In this paper, a generalised version A of the celebrated Ackermann encoding of the hereditarily finite sets, aimed at assigning a real number also to each hereditarily finite hyperset and multiset, is studied. Such a mapping establishes a significant link between real numbers and the theories of such generalised notions of set, so that performing set-theoretic operations can be translated into their number-theoretic equivalent. By appropriately choosing a parameter , both the Ackermann encoding and the less known map R arise as special cases; a bijective encoding of a subuniverse of hereditarily finite multisets occurs whenever this parameter is chosen among natural numbers, while if it is taken transcendental and within a peculiar interval of the real positive line, then the function is surmised to ensure an injective mapping of both the aforementioned universes.
In 1937, Wilhelm Ackermann defined an encoding of the hereditarily finite sets – namely, finite sets
whose construction involves only finite sets at any nesting depth – into natural numbers (see [
A hyperset admits cycles in the membership relation, so it violates the so-called well-foundedness
principle. Despite this, a strongly restrained equality notion – namely, bisimilarity (see [
Diferently from the previous case, a multiset allows each of its elements to occur with multiplicity
higher than one, while the order of those elements is still regarded as irrelevant. Despite their plain
application in computer sciences, a formal theory of pure multisets is not uniquely established yet –
nonetheless, an attempt can be found in [
The aim of this paper, resulting from a master’s degree thesis,1 is to generalise the Ackermann
encoding and its variant R – already introduced and studied in [
This paper is structured as follows. In Section 1, it introduces the notions of hereditarily finite (hyper-, multi-) sets; Section 2 focuses on the Ackermann encoding and the map R. In Section 3, the encoding scheme A is introduced, while the cases ∈ N and − ≤ ≤ 1/ are analysed in Sections 4 and 5 resp. – the latter being an interval whose significance emerges from a theorem by Euler. Section 6 is devoted to show two meaningful examples, and Section 7 to point out the conclusions of this study.
Standard set-theoretic notations will be adopted throughout the paper; in particular, P(· ) will represent
the powerset operator. Recall that, if it exists, the transitive closure of a set is the collection of all its
elements, together with their own elements at any nesting depth:
trCl() = ∪
⋃︁ trCl(′) .
′∈
Notice that its existence is always guaranteed if well-foundedness, together with a minimal axiomatic
equipment, is assumed on the membership relation (see, e.g., [
To denote numerical sets, N, R and so on will be used; the abbreviations N+ := { ∈ N | > 0}, 0} and similar are adopted. For classical numerical operations, standard notations ℶ (0) = 0, ℶ ( + 1) = ℶ () = · · · for ∈ R+, ∈ N is justified by the frequent use of iterated exponentials along this paper.
The following definitions are given by means of the concept of cumulative hierarchy, meaning that they are built up by introducing a sequence of levels or layers such that each one of them is strictly contained in the subsequent one. The first and smallest universe of sets to be introduced is the following.
HF = {︃ ∅
Therefore, hereditarily finite (from now on, often abbreviated as h.f.) sets are finite at any nesting depth; observe that this universe is well-founded by construction. The rank of a h.f. set expresses also the maximum depth at which the empty set is nested inside it.
Diferently from standard sets, the universe of multisets allows each element of any of its members to occur more than once.
Definition 1.2 (Multisets). Let 1, . . . , be distinct objects and let 1, . . . , ∈ N+ be positive integers; then the list or equivalently
1⏞ = [1, . . . , 1, . . . , , . . . , ], ⏟ ⏟
⏞ = {11, . . . , }, up to any permutation, defines the multiset containing the objects 1, . . . , with multiplicities 1, . . . , respectively, i.e. containing exactly occurrences of for every ∈ {1, . . . , }.2 The multiplicity map of and its multiset membership relation are then defined as () = ⇐⇒ ∈ .
Multisets are introduced and often used without much care about formal aspects (see [
P () = {︀ {11, . . . , } | 1, . . . , ∈ ∧ (∀ ̸= )( ̸= ) ∧ 1, . . . , ∈ N+ ∧ ∈ N︀} .
HF = {︃ ∅ P (HF− 1) if ∈ N+, defines the cumulative hierarchy of the hereditarily finite multisets .
Given ∈ HF , its rank rk() is the least integer such that ∈ HF+1.
Some arithmetical operations such as sum, multiplication by a positive integer, product and
exponentiation are defined inside HF , so that polynomials of multisets can be defined, too (see [
Diferently from the previous cases, the following universe is not defined by means of a cumulative
hierarchy but by means of set systems describing the transitive closures of its members; the
equality criterion between two hypersets generalises from one-to-one correspondence of their elements
(extensionality, see [
Definition 1.5 (Bisimulation). A dyadic relation ♭ on the finite set of the nodes of an directed graph ℳ = (, ) is said to be a bisimulation on ℳ if 0♭1 always implies that • for every child 1 of 1, 0 has at least one child 0 s.t. 0♭1, and • for every child 0 of 0, 1 has at least one child 1 s.t. 0♭1.
The largest of all bisimulations on ℳ (relative to inclusion) is the following equivalence relation.3
Definition 1.6 (Bisimilarity). The bisimilarity of a digraph ℳ whose set of nodes is finite is the
dyadic relation ≡ ℳ over such that ≡ ℳ holds between , in if and only if ♭ holds for
some bisimulation ♭ on ℳ.
2Following [
⏟ ⏞ 3See [3, pp. 20–22].
In the wording of [15, pp. 78–80], bisimilarity induces the coarsest partition of that is stable w.r.t. ℳ. The graphs taken into account here are associated with systems of set equations involving unknowns (acting as nodes); each equation = {,1, . . . , , } brings the edges ⟨, ,1⟩, . . . , ⟨, , ⟩ into . Definition 1.7 (Hereditarily finite rational hypersets) . A hereditarily finite rational hyperset is the solution 0 (unique, up to bisimilarity) of a finite set system with , ∈ {0, 1, . . . , } for every ∈ {0, 1, . . . , } and ∈ {1, . . . , }. The class of h.f. rational hypersets is denoted by HF1/2.
Example 1.1. The hyperset Ω = {{{· · · }}}
is the solution of the equation = {}.
Notice that also h.f. well-founded sets can be defined by finite set systems, with the constraint that the elements of can only be chosen from among +1, . . . , . Moreover, by allowing multiple occurrences of the same unknown, h.f. multisets can be defined in this way too. Multihypersets, i.e., non-well-founded multisets, are neither treated in this paper nor – to the best of authors’ knowledge – elsewhere.
N(ℎ) = ∑︁ 2N(ℎ′) ℎ′∈ℎ if ℎ ∈ HF recursively defines the Ackermann encoding of hereditarily finite sets.
The Ackermann encoding has a number of well-known and interesting properties, which allow an exact match not only between HF and N, but also between the related operations and theories. Some of these properties are shown below.4 • N is a bijection between HF and N. • ℎ′ ∈ ℎ ∈ HF if and only if there is a ‘1’ at position N(ℎ′) of the binary expansion of N(ℎ). • N gives a natural, total ordering to HF: this is established as ℎ ≺ ℎ′ ⇔ N(ℎ) < N(ℎ′).
To extend the domain of an encoding map so as to embrace also h.f. hypersets, in [
Notice that this definition allows the codes of h.f. hypersets to be finite, despite not so evidently
guaranteeing it. As a very intuitive extension of this encoding, the following definition is introduced.
4For a proof see, e.g., [
Definition 2.3 (R-code). The R-codes of hereditarily finite multisets are defined as follows: R() = ∑︁ (′) · 2− R(′)
for ∈ HF .
′∈ Clearly, if the multiplicities are all equal to 1 at any nesting depth, the multiplicity map is unnecessary Observe that this encoding is not injective over the whole HF , indeed and the resulting formula is the same as the previous one, thereby showing that R⃒⃒ HF1/2 = R.
R(︀ [[∅], [∅]]︀) = 2 · 2− 1 = 1 = 20 = R(︀ [∅]︀) .
Besides, a suitable subuniverse of it is defined in [
Although this conjecture has some relevant consequences over HF in the first place (see [
already known.
Definition 3.1 (A -code). Let ∈ R+
∖ {1}; then Consider the following family of encodings of hereditarily finite (hyper-, multi-) sets, including those ∑︁ ℎ′∈ℎ A (ℎ) def = ℎ(ℎ′) · A (ℎ′) for ℎ ∈ HF1/2 ∪ HF defines the A -codes of the hereditarily finite (hyper-, multi-) sets.
Remark 1. Although the codes may vary significantly by changing the basis , two hereditarily finite sets trivially have an established and constant code; they are ∅ →↦− 0, {∅} →↦− 1.
As a consequence, the codes of the multisets containing just the empty set but with any multiplicity range over all natural numbers: (∀ ∈ R + ∖ {1})(∀ ∈ N) A (︀ ∅})︀ = .
{
︁(
︁)
6This notation is an adaption from H1∞ introduced in [
Notice that the definition, as it stands, is not insisting in any way on the injectivity of the codes; indeed, the following two particular cases arise.
Example 3.2. Let = 2; then A2⃒⃒ HF
injective over its whole domain, which is a much wider family of generalised sets; in particular (see = N. Since the latter is bijective onto N, the former cannot be
+ (︁ (∀ ∈ N ) A (ℎ) = A (︀ {∅})︀ .
︁) Example 3.3. Let = 1/2; then A1/2 = R. As has been already seen, R(︀ [[∅], [∅]]︀) = 1.
A remarkable general property is that any basis makes the codes of h.f. sets grow beyond any bound.
Proposition 3.1. Let ∈ R+
∖ {1}; then, the set of A -codes of HF is superiorly unbounded.
Proof. Fix ∈ N. For any odd number ∈ N, consider ℎ as the -th h.f. set with respect to the A (ℎ ) = ∑︁ A (ℎ′) = ∑︁ A (︀
{ℎ′})︀ ≥ 1. ℎ′∈ℎ ℎ′∈ℎ
Let = 2⌈ − 1⌉ and consider the h.f. set ℎ = {{ℎ′ } | ′ ≤ }. Thus, A (︀
{ℎ })︀ = A (ℎ) ≤ .
A (︀
{{ℎ }}︀) = A ({ℎ}) ≥ > .
A (ℎ) = ∑︁ A (︀
{{ℎ′ }}︀) ≥ Case > 1. Let > 1; then,
A (︀ {ℎ })︀ = A (ℎ) > .
Consider again = 2⌈ − 1⌉, and ℎ = {ℎ′ | ′ ≤ }. As in the previous case, ′=0
′=0 A (ℎ) = ∑︁ A (︀ {ℎ′ })︀ ≥
∑︁ A (︀ ′=0 ′ odd {{ℎ′ }}︀) > · 2
= .
′=0 ′ odd ∑︁ A (︀ {ℎ′ })︀ > · 2 = .
Generalising the Ackermann map from h.f. sets to natural numbers, consider a basis ∈ N+, ≥ 2; even when >
2, the Ackermann ordering is clearly kept over HF, but there may be a gap between two consecutive codes; e.g., if = 3, their codes are 0, 1, 3, 4, 27, . . . .
A rather intuitive feature that any of these encodings share with Ackermann’s is that the expression of the codes as base- numbers shows the membership relation as the presence of a ‘1’ at the position given by the Ackermann ordering; therefore, codes of h.f. sets are sequences of just 0s and 1s in that representation. Notably, the missing codes can be filled with multisets with multiplicities at most − 1 at any nesting depth; in this way, A over a proper subfamily of HF turns out to be bijective, and the code of a multiset as expressed as a base- number defines completely its members with their appropriate multiplicities.
To achieve a reasonable definition of the subuniverse of HF that can be encoded with a natural basis, consider an alternative version of powerset which is compatible with multisets. Definition 4.1 (-powerset). Given a multiset and ∈ N+, define
By using the classical powerset operation and the multiplication of a multiset by an integer (see
[
With this new operator, the h.f. multisets that can be properly encoded by A with ∈ N+ ∖ {1} can reasonably be delimited.
Definition 4.2 (Hereditarily finite -multisets). Let ∈ N+ ∖ {1}. Then,
HF() = {︃ ∅ P(− 1)(HF(−)1) if ∈ N+, defines the cumulative hierarchy of the hereditarily finite -multisets.
The simplest case occurs when = 2, since HF = HF(2) as subfamilies of h.f. multisets. More generally, it is guaranteed that the elements of each ℎ ∈ HF() have multiplicities at most − 1 at any nesting depth. Each HF() ranges over every layer HF and HF = ⋃︀∈N+∖{1} HF(), in complete analogy with the relationship between P and P(). Moreover, the rank of an h.f. multiset inside HF() is the same as inside HF :
HF() = HF() ∩ HF.
Theorem 4.1. Let ∈ N+ ∖ {1}. Then the encoding map
A⃒⃒ HF() :
HF() →−
N is bijective.
Proof. Given any ℎ ∈ HF(), its code is well-defined by the recursive construction of the map A. On the other hand, any ∈ N can be expressed as a sum of powers of : = 0 + 1 + 22 + · · ·
+ , with ∈ {0, . . . , − 1} and ∈ N.
0 = A(︀ [∅, . . . , ∅])︀ .
⏟ 0⏞
This meaningful property ensures a total ordering of each HF(); a total ordering of the whole HF would be a limit case of all such encodings, but it cannot be implemented without introducing transfinite ordinals. Moreover, since HF() is naturally embedded into HF(+1), the ordering given by A to the former is kept by the ordering given by A+1 to the latter.
Despite these promising results about A over HF(), no = would remain acceptable over the whole families of h.f. sets and multisets, due to violated injectivity: with the given definition, every A-code of a set in HF ∖ {∅, {∅}} is the code of at least one multiset in HF ∖ HF.
Moreover, recalling what this paper is aimed at, the most significant reason to abandon any encoding attempt with an integer basis is that there is no convergence on the codes of hypersets. 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 0 1 2 10 11 12 20 21 22 100 101 102 110 111 112 120 121 122 200 201 0 30
5.1. Over hypersets
A most remarkable result concerning iterated exponentials, first discovered by L. Euler in 1777 and
subsequently re-discovered and proven in several papers, is shown below.7 Here, its statement is taken
with slight changes from [
Theorem 5.1. The function = () = lim→∞ ℶ() = · converges when − ≤ ≤ 1/ and diverges for all other positive outside this interval. On this interval is the partial inverse of [namely, () = 1/], that is, · · ︀( ())︀ = (︀ ())︀ = if − ≤ ≤ 1/, if − 1 ≤ ≤ [ = 2.71828 . . . is the Euler number]. In particular, four nontrivial modes of convergence and divergence occur.
Case 1: > 1. The sequence of hyperpowers increases monotonically: ℶ(1) < ℶ(2) < ℶ(3) < · · · .
Subcase 1c: 1 < ≤ 1/. The sequence is bounded by , and so () converges.
Definition 5.1. = , i.e.
Subcase 1d: 1/ < . The sequence increases without bound, and so () diverges.
Case 2: < 1. The sequence of hyperpowers oscillates: ℶ(2) < ℶ(2 − 1) for > 1 and, moreover, the two subsequences ℶ(2) < ℶ(4) < ℶ(6) < · · · and · · · < ℶ(5) < ℶ(3) < ℶ(1) each converge.
Subcase 2c: − ≤ < 1. The preceding two subsequences of odd and even hyperpowers converge to the same value, and so () converges.
Subcase 2d: < − . The preceding two subsequences each converge separately to diferent values, and so () diverges.
Notice that, if the infinitely iterated exponential function converges, then its limit is a solution of
the exponential equation = ; the function (, ) = − has a unique zero if 0 < ≤ 1, two
zeros if 1 < ≤ 1/ – actually, for = 1/ there is a unique zero with multiplicity 2, namely = –
and no zeros if > 1/. In the second case, the limit of the function is the lower of such zeros (see,
e.g., [
Let ∈ R+, ≤ 1/; then Ω − 1 is defined as the lower solution of the equation Ω − 1 = min { | = }.
∈R+ Corollary 5.1. Let ∈ R+, = ℶ () = A ({∅}).
Case 1: − ≤ < 1. The following hold true.
0 = 0 < 2 < · · · < 2 < · · · < Ω − 1 < · · · < 2+1 < · · ·
< 3 < 1 = 1, lim 2 = Ω − 1 = lim 2+1.
→∞ →∞ Case 2: 1 < ≤ 1/. The following hold true.
0 = 0 < 1 = 1 < = 2 < · · · < < · · ·
< Ω − 1 , lim = Ω − 1 .
→∞
For a further generalisation of A -codes to HF1/2, which follows naturally from the previous corollary
on the code of Ω, consider the following extension of the concept of code system (Definition 4, [
C (0, 1, . . . , ) = ⎧⎪0 = 0,1 + · · · ⎪ ⎪⎨⎪1 = 1,1 + · · · . .
. ⎪ ⎪ ⎪ ⎩⎪ = ,1 + · · · + 0,0 + 1,1 + , , where , is a shorthand for S (,) for ∈ {0, 1, . . . , } and ∈ {1, . . . , }.
S (0, 1, . . . , ) once assigned to 0, 1, . . . , .
Definition 5.3
(Normal set systems). A set system S (0, 1, . . . , ) is said to be normal if there exist + 1 pairwise non-bisimilar hypersets ℏ0, ℏ1 . . . , ℏ ∈ HF1/2 that satisfy all the equations in
The following definitions are taken from [
Given a normal set system S (0, 1, . . . , ), two distinct unknowns and ′ , with , ′ ∈ {0, 1, . . . , }, are said to be distinguished at step > 0 by its multiset approximating sequence if
Some significant properties of set and multiset approximating sequences are shown in [
Lemma 5.1. Given ∈ R+, < 1, (∀, ∈ R)(︀ || ≤ | | ∧ ≤ 0 ⇒ | − 1| ≤ ⃒⃒ − − 1⃒⃒ )︀ .
Proof. Let , ∈ R such that ≤ 0, and suppose || ≤ | |. Then, since < 1,
1 ≤ −| | ≤ −| |. 1 ≤ ≤ − ⇒ 0 ≤ − 1 ≤ − − 1 ⇒ | − 1| ≤ | − − 1|.
Suppose ≤ 0 ≤ , then
Otherwise, i.e. ≤ 0 ≤ , 1 ≤ − ≤ ⇒ − ≤ ≤ 1 ⇒ − − 1 ≤ − 1 ≤ 0 ⇒ 0 ≤ | − 1| ≤ | − − 1|. 8If two unknowns are distinguished at a certain step, then they are distinguished at every subsequent step; see Lemma 2 (a),
+1 = ∑︁ A (,)(︀ , − 1 ︀)
where , = S (,)
≥ ) ⇒ l→im∞ > 0.
Lemma 5.2. Given ∈ R+
≤ ≤ 1/, let S (0, 1, . . . , ) be a normal set system with (⟨ | 0 ≤ ≤ ⟩)∈N. Then, for every ∈ {0, 1, . . . , } and ∈ N, the following facts hold true. index map S , multiset approximating sequence (⟨ | 0 ≤ ≤ ⟩)∈N and A -code increment sequence =0 =1 ∑︁ =1 A (+1) = ∑︁
0 = Moreover, if < 1, while, if > 1, =0 = = ∑︁ =1 (1) (2) (3) (4) (5) (6) (7) (8)
=1
Claim (2). This is another trivial consequence of the given definitions. Indeed, the codes of the first step of multiset approximating sequence are the sum of as many 1s as the elements of the considered hyperset, the empty set being the step 0 of whatever sequence.
+1 = ∑︁ A (,+1) − A (,) = ∑︁(︁ A (,+1) − A (,)︁) =1 =1 = ∑︁(︁ A (,)+ , − A (,)︁) A (,) · ︀( , − 1︀) . the result is proven.
1; since this is also the case of A1/2 = R, they
conclude the part treated also by Lemma 4, [
Claim (4). It follows by (2) and (3), since 0 ≥ 0 for every ∈ {0, 1, . . . , }, so that 1 is a nonalternatively non-negative and non-positive values, for even and odd indices respectively. positive sum, because 0 − 1 ≤ 0. Therefore, by induction, the A -code increment sequence assumes
· ⃒⃒⃒ +1 − 1⃒⃒⃒ ≤ ⃒ ⃒⃒ Proof. Fix an index ∈ {0, 1, . . . , }; along the proof, the following shorthand notation will be adopted. , = S (,),
, = S (,), , = S (,).
Claim (1). It is inductively proven by using the definition of A -code increment sequence:
A (1) = 0 + A (0) = 0 A (+1) = + A ( ) = + ∑︁
= ∑︁ . which will now be proven by induction on ∈ N, for every ∈ {0, 1, . . . , }. Since
0 by claim (4), the base case of (8) is proven by This completes the proof of the claim (8). Claim (5) then follows for > 0 as an immediate by-product of its proof, while in the case = 0, it amounts to (9).
1.
Claim (6). The A -code increment sequence is positive, since +1 = A (,) ︀( , − Then, to move on from the induction hypothesis (8) to the next value of , we argue as follows: · ⃒ ⃒ · · · − · ⃒ − 0, 1⃒ = (1 − , ) ≤ = ⃒⃒ 0⃒⃒ ,
(9) − − − ︀) − − −
︀) 0, focus now on the sum between parentheses.
∑︁ =1 A (,+1) · (︀
,+1 A (,) , ︀(
· A (,) ︁(
, (︀ A (,) ︀( ,+ , +1 2 , + 1)︀ . being 0 = ≥
0 non-negative in the first place.
Claim (7). Assume +1
; it follows ,+ , +1 − 2 , + 1 ≥ 2 , 2 , + 1 = (︀ , 1 ︀) 2 0 Therefore, the A -code increment sequence is either stable or increasing from the -th step on. − ⃒ ⃒ ∑︁ =1 − 1⃒ ⃒ ⃒ ∑︁ =1 ∑︁ =1
︀) 1 − A (,) ︀(
, · A (,) ︀(
· ∑︁ =1 Remark 2. Observe that, despite not being proven over HF1/2 yet, convergence on the codes of h.f. wellfounded sets and multisets is guaranteed by the convergence of the multiset approximating sequence within a number of steps equal to their rank, so that the A -code increment sequence is constantly 0 for all the subsequent steps.
If − on the A -codes of the other rational hypersets ℏ such that ℏ ∈ ℏ. Otherwise, if 1 < ≤ 1/ the convergence is not guaranteed, since whenever +1 ≥ the code sequence ⟨A ( )⟩∈N diverges due to statement (7) of the preceding lemma. An initial result is presented below, showing a minimum 1, the convergence on the A -code of Ω appears to imply at least the convergence requirement to get 1 < 0. then 1 < 0.
Proposition 5.1. Given >
1, consider the set system S (0, 1, . . . , ) with index map S ; let ∈ {0, 1, . . . , } be such that > 0 and define , = max∈{1,...,}{,}. If , < log 2, Proof. Recalling that 0 = = A (1), , < log 2 implies 1 = ∑︁ =1 A (0,)( 0, − 1) = ∑︁( 0, − 1) = ∑︁( , − 1) =1 =1 ≤ ( , − 1) < (2 − 1) = = 0.
A -code. − and 1 can be conjectured.9
Observe that the above constraint appears to be quite restrictive when dealing with the convergence of the multiset approximating sequence’s A -codes, also because it has to be strengthened at every subsequent step. Although a further analysis needs to be done on the convergence on A -codes of HF1/2, the existence and uniqueness of A -codes of all rational hypersets for a basis chosen between Conjecture 5.1. Consider ∈ R, − ≤ <
1, and ℏ ∈ HF1/2. Then, there exists and is unique its (∀ℏ ∈ HF1/2)(∀ ∈ R)(︀ − ≤ < 1 ⇒ ∃!A (ℏ) ∈ R+)︀ .
The challenging question of injectivity is still open; it will follow from the injectivity of the map over HF , by the method of multiset approximating sequence. 5.2. Over multisets at the very first levels of the hierarchy, since analog of ℋ2 has to be introduced as domain for this map.
Definition 5.6. Let ∈ N+
∖ {1} and For what concerns the application of the map A to h.f. multisets, observe that in all the cases = 1/ with ∈ N+ ∖ {1}, analogues of the properties valid for R can be found too. Injectivity is violated A1/ { ︀( {∅}}
︀) = · 1/ = 1, so that to require it an ℋ, def = {︃{︀ ∈ HF | ({∅}) ≤ − 1
︀} ︀{ ∈ ℋ,0 | ⊆ ℋ ,− 1︀} − 1 at any nesting depth. thus ℋ is the collection of multisets containing no occurrences of {∅} with multiplicity larger than
A1/-code. to
Further, by developing tools analogous to the ones of [
∖ {1}, the coding map A1/ is injective over the collection ℋ.
Theorem 5.2. Under Conjecture 5.2, every hereditarily finite set of rank at least 4 has a transcendental
Proof. It is a rearrangement of the proof of Theorem 4.12, [
Since multisets introduce multiple occurrences of their elements, for every algebraic basis − 1 there are issues similar to the ones already encountered for = 1/ with natural, the latter being a subcase of the former. Indeed, since by definition an algebraic number is the root of a polynomial with integer coeficients, an algebraic basis satisfies ( ) = 0 where () = 0 + 1 + 22 + · · · + ∈ Z[].
Therefore, in these cases it would be necessary to introduce a proper subfamily of HF to exclude the possibility that the same polynomial is reproduced by the code of any multiset in it. The case of interest is the one in which 0 < 0, ≥ {1{∅}, 2 2 { ∅}, . . . , {
∅}} would coincide.
choice appears to be − 1 = 1/ ∼
logarithm (see, e.g., [
These observations lead to focus on transcendental real numbers within − and 1. The most obvious 0.36788 . . . , since is one of the most studied transcendental mathematical constants and is used for both the definitions of the natural logarithm and the product
Despite being the only possible choice to encode completely both HF and HF1/2, − 1 and any other transcendental basis sufer of a lack of knowledge about their behaviour when involved in (iterated) exponentiation. Nonetheless, having already excluded all the algebraic numbers from count, the following conjecture is stated as a motivation for future research.
Conjecture 5.3. The A− 1 encoding of h.f. multisets and hypersets is injective over the whole universe 0 for ∈ {1, . . . , }, so that the A -codes of {− 0∅} and HF1/2
∪ HF .
Example 6.1 (Non-injectivity of R). Consider = 1/2; then, A1/2(ℏ) = R(ℏ) = 1, where is the solution of the set system ℏ = {{{{. . . }, {∅}}, {∅}}, {∅}} S (0, 1, 2) = ⎨ ⎧⎪0 = {0, 1}
1 = {2} ⎪⎩2 = {}.
A1/2-code is the solution of = 2− + 2− 1, which countably many non-bisimilar hypersets with A1/2-code 1 are obtained. is trivially 1. Moreover, notice that by iteratively putting ℏ0 = ℏ and ℏ = {ℏ, ℏ− 1} for ∈ N+, Example 6.2 (Convergence and divergence at the two extreme cases). Consider the hyperset solution of
ℏ = {{{{. . . }, ∅}, ∅}, ∅}, S (0, 1) = {︃0 = {0, 1}
1 = {}.
By considering its multiset approximating sequence and the corresponding A -code approximating sequence for the bases
= − and = 1/, it turns out that the former guarantees convergence to a ifnite real number, while the latter does not. This is easily explained by using an analytical approach: depends on the behaviour of the function () = ( − 1)1/. To find out its extrema, consider indeed, observe that the A -code of ℏ is the root of the equation = + 1, so that its existence () = ( − 1)1/− 1(︀ − ( − 1) ln( − 1))︀ 2 . that is zero at the point = (1/)+1 + 1 ∼ 4.59112 . . . s.t. = () = ( (1/)+1)1/( (1/)+1+1) where (· ) is the principal branch of the product logarithm. This last value, representing the maximum value of (), is the greatest that can be assigned to to ensure the convergence on the A -code of ℏ.
A parameterised encoding scheme has been defined, in such a way so as to embrace both the celebrated Ackermann encoding and the new and less known R map, the latter being a version of the former that permits an extension of its domain from hereditarily finite well-founded sets to the universes of h.f. ill-founded sets (hypersets) and h.f. (well-founded) multisets. This generalised Ackermann encoding, defined as
A (ℎ) = ∑︁
ℎ(ℎ′) A (ℎ′) ℎ′∈ℎ – where ℎ(ℎ′) expresses the multiplicity of ℎ′ inside ℎ – depends on a real, positive and non-1 basis , whose choice turns out to be significant to obtain both an everywhere convergent and injective map, at least on a subuniverse.
Selecting such a basis among natural numbers has shown how one can define a subfamily of h.f. multisets over which the map is bijective. This mapping then gives a total ordering of this subfamily, and given the code of a multiset in it one can extrapolate which multisets belong to it with its corresponding multiplicity. This result appears to be promising in algorithmics to encode multisets whose maximum multiplicity is known, at any nesting depth. Despite this, all these encodings cannot apply to h.f. hypersets, since all their codes would be missing.
Given this first example, the focus has shifted to an interval – including 1 – on which a theorem by Euler ensures at least the convergence on the code of the hyperset Ω = {Ω}. Following the example of a previous paper concerning R, a way to approximate hypersets and their codes has been shown (with a basis within this interval); some properties of these approximating sequences have been proven, concluding that it is plausible that any of such bases can encode properly the universe of h.f. hypersets and multisets if lower than 1.
Once the most promising interval where to look for was found, the algebraic bases have been excluded, due to issues regarding injectivity over h.f. multisets. This suggested that one should adopt a transcendental basis, e.g., the inverse Euler number − 1, for future studies; the main dificulty in this case is the lack of knowledge on the behaviour of iterated exponentials with a transcendental basis.
Despite the results obtained about this encoding scheme, several problems are still left open. The challenge of proving existence and uniqueness of codes of all the h.f. rational hypersets is still there, even when choosing a basis within the aforementioned interval and lower than 1. Moreover, determining the range in which a > 1 must lie in to ensure existence of the code of a h.f. hyperset might be a way to introduce a non-arbitrary concept of rank for the universe of such aggregates. Nonetheless, this would follow from a deep study of nested exponential equations in two variables, which are made possible just with a significant knowledge of real analysis.