On The Model Of Hyperrational Numbers With Selective Ultrafilter
aa r X i v : . [ m a t h . L O ] A p r MSC 03H05
On The Model Of Hyperrational NumbersWith Selective Ultrafilter
A. GrigoryantsMoscow State University Yerevan BranchApril 6, 2020
Abstract
In standard construction of hyperrational numbers using ultra-power we assume that the ultrafilter is selective. It makes possible toassign real value to any finite hyperrational number. So, we can con-sider hyperrational numbers with selective ultrafilter as extension oftraditional real numbers. Also proved the existence of strictly mono-tonic or stationary representing sequence for any hyperrational num-ber.
Keywords: hyperrational number, selective ultrafilter, non-standardanalysis, ultrapower
We use standard set theoretic notation (see [3]). Let us give somewell-known definition for convenience.
Partition of a set S is a pairwise disjoint family { S i } i ∈ I of nonemptysubsets such that S i ∈ I S i = S .Let F ⊂ ω be non-principal ultrafilter on ω . We’ll call elementsin F big subsets (relative to F ) and elements not in F small subsets (relative to F ). F is called selective ultrafilter if for every partition { S n } n ∈ ω of ω into ℵ pieces such that S n / ∈ F for all n ( small partition ) thereexists B ∈ F ( big selection ) such that B ∩ S n is singleton for all n ∈ ω . Equivalently, F is selective if for every function f : ω → ω such hat f − ( i ) is small for every i , there exists a big subset B such thatrestriction f | B is injective.The restriction of F to a big subset B ⊂ ω defined as F B = { J ∩ B | J ∈ F } ⊂ B is selective ultrafilter on B .Continuum hypothesis (CH) implies existence of selective ultrafil-ters due to Galvin [3, theor. 7.8]. The result of Shelah [4] shows thatexistence of selective ultrafilters is unprovable in ZFC. So, we continuewith ZFC & CH to ensure the existence of selective ultrafilter.Let [ S ] k = { X ⊂ S : | X | = k } is the set of all subsets of S thathave exactly k elements. If { X i } i ∈ I is a partition of [ S ] k then a subset H ⊂ S is homogeneous for the partition if for some i : [ H ] k ⊂ X i .The following fact is special case of Kunen’s theorem proven in [5,theor. 9.6]: Theorem 1.
An ultrafilter on ω is selective if and only if for everypartition of [ ω ] into two pieces there is a big homogeneous set. Due to this theorem selective ultrafilters are also called Ramseyultrafilters. We’ll give simplified proof of theorem 1.We say that filter F ⊂ ω is normal if for any collection { A i } i ∈ ω ⊂ F there exists B ∈ F such that for any i, j ∈ B : i < j = ⇒ j ∈ A i .Equally, we can say that F is normal if there exists I ∈ F such thatabove definition holds for any collection { A i } i ∈ I ⊂ F . Indeed we canexpand given collection adding A k = ω for k / ∈ I , then apply definitionto { A i } i ∈ ω and intersect obtained B with I .We continue with fixed selective ultrafilter F on ω .Let us denote Q ∗ = Q ω / ∼ F the quotient of Q ω = { ( x i ) i ∈ ω | ∀ i : x i ∈ Q } by the following equivalence relation:( x i ) ∼ F ( y i ) ⇔ { i ∈ I | x i = y i } ∈ F . This is well-known ultrapower construction widely used in model the-ory and in Robinson’s non-standard analysis (see [1], [2]). We onlyadded the property of selectivity to F . So, we call elements of Q ∗ hyper-rational numbers . There is natural embedding ι : Q → Q ∗ where ι ( q )is the equivalence class of constant sequence ( q, q, . . . , q, . . . ). We’llidentify Q with ι ( Q ). Also Q ∗ satisfies the transfer principle. So, alltrue first order statements about Q are also valid in Q ∗ . In particular, Q ∗ is ordered field.We call element x ∈ Q ∗ infinitely large if | x | > | q | for all q ∈ Q , infinitesimal if | x | < | q | for all q ∈ Q . Otherwise x is called finite . Wewrite x < ∞ if x is finite or infinitesimal.We use short notation x n for some representative ( x n ) n ∈ ω of equiv-alence class x = [( x n ) n ∈ ω ] ∈ Q ∗ . So, we can make arbitrary changesto sequence x n on arbitrary small set without changing appropriate ∈ Q ∗ . Allowing some inaccuracy we’ll talk about hyperrational num-ber x = x n . Let us call subsequence x n k = x J of x n big subsequence if the subset J = { n k : k ∈ ω } of indexes is big.The hyperrational number x = x n is infinitely large if and only ifany big subsequence x n k is unbounded, is infinitesimal if and only if1 /x is infinitely large. Theorem 2.
For any x = x n ∈ Q ∗ , there exists big subsequence x n k ,which is strictly increasing or strictly decreasing or stationary. Thecases are mutually exclusive. In case of x < ∞ the subsequence is fun-damental and any two such subsequences x n k and x m k are equivalentin traditional metric sense lim k →∞ ( x n k − x m k ) = 0Hyperrational numbers given by increasing (decreasing) sequenceswe call left ( right ) numbers. Hyperrational numbers given by station-ary sequences are exactly rational numbers.Let Q ∗ fin = { x ∈ Q ∗ | x < ∞} be the set of all finite hyperrationalnumbers and infinitesimals. Theorem 3.
The set Q ∗ fin is local ring, whose unique maximal idealis the set of infinitesimals I . The factor ring of Q ∗ fin modulo I is thefield isomorphic to the field of real numbers: Q ∗ fin / I ≃ R We give the proof of the theorem 1 which is a bit more simple thanthe one from [5] in part of implication ”selective ⇒ normal”. Proof of theorem 1.
We use following proof schema: selective ⇒ nor-mal ⇒ Ramsey ⇒ selective.selective ⇒ normal. Let { A i } i ∈ ω ⊂ F be arbitrary collection of bigsets. We can assume with no loss of generality that ∀ i ∈ A k : i > k ,because we can replace A k with big subsets A ′ k = { i ∈ A k | i > k } . Letus define a mapping f : ω → ω, f ( i ) = min { j ∈ ω | i / ∈ A j } . Thus f ( i ) ≤ i because i / ∈ A i . Obviously, f − ( j ) ∩ A j = ∅ and, so, sets f − ( j ) are small for all j . Then there exists big set B such that f isinjective on B .We’ll construct big set A with the property: ∀ i, j ∈ A : i < j ⇒ i < f ( j ) ≤ j . Such a set satisfies the conditions of the statement. ow let us construct subsets P k of B as follows: m = f ( b ) = min f ( B ) , P = { b } ,S k = { s ∈ f ( B ) | s > max k − [ i =0 P i } , k ≥ ,m k = f ( b k ) = min S k , k ≥ ,P k = { b ∈ B | m k − < f ( b ) ≤ m k } , k ≥ P k are finite because f | B is injective. Obviously, ∪ ki =0 P i = { b ∈ B | f ( b ) ≤ m k } . All S k are infinite because f ( B ) is infinite. Thus S k = ∅ and all m k are correctly defined. Note that S k +1 ⊆ S k and so m k ≤ m k +1 for all k .In fact m k = f ( b k ) ≤ b k < m k +1 because b k ∈ P k and m k +1 > max ∪ ki =0 P i . So, the sequence m k is strictly increasing. The sets P k are not empty because at least b k ∈ P k .Now if y ∈ P k +2 then max( ∪ ki =0 P i ) < m k +1 < f ( y ) and so x
0. So, i < f ( j ) ≤ j and j ∈ A i .normal ⇒ Ramsey. Let [ ω ] = P ` Q be some partition of [ ω ] .We consider following subsets of ω : P i = { j ∈ ω | { i, j } ∈ P and j > i } Q i = { j ∈ ω | { i, j } ∈ Q and j > i } Obviously, ω = P i ` Q i ` { , . . . , i } for all i . So, for fixed i oneand only one of P i and Q i is big. Let B = { i ∈ ω | P i ∈ F } and C = { i ∈ ω | Q i ∈ F } . One of B and C is big. Let it be B . So, wehave family { P i } i ∈ B of big sets. By the definition of normal ultrafilterthere exists big set A such that for any two elements i < j from A wehave j ∈ P i which means { i, j } ∈ P . So, A is homogeneous.Ramsey ⇒ selective. Let { S i } i ∈ ω be a small partition of ω . Con-sider Q = {{ i, j } ∈ [ ω ] | ∃ k : i ∈ S k and j ∈ S k } and P = ω − Q .There exists big subset H ⊂ ω such that [ H ] ⊂ Q or [ H ] ⊂ P . But[ H ] ⊂ Q implies H ⊂ S k for some k which is impossible because S k is small. Thus, [ H ] ⊂ P and the intersection H ∩ S k can not containmore than one element for any k . We can add elements to H if someof the intersections are empty. So, H is the desired big selection. Lemma 1.
For any injection π : ω → ω there exists big subset B suchthat π | B is increasing. roof. We define the partition [ ω ] = P ` Q as follows P = {{ i, j } ∈ [ ω ] | i < j and π ( i ) < π ( j ) } and Q = [ ω ] − P . There exists homo-geneous big set B for this partition. But [ B ] can not be subset of Q because there is no infinitely decreasing sequences in ω . So, [ B ] ⊂ P and π | B is increasing. Proof of theorem 2.
Let x = x n ∈ Q ∗ be arbitrary hyperrational num-ber. First of all let us consider the equivalence relation on ω : n ∼ k if and only if x n = x k . If the partition corresponding to this relationhave big subset then there is big stationary subsequence of x n .Otherwise, we consider D ⊂ ω be big selection from this partition.So, for k, n ∈ D if k = n then x k = x n . For any k ∈ Z we definesubsets I k = { n ∈ D | x n ∈ ( k, k + 1] } . It is clear that choosingnonempty subsets I k we get the partition of D . If this partition issmall then there exists big selection B . One and only one of two sets B = { n ∈ B | x n > } and B = { n ∈ B | x n < } is big. Note thatthe set { x n | n ∈ B i } for big B i has order type ω in case of B and ω ∗ in case of B .Otherwise, I k is big for some k . Thus, x I k is big bounded subse-quence and number x is finite or infinitesimal. Only one of two sets E = { n ∈ B | x n < x } and E = { n ∈ B | x n > x } is big.Let show that if E is big then there exists big subset B ⊂ E such that { x n | n ∈ B } has order type ω and x B is fundamental.For this purpose let us define the partition of E as follows: J s = { i ∈ E | x − s ≤ x i < x − s +1 } . J s subsets are small for all s because if J s is big for some s then the number x ′ given by x J s isequal to x but on other hand x ′ < x . Contradiction. Thus, we canget big selection B from the partition { J s } s<ω . The sequence x B isobviously fundamental and the set { x n | n ∈ B } has order type ω .Similarly If E is big we can get big subset B ⊂ E such that thesequence x B is fundamental and the set { x n | n ∈ B } has order type ω ∗ . If x K and x L are big fundamental subsequences then x K ∩ L is bigfundamental subsequence of both x K and x L . Thus, x K and x L areequivalent in traditional metric sense.Let C be the only big subset from subsets B i and X = { x n } n ∈ C ⊂ Q subset of sequence elements. We define injection π : ω → ω asfollows: π ( n ) = ( index of min( X − { x π (1) , . . . , x π ( n − } ) , if C = B or C = B index of max( X − { x π (1) , . . . , x π ( n − } ) , if C = B or C = B Thus, π is descending or ascending ordering of X and for any i < j we have x π ( i ) < x π ( j ) in first case and x π ( i ) > x π ( j ) in second case. ccording to lemma 1 there exists big subset E ⊂ ω such that π E isincreasing and subsequence x E ∩ C is monotonic.We call ν ( x ) ∈ R the value of number x ∈ Q ∗ fin . Proof of theorem 3.
Let us define mapping ν : Q ∗ fin → R . For any x = x n ∈ Q ∗ fin we set ν ( x ) equal to limit of some big fundamentalsubsequence of x n . This limit is uniquely defined as follows from thetheorem 2. It is easy to see that ν is epimorphism of Q -algebras andker ν = I is the ideal of all infinitesimals in Q ∗ fin . All elements ofcompliment of I in Q ∗ fin are invertible. Thus, I is only maximal idealin local ring Q ∗ fin and Q ∗ fin / I ≃ R . References [1] Robinson, Abraham “Non-standard analysis” , Proc. Roy. Acad.Sci. Amst., ser. A 64 (1961), 432-440[2] Robinson, Abraham “Non-standard analysis” , Princeton Univer-sity Press, 1996, ISBN: 978-0-691-04490-3[3] Jech, Thomas “Set theory” , Springer, 2006, ISBN 3-540-44085-2[4] Wimmers, Edward (March 1982), “The Shelah P-point indepen-dence theorem” , Israel Journal of Mathematics, Hebrew Univer-sity Magnes Press, 43 (1): 28–48, doi:10.1007/BF02761683[5] Comfort W. W., Negrepontis S., “The theory of ultrafilters” ,Springer-Verlag Berlin Heidelberg, 1974, ISBN-10: 0387066047,Springer-Verlag Berlin Heidelberg, 1974, ISBN-10: 0387066047