i ).It is a contradiction. A set u is connected in C iff for each nonempty proper subset v of u there exist x ∈ v and y ∈ u \ v which satisfy x . = C y . To notify a set u is connected, letus denote Cntd ( u ). A figure X is connected iff for each x, y ∈ X there is aconnected set u ⊆ X which satisfies x, y ∈ u . A continuum C is connected ifffor all figures of C is connected. Theorem 16.
A continuum C is connected iff C has no clopen figures besides C and ∅ .Proof. Suppose C has a clopen figure X , then, C \ X is set-theoretically defin-able, thus, clopen. It contradicts with the C ’s connectedness.Conversely, suppose that C is not connected, then, there exists at least oneclopen figure X . It is a contradiction. Theorem 17.
Let C be a continuum and X ⊆ C be a set-theoretically definableclass. X is connected iff there exists no clopen figure Y in X = h X, . = C ↾ X i which satisfies X ∩ Y = ∅ and X \ Y = ∅ . roof. Suppose there exists a clopen figure Y which satisfies X ∩ Y = ∅ and X \ Y = ∅ . Then, for all x ∈ X \ Y ⊆ C \ Y and y ∈ X ∩ Y , it is satisfied that ¬ ( x . = C y ). It contradicts with X ’s connectedness.Conversely, suppose X is not connected. Then, there exists a pair of points x, y ∈ X which has no subset u ⊆ X which is connected and include them.It implies that { x } and { y } are separable in X , so that there exists a set-theoretically definable figure Y in X which satisfies cl X ( x ) ⊆ Y and cl X ( y ) ∩ Y = ∅ . Since X is set-theoretically definable, both Y ∩ X and X \ Y are alsoset-theoretically definable and nonempty. It is a contradiction. Theorem 18.
Let C be a continuum and X be a closed figure. Then X isconnected iff there exists no pair of closed figures Y and Y such that X ⊆ Y ∪ Y Y ∩ Y ∩ X = ∅ Y ∩ X = ∅ Y ∩ X = ∅ . Proof.
Suppose Y and Y satisfy the condition, there exist mutually disjointsets y , y ⊆ X which satisfy fig C ( y ) = Y ∩ X and fig C ( y ) = Y ∩ X . Sincefig C ( y ) ∩ fig C ( y ) = ∅ , for all a ∈ y and b ∈ y , ¬ ( a . = C b ) holds. It contradictswith X ’s connectedness.Conversely, if X is not connected, there exists mutually disjoint sets y and y which satisfies y ∪ y = x where fig C ( x ) = X and ¬ ( a . = C b ) for all a ∈ y and b ∈ y . Then, Y = fig C ( y ) and Y = fig C ( y ) satisfy all the fourconditions. Corollary 19.
If continuum C has no clopen figures, there exist connected setsfor any pair of sets x, y ∈ C . Let C be a continuum. A metric d is a function d : C × C → Q which satisfies(1) d ( x, y ) ≥ x, y ∈ C ,(2) d ( x, y ) = 0 iff x = y for all x, y ∈ C ,(3) d ( x, y ) = d ( y, x ) for all x, y ∈ C ,(4) for all x, y , and z ∈ C , the inequality d ( x, z ) ≤ d ( x, y ) + d ( y, z ) holds.A pair h C, d i is a metric space where d is a metric of C . A distance between x and y of C is given as d ( x, y ).Let a ∈ C and e > ball of radius e centered at the position a is defined as B ( a ; e ) = [ i ∈ FN (cid:26) x : ( x ∈ C ) ∧ (cid:18) d ( a, x ) < e − i (cid:19)(cid:27) Theorem 20.
There exists an indiscernibility equivalence . = B which makes B ( a ; e ) an open class for any a ∈ C and e ∈ FQ. roof. Let ( R n ) n ∈ FN be a sequence of set-theoretically definable class as R n = (cid:26) h a, b i ∈ C × C : (cid:18) d ( a, b ) < e − n (cid:19) ∨ ( d ( a, b ) > n ) (cid:27) . Then, ∩ n ∈ FN R n forms an indiscernibility equivalence . = B .Let C B be a continuum h C, . = B i . Given a ∈ C and e ∈ FQ , define a class A = { x ∈ C : d ( a, x ) < e } . Then, the ball B ( a ; e ) is given as int C B ( A ). Let R = h Q, . = i be a real continuum where . = is a indiscernibility equivalencedefined as . = ≡ T n ∈ FN R n in which R n = (cid:26) h a, b i ∈ Q × Q : (cid:18) d ( a, b ) < n (cid:19) ∨ ( d ( a, b ) > n ) (cid:27) , and let R = BQ / . = be a class of real numbers.Notice that a real continuum R contains ∞ ≡ mon ( α ) and −∞ ≡ mon ( − α )for α ∈ N \ FN as its elements, while R doesn’t.For each bounded rational q ∈ BQ there exists a unique real mon ( q ) ∈ R .Let us denote such a real simply as q ∈ R . Conversely, for each real r ∈ R thereexists a bounded rational number q ∈ BQ which satisfies r = mon ( q ). Let usalso denote such a rational simply as r ∈ BQ for notational ease, hereafter. Let a, b ∈ R which satisfy a < b . Then, the real intervals ( a, b ], ( a, b ), [ a, b ] and [ a, b )are given respectively as( a, b ] = cl ( { q ∈ Q : a < q < b } ) \ mon ( a )= int ( { q ∈ Q : a < q < b } ) ∪ mon ( b )( a, b ) = int ( { q ∈ Q : a < q < b } )[ a, b ] = cl ( { q ∈ Q : a < q < b } )[ a, b ) = cl ( { q ∈ Q : a < q < b } ) \ mon ( b )= int ( { q ∈ Q : a < q < b } ) ∪ mon ( a )To make sure that this construction of R is really identical with that of realnumbers, let us next examine its characteristics of algebraic structures.Arithmetic operations on R are given as same as that on Q , that is: for all a, b ∈ BQ mon ( a ) + mon ( b ) = mon ( a + b )mon ( a ) · mon ( b ) = mon ( a · b ) . An ordered relation is also given as the same manner, that is: for all a, b ∈ BQ mon ( a ) ≤ mon ( b ) ⇔ a ≤ b. It is evident that ( R, ≤ , + , · ) constitutes an ordered field.11 efinition 21. D ⊆ R is called Archimedean iff for each x ∈ D there exists an n ∈ FN such that x < mon ( n ). Otherwise, D is called non-Archimedean . Theorem 22. R is an Archimedean.Proof. For any given a ∈ R there exists q ∈ BQ such that a = mon ( q ). Since q is bounded there always exists n ∈ FN in which n > q . It implies that a < mon ( n ).Let D ⊆ R . The element ℓ ∈ D is an upper bound of the nonempty class A ⊆ D iff x ≤ ℓ for all x ∈ A . Moreover, if no m ∈ D for which m < ℓ is anupper bound of A , ℓ is said to be the least upper bound of A . Definition 23. D is complete iff every nonempty subclass of D that has anupper bound has a least upper bound. Theorem 24. R is a complete ordered field.Proof. Let us check that R is complete. Let C ⊆ R , C = ∅ and C has an upperbound in R .Let B = { b ∈ Q : mon ( b ) is an upper bound of C } . Since R is Archimedean,for all x ∈ C there exists q ∈ Q and n ∈ FN satisfying that x = mon ( q ) < mon ( n ), so that B = ∅ . Conversely, let A = Q \ B . Then A is not empty toosince for every x ∈ C there exists q ∈ A such that x = mon ( q ).Choose a ∈ A and b ∈ B arbitrarily. Let c = b , and c i +1 = ( c i + a − b i +1 if c i + a − b i +1 ∈ Bc i otherwise for all i ∈ τ + 1 ∈ N \ FN . Then, since c α +1 − c α ≤ a − b α +1 . = 0 for all α ∈ τ \ FN , the equation holds:mon ( c α ) = mon ( c τ ) . Suppose that there exists z ∈ B which satisfies z < c τ and z . = c τ . Then, thereexists ℓ ∈ FN which satisfies a − b ℓ < z − c τ ≤ a − b ℓ +1 . It implies that there exists m ≤ ℓ which satisfies c m + a − b m +1 ∈ B but c m +1 = c m .It contradicts with the way c i is built.Therefore, it is concluded that mon ( c τ ) is the least element of B/ . =, that is,the least upper bound of C .The theorem assures us that the construction of R in AST is isomorphic to,say, R of ZFC .
10 Morphisms
Let C and C be continua, and F : C → C be a function. F is continuous iff for all a, b ∈ C which are mutually indiscernible, that is, a . = C b , F ( a ) and F ( b ) are also indiscernible, F ( a ) . = C F ( b ). Two continuous functions F and12 are indiscernible , denoted simply as F . = G , if for all x ∈ C , F ( x ) . = G ( x )follows .The morphism F between two continua is defined as follows .(1) A morphism from C to C is a monad mon ( F ) denoted simply as [ F ] forsome continuous function F from C to C .(2) If C i ( i = 1 ,
2) are continua, the notation F : C → C means that F is amorphism from C to C .(3) If F is a morphism, then the expression G ∈ F means G . = F in which F = [ F ]. If G ∈ F , we say that the morphism F is represented by G and G represents F .(4) If F and G are morphisms represented respectively by F and G , then theexpression F = G means F . = G .It is essential that morphisms are defined as the monads of continuous function.When a set-theoretically definable function F : C → C has an indiscerniblegap at a position x ∈ C , that is, ¬ ( F ( x ) . = F ( y )) for some y . = x , its value atits point mon C ( x ) cannot be determined uniquely since x . = C y but F ( y ) . = C F ( x ), thus, mon C ( F ( x )) ∩ mon C ( F ( y )) = ∅ .It is also worth mentioning that continuity of F cannot guarantee its mor-phism F ’s continuity , which will be defined later, since it does not guaranteecontinuous change at its value outside its monad. For example, the morphism F : R → R is not continuous. F ( x ) = ( x = 00 otherwisebut any function F which satisfies [ F ] = F is continuous since for each q ∈ Q ,it is satisfied that F ( q ) = 0 except F ( q ) = 1 = F (0) when q . = 0. To capturemorphism’s continuity, stronger conditions are needed.
11 Continuous Morphisms
Let F : C → C be a morphism from C to C . For simplicity, let F denote arepresentative of F = [ F ] hereafter.A morphism F : C → C is continuous iff( ∀ u ⊆ dom F ) (Cntd ( u ) ⇒ Cntd ( F ( u ))) . (1) To be precise, put r i ≡ (cid:26) h f, g i ; ( f, g ∈ Fun ( c , c )) ∧ ( ∀ x ∈ c ) (cid:18) f ( x ) − g ( x ) ≤ i (cid:19)(cid:27) , in which C i ⊆ c i for i = 1 , C is a semiset, otherwise H ( C i ) ⊆ c i for a given similarityendomorphism H : V → D in which D is a semiset (for the definition and its existence seep.111 of Vopˇenka [5]), and . = c c ≡ T i ∈ FN r i .For each pair of functions F, G ∈ Fun ( C , C ), let us denote F . = G iff there exists their (ortheir similar classes’) prolonged sets of functions f, g ∈ Fun ( c , c ) which satisfies F = f ↾ C (or H ( F ) = f ↾ H ( C )), G = f ↾ C (or H ( G ) = g ↾ H ( C )) and f . = c c g . Contrary to the framework of Tsujishita [4], in which the morphisms are not guaranteedto be classes, they are in AST. δ is a motion of a position in a time θ of C , in which θ ∈ N , iffdom( δ ) = θ + 1 and for each α < θ , δ ( θ ) . = δ ( θ + 1). If δ is a motion of aposition then rng( δ ) is called a trace of δ .Then, a morphism F : C → C is continuous iff( ∀ δ ) ((rng ( δ ) ⊆ dom ( F )) ⇒ ( F ( δ ) is a motion of a position of C )) . (2)It is easy to examine these two conditions (1) and (2) are equivalent. Tomake sure that it is true, two theorems of Vopˇenka at the Section 1 of Chapter4 are useful. Theorem 25 (the first Theorem of Vopˇenka [5] at Section 1 of Chapter 4) . Thetrace of a motion of a position is a connected set.
Theorem 26 (the second Theorem of Vopˇenka [5] at Section 1 of Chapter 4) . For each nonempty connected set u there is a motion of a position such that u is the trace of δ . As easily seen by Theorem 25, (2) ⇒ (1) is evident. (1) ⇒ (2) is also evidentby Theorem 26.Let us next examine relationship between motions and sequences. It is easilyverified that within a connected continuum these two concepts coincide. Theorem 27.
Let C be a connected continuum. Then, following two conditionsare equivalent:(1) there exists a motion δ : τ + 1 → C which satisfies δ (0) = a and δ ( τ ) = a ,(2) there exists a sequence (mon C ( a i )) i ∈ FN with its limit lim i ∈ FN mon C ( a i ) =mon C ( a ) .Proof. To show (1) ⇒ (2), arbitrarily choose one position a i ∈ Z i ( a ) ∩ rng ( δ )for each i ∈ FN . Then, the sequence (mon C ( a i )) i ∈ FN converges to mon C ( a )since for each i ∈ FN it has a corresponding position a i satisfying h a i , a i ∈ R i .To show a converse case is straightforward. Since C is connected, betweenany two positions, there exists motions. Thus, motion from a to a alwaysexists.By Teorem 27, it is equivalent to define continuity by way of converging se-quences: a morphism F : C → C is continuous iff for all converging sequences,say lim i ∈ FN (mon C ( a i )) = mon C ( a ), the following holds:lim i ∈ FN mon C ( F ( a i )) = mon C ( F ( a )) . Or equivalently,( ∀ k ∈ FN )( ∃ j ∈ FN )(( h a i , a i ∈ R j ) ⇒ ( h F ( a i ) , F ( a ) i ∈ R k )) . The well known property of continuity described below also holds.
Theorem 28.
Let C = h C , . = i and C = h C , . = i be two continua, C and C be set-theoretically definable class, and F : C → C be a morphism. Then,the following three conditions are equivalent. i) For any open class X of C , F − ( X ) is also an open class of C .(ii) For any closed figure Y of C , F − ( Y ) is also a closed figure of C .(iii) F is continuous.Proof. To prove (i) ⇒ (ii), let us remind that for every X ⊆ C , the followingequation holds. F − ( C \ X ) = C \ F − ( X ) (3)Given that (i) holds and X is open, then F − ( X ) is also open. Since X is openand C is set-theoretically definable, C \ X is closed, so too is C \ F − ( X ).It implies that F − ( C \ X ) is closed by the equation (3).(ii) ⇒ (i) follows by a similar argument.To prove (iii) ⇒ (ii), let Y be a closed figure of C . Then, there exists v ⊆ Y which satisfies fig ( v ) = Y , and fig (cid:0) F − ( v ) (cid:1) is a figure in C . For every a ∈ fig (cid:0) F − ( v ) (cid:1) there exists y ∈ v which satisfies a . = F − ( y ). By continuityof F , a satisfies F ( a ) . = y , which means that F ( a ) ∈ mon ( y ) ⊆ Y , thus, a ∈ F − ( Y ). Consequently, fig (cid:0) F − ( v ) (cid:1) = F − ( Y ), thus, F − ( Y ) is closed.To prove (ii) ⇒ (iii), let u ⊆ C be a connected set of C . Then, for every a ∈ u , the inverse image of a monad of y = F ( a ), that is, F − (mon ( y )) isa closed figure of C . Since y = F ( a ), F − (mon ( y )) contains mon ( a ) as itssubclass. It means that all the indiscernibles of a ∈ u are included in the figure F − (mon ( y )). Thus, F (mon ( a )) ⊆ mon ( F ( a )) follows. Since u is connected,there exists θ ∈ N and a motion δ : θ + 1 → u which satisfies rng ( δ ) = u and δ ( α ) . = δ ( α + 1) for all α ∈ θ + 1. δ traces along indiscernibles, one byone, so does F ( δ ) which means that F ( δ ) is a trace of C . Therefore, F ( u ) isconnected.When morphisms are defined on metric continua, continuity can be defineby a standard ε - δ manner. A morphism both from and to metric continua F : C → C is continuous iff( ∀ e ∈ Q ) ( ∃ d ∈ Q ) ( d > ( d ( x, a )) ⇒ ( e > d ( F ( x ) , F ( a )))) . Lastly, let us claim that continuous morphisms of AST are uniform too,where a morphism F : C → C is said to be uniformly continuous iff( ∀ k ∈ FN )( ∃ j ∈ FN )( ∀ x, y ∈ C )(( h x, y i ∈ R j ) ⇒ ( h F ( x ) , F ( y ) i ∈ R k ))where ( R j ) j ∈ FN and ( R k ) k ∈ FN are generating sequences of . = and . = respec-tively. Theorem 29.
Let F : C → C be a continuous morphism. Then F is alsouniformly continuous.Proof. Suppose F is not uniformly continuous. Then there exists k ∈ FN whichsatisfies for all j ∈ FN there exists some x, y ∈ C ,( h x, y i ∈ R j ) ∧ ( h F ( x ) , F ( y ) i / ∈ R k ) . It implies that x . = y but F ( x ) . = F ( y ). It contradicts with F ’s continuity.15 The paper puts in order (1) a way to define basic concepts of topology in AST,and shows (2) their correspondence with those of conventional ones, and (3)isomorphicity of a system of real numbers in AST to that of conventional one.As it is widely known in model theory, there are many equivalent systemsof real numbers other than R constructed within an axiomatic system of ZFC.One of renowned examples may be that of nonstandard analysis, see Robinson[2] or Davis [1] for more detailed information. Its way to construct a system isessentially the same as AST’s, that is, regarding monads as real numbers.At odds with that popularity, the attempts regarding morphisms as realfunctions in the studies of nonstandard analysis are rarely seen except Tsujishita[4], as far as we know. However, this line of investigations may seem fruitful,especially when one tries to apply them to capture the phenomena in the realworld.One benefit of that rests on its ability enabling us to see continuity of func-tions, or morphisms, from the more natural perspective. Remind us here the ε - δ arguments, which regard continuity of functions indirectly as having no gap in its graph, since it says only that however small the gap is in the range, therealways remains enough space left on the domain of the function, assigning thevalue picked from that domain to the function always resulted in the valuestaying within that range. And the process confirming whether there remainsunchecked gaps in the range left will not end and last forever by definition.However, as we saw in the last section, by contrast, viewing continuity byway of motions enables us to check it directly. The definition requires thatmorphisms F to preserve properties of motions, that is, for any motion δ ,composed of . =-chains with its trace, it must be satisfied that F ( δ ) is also amotion. It simply says that morphism is continuous since its graph is connected.The key difference is that AST has a way to represent direct connectionbetween two objects, enabled by indiscernibility equivalences . =, while ε - δ ar-gument can only judge two are separated, enabled by open sets, which do notguarantee that they are connected in general.This same ability of AST enables the authors to deal with the problem ofe-mail game, that is, how agents can come to know that contents of e-mails areshared, in Sakahara and Sato [3]. They come to know that they share the sameinformation, not because there remains no possibility of not knowing, but theysimply ignore indiscernible differences how many times they confirmed that.They did it by, say, jump . It is the same jump which enable us to judge themotion of points has no gap in its trace.It is also worth pointing out that real functions and their representationsare defined on the same continua. Consequently, results acquired from themreside in the same domain. In contrast, when one works within a nonstandardframework, the system of reals and its extensions reside in different domains.The result concerning, say, external elements, thus, have no room to reside inthe standard domain. They are simply nonexistent.Nevertheless, it is not, of course, an essential problem of nonstandard anal-ysis, since the arguments provided here can also be applied to nonstandardanalysis. It can also be said that the perspective this paper presents benefitsnonstandard analysis too. 16 eferences [1] Martin Davis. Applied nonstandard analysis . Wiley New York, 1977.[2] Abraham Robinson.
Non-standard Analysis . North-Holland Publishing Co.,Amsterdam, 1966.[3] Kiri Sakahara and Takashi Sato. An Alternative Set Model of CognitiveJump. arXiv e-prints , arXiv:1904.00613, Apr 2019.[4] Toru Tsujishita. Alternative Mathematics without Actual Infinity. arXive-prints , arXiv:1204.2193v2, Jun 2012.[5] Petr Vopˇenka.
Mathematics in the Alternative Set Theory . Teubner Verla-gagesellshaft, Leipzig, 1979.[6] Petr Vopˇenka and Kateˇrina Trlifajov´a. Alternative set theory. In Christodou-los A. Floudas and Panos M. Pardalos, editors,