Encoding Sets as Real Numbers Domenico Cantone 1 Alberto Policriti 2 Dept. of Mathematics and Computer Science, University of Catania, Italy Dept. of Mathematics, Computer Science, and Physics, University of Udine, Italy Domenico Cantone, Alberto Policriti Encoding Sets as Real Numbers 1/18
Sets Paul Halmos “Naive Set Theory” Every mathematician agrees that every mathematician must know some set theory; the disagreement begins in trying to decide how much is some. Sets, as they are usually conceived, have elements or members. An element of a set may be a wolf, a grape, or a pidgeon. It is important to know that a set itself may also be an element of some other set. [...] What may be surprising is not so much that sets may occur as elements, but that for mathematical purposes no other elements need ever be considered. Domenico Cantone, Alberto Policriti Encoding Sets as Real Numbers 2/18
Definition (Hereditarily finite sets) HF := � n ∈ N HF n is the collection of all hereditarily finite sets, where � HF 0 := ∅ , HF n + 1 := P ( HF n ) , for n ∈ N . Domenico Cantone, Alberto Policriti Encoding Sets as Real Numbers 3/18
Ackermann 1937 Definition N A ( x ) = Σ y ∈ x 2 N A ( y ) Domenico Cantone, Alberto Policriti Encoding Sets as Real Numbers 4/18
Ackermann 1937 Definition N A ( x ) = Σ y ∈ x 2 N A ( y ) Example 1 1 0 1 0 is the code (in base 2) of the 26th (non-empty) set, whose elements are the second, the 4th and the 5th sets in Ackermann order (numbering). Domenico Cantone, Alberto Policriti Encoding Sets as Real Numbers 4/18
Ackermann 1937 Definition N A ( x ) = Σ y ∈ x 2 N A ( y ) Ackermann Order h i ≺ h j ⇔ N A ( h i ) = i < j = N A ( h j ) Ordering h i ≺ h j ⇔ N A ( h i ) = i < j = N A ( h j ) ⇔ max ( h i \ h j ) ≺ max ( h j \ h i ) ... which is as comparing the binary codes of i and j Domenico Cantone, Alberto Policriti Encoding Sets as Real Numbers 4/18
Equality and Extensionality Uniqueness of (binary) positional notation Two natural numbers are equal if and only if they have the same binary code. Axiom of Extensionality Two sets are equal if and only if they have the same elements. Domenico Cantone, Alberto Policriti Encoding Sets as Real Numbers 5/18
What about hypersets ? Definition Ω = { Ω } Questions Extensionality? Can we propose a numerical encoding for hypersets? Domenico Cantone, Alberto Policriti Encoding Sets as Real Numbers 6/18
What about hypersets ? Definition Ω = { Ω } Questions Extensionality? Can we propose a numerical encoding for hypersets? Domenico Cantone, Alberto Policriti Encoding Sets as Real Numbers 6/18
Extending Ackermann code: A first version Definition (Dyadic rational numbers) n / 2 m n , m ∈ N 11010 , 101 Domenico Cantone, Alberto Policriti Encoding Sets as Real Numbers 7/18
Extending Ackermann code: A first version Definition (Dyadic rational numbers) n / 2 m n , m ∈ N 11010 , 101 Mapping on dyadic rational numbers Find an order of (proper) non well-founded h.f. sets and put them on the right: let Z A the result. Use Z A , that is N A on the left and Z A on the right. Domenico Cantone, Alberto Policriti Encoding Sets as Real Numbers 7/18
Extending Ackermann code: A first version Definition (Dyadic rational numbers) n / 2 m n , m ∈ N 11010 , 101 Mapping on dyadic rational numbers Find an order of (proper) non well-founded h.f. sets and put them on the right: let Z A the result. Use Z A , that is N A on the left and Z A on the right. Example 1 1 0 1 0 , 1 0 1 is the code of h 26 ∪ { � 1 , � 3 } with, in addition, the first and the third non well-founded sets . Domenico Cantone, Alberto Policriti Encoding Sets as Real Numbers 7/18
Extending Ackermann code Definition R A ( x ) = Σ y ∈ x 2 − R A ( y ) Domenico Cantone, Alberto Policriti Encoding Sets as Real Numbers 8/18
Extending Ackermann code Definition R A ( x ) = Σ y ∈ x 2 − R A ( y ) Ω = { Ω } ξ = 2 − ξ . (1) Domenico Cantone, Alberto Policriti Encoding Sets as Real Numbers 8/18
Extending Ackermann code Definition R A ( x ) = Σ y ∈ x 2 − R A ( y ) Ω = { Ω } ξ = 2 − ξ . (1) y y = x y = 2 − x x Domenico Cantone, Alberto Policriti Encoding Sets as Real Numbers 8/18
Extending Ackermann code What kind of number is Ω ? Ω is irrational. Ω is transcendental: (Gelfond-Schneider theorem: a b is transcendental if a and b are algebraic, 0 � = a � = 1, and b is irrational) Ω algebraic ⇒ − Ω algebraic ⇒ (G.S.) 2 − Ω = Ω would be transcendental. √ 2 of {∅} 4 is transcendental. The R A -code 2 − 1 / ... Domenico Cantone, Alberto Policriti Encoding Sets as Real Numbers 8/18
Conjectures Conjecture 1 The function R A is injective on HF 0 . Conjecture 2 The function R A is injective on HF 1 / 2 . Domenico Cantone, Alberto Policriti Encoding Sets as Real Numbers 9/18
The domain of R A Definition ( Set systems ) A set system S ( x 1 , . . . , x n ) in the set unknowns x 1 , . . . , x n is a collection of set-theoretic equations of the form: x 1 = { x 1 , 1 , . . . , x 1 , m 1 } . . (1) . x n = { x n , 1 , . . . , x n , m n } , with m i ≥ 0 for i ∈ { 1 , . . . , n } , and where each unknown x i , u , for i ∈ { 1 , . . . , n } and u ∈ { 1 , . . . , m i } , occurs among the unknowns x 1 , . . . , x n . A set system S ( x 1 , . . . , x n ) is normal if there exist n pairwise distinct (i.e., non bisimilar) hypersets � 1 , . . . , � n ∈ HF 1 / 2 such that the assignment x i �→ � i satisfies all the set equations of S ( x 1 , . . . , x n ) . Domenico Cantone, Alberto Policriti Encoding Sets as Real Numbers 10/18
Our result R A is well-given Any finite collection � 1 , . . . , � n of pairwise distinct sets in HF 1 / 2 satisfying S ( x 1 , . . . , x n ) , univocally determines real numbers R A ( � 1 ) , . . . , R A ( � n ) satisfying: R A ( � 1 ) = � m 1 k = 1 2 − R A ( � 1 , k ) . . . R A ( � n ) = � m n k = 1 2 − R A ( � n , k ) . Domenico Cantone, Alberto Policriti Encoding Sets as Real Numbers 11/18
Proof sketch Definition The multi-set approximating sequence for (the solution of) S ( x 1 , . . . , x n ) is � ∅ | 1 ≤ i ≤ n � if j = 0 � � µ j i | 1 ≤ i ≤ n := �� � � µ j − 1 i , 1 , . . . , µ j − 1 | 1 ≤ i ≤ n if j > 0 , i , m i Domenico Cantone, Alberto Policriti Encoding Sets as Real Numbers 12/18
Proof sketch Definition � R µ A ( µ j i ) | 1 ≤ i ≤ n }� � � Code approximating sequence j ∈ N by recursively putting: R µ � A ( µ 0 i ) = 0 (2) u = 1 2 − R µ i , u ) . A ( µ j R µ A ( µ j + 1 � m i ) = i We also define the corresponding code increment sequence � δ j i | 1 ≤ i ≤ n � � � j ∈ N by putting, for i ∈ { 1 , . . . , n } and j ∈ N : δ j i = R µ A ( µ j + 1 ) − R µ A ( µ j i ) . (3) i Domenico Cantone, Alberto Policriti Encoding Sets as Real Numbers 12/18
Proof sketch Lemma (i) R µ A ( µ j + 1 i + · · · + δ j ) = δ 0 i , i (ii) δ 0 i = m i , u = 1 2 − R µ i , u ) · ( 2 − δ j A ( µ j (iii) δ j + 1 = � m i i , u − 1 ) , i (iv) δ 2 j + 1 ≤ 0 ≤ δ 2 j i , i (v) | δ j + 1 | ≤ | δ j i | , and i (vi) lim j →∞ δ j i = 0 . Domenico Cantone, Alberto Policriti Encoding Sets as Real Numbers 12/18
Theorem For any given normal set system, the corresponding code system admits always a unique solution. Domenico Cantone, Alberto Policriti Encoding Sets as Real Numbers 13/18
Theorem For any given normal set system, the corresponding code system admits always a unique solution. Remark µ 0 � � is 0 for any i ∈ { 1 , . . . , n } , the value of While the value of R A i � µ 1 � —first approximation of R A ( µ i ) —is the cardinality of µ i , R A i and the subsequent approximations oscillate within the interval [ 0 , | µ i | ] . Domenico Cantone, Alberto Policriti Encoding Sets as Real Numbers 13/18
Partial results: super-singletons Definition The elements of the family S of super-singletons {∅} i | i ∈ N � � S = , are defined recursively as follows: {∅} 0 = ∅ and {∅} n + 1 = {{∅} n } . 1 1 √ 0 Ω 1 2 2 s 0 s 2 s 4 s 5 s 3 s 1 Domenico Cantone, Alberto Policriti Encoding Sets as Real Numbers 14/18
Partial results: super-singletons Definition The elements of the family S of super-singletons {∅} i | i ∈ N � � S = , are defined recursively as follows: {∅} 0 = ∅ and {∅} n + 1 = {{∅} n } . Proposition The following hold: 0 = s 0 < · · · s 2 i < s 2 i + 2 · · · < Ω < · · · s 2 i + 3 < s 2 i + 1 · · · < s 1 = 1 , and i →∞ s 2 i = lim lim i →∞ s 2 i + 1 = Ω . Domenico Cantone, Alberto Policriti Encoding Sets as Real Numbers 14/18
Partial results: successors, density, ... Proposition For all i ∈ N : 1 R A ( h i ) � = R A ( h i + 1 ) ; 2 R A ( h i ) � = R A ( h i + 2 ) . Domenico Cantone, Alberto Policriti Encoding Sets as Real Numbers 15/18
Partial results: successors, density, ... Proposition For all i ∈ N : 1 R A ( h i ) � = R A ( h i + 1 ) ; 2 R A ( h i ) � = R A ( h i + 2 ) . Proposition { R A ( h ) | h ∈ HF } is dense in R . Domenico Cantone, Alberto Policriti Encoding Sets as Real Numbers 15/18
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