The Third Generalization of the Hausdorff Dimension Theorem for Fuzzy Fractals

1. Introduction

1.1. Fuzzy Fractals

Fuzzy set theory began with Lotfi A. Zadeh in 1965 [1], who replaced the two-valued indicator of a classical set with a membership function valued in $[0,1]$, so that a point may belong to a set not absolutely but to an intermediate degree—turning crisp inclusion into a graded notion suited to imprecisely bounded objects. Fractals entered this framework through the iterated function system: building on Hutchinson’s 1981 self-similar attractors [2] , Cabrelli introduced the iterated fuzzy set system in 1992 [3], whose contractive fuzzy Hutchinson–Barnsley operator admits a unique fixed point—an invariant, normal, upper-semicontinuous fuzzy set that they identified as the fuzzy fractal—and they did so precisely to address the inverse problem for fractals. The present work proceeds along a more general route: rather than realizing a fuzzy fractal as the self-similar attractor of such a system, it adopts a Mandelbrot-style criterion [4,5,6] under which a fuzzy set is a fuzzy fractal whenever one of its $\alpha$-cuts has Hausdorff dimension exceeding its topological dimension, and it builds such sets directly from prescribed fractal cores—through indicators or distance-kernel decorations—without any self-similarity requirement. Since these beginnings the subject has matured, resting on the metric theory of spaces of fuzzy sets [10,11], with results on the existence, uniqueness, and continuity of fuzzy attractors [12], algorithms for rendering them [13], and extensions to hyperfractals and to fuzzy metric spaces [14,15]. Across this body of work the guiding concern has been to construct fuzzy fractals and to understand the attractors so produced; the complementary question—which prescribed quantitative data a fuzzy fractal can be made to realize—belongs to the inverse problem, to which we now turn.

1.2. Inverse Problem

The inverse problem—establishing the existence of fractals carrying a prescribed set of quantitative features—reverses the usual direction of fractal geometry: rather than measuring the dimension and Lebesgue measure of a given set, one fixes those quantities in advance and asks whether a fractal realizing them exists, how many do, and what structure the family of all such fractals possesses [16,17,18,19,20]. Constructive answers of this kind matter well beyond geometry, echoing long-standing questions in the foundations of mathematics about when the existence of an object guarantees a means of building it, and they recur throughout computer science [21,22]. Their urgency for fractals comes from application: fractal models find application across an expansive spectrum of scientific disciplines [23,24,25,26], where the target dimension and size of a structure are often known before any explicit set attaining them is in hand. The fuzzy category extends this program to a setting closer to physical reality. Natural objects seldom carry the sharp boundaries a deterministic fractal presumes; their membership is graded, an imprecision Zadeh placed at the center of mathematical modeling [1]. A fuzzy fractal absorbs this gradation, and its quantitative signature is correspondingly richer—no longer a single dimension–measure pair but the level-indexed fuzzy fractal profile. The inverse problem here thus seeks fuzzy fractals of prescribed profile, a question at once more expressive and more delicate than its deterministic and random forerunners, and the one taken up in this paper.

1.3. Motivation

The inverse problem for fractals in ${\mathbb{R}}^n(n\ge 1)$ was first taken up in the mid-2000s for thin deterministic fractals [16], and more recently, in the early 2020s, for generalized deterministic fractals [19] and their random counterparts [20]. No such existence statement is yet available in the fuzzy category, and this case is genuinely more delicate: as we shall see, the prescribed data are no longer a single numerical pair but an entire level-indexed family of pairs, namely the fuzzy fractal profile. Beyond merely filling this gap, we are guided by two questions that the deterministic and random settings bring into view. The first concerns invariance: whether the maximal cardinality of aleph-two, attained by the available families in those settings, persists across the fuzzy category as well, so that this cardinality is independent of whether the prescribed setting is deterministic, random, or fuzzy. The second concerns regularity: the deterministic and random treatments leave implicit the way a regularity condition imposed on the prescribed setting governs the cardinality of the resulting family, and we make this "regularity-cardinality" interplay explicit here in the fuzzy setting.

1.4. Study Outline

This work provides a fuzzy counterpart to the existing deterministic and random existence results, constructing fuzzy fractals of prescribed Hausdorff dimension and Lebesgue measure in n-dimensional Euclidean space, and shows that the cardinality of the resulting family is independent of the construction setting. The outline is as follows. Section 2 collects the required preliminaries: the basic definitions and remarks of fractals in the fuzzy setting (Section 2.1), together with the key existence results and their hierarchy in the deterministic and random settings (Section 2.2). Section 3 develops the auxiliary results that bridge these preliminaries to the main theorems. Section 4 then leverages them to establish the existence results for fuzzy fractals (Section 4.1) and for strong fuzzy fractals (Section 4.2). Finally, Section 5 discusses the contributions to fractal geometry across the deterministic, random, and fuzzy settings (Section 5.1), outlines directions for future work (Section 5.2), and concludes the discussion (Section 5.3).

2. Preliminaries

The reader familiar with the theory of fractals [2,5,6,9,19,20] and fuzzy sets [1,8,10,11] will recognize that the present framework simultaneously unifies five foundational dimensions of upcoming analysis: (i) the Mandelbrot criterion of non-integer Hausdorff dimension [4], (ii) the α-cut geometry of fuzzy membership [1,8], (iii) the iterated fuzzy set systems of Cabrelli and Molter [3], (iv) the parallel-body and measure-theoretic foundations of Falconer and Tricot [6,9], and (v) the inductive topological dimension theory of Hurewicz and Wallman [7]. Throughout, the underlying space carries the Euclidean topology induced by the conventional metric $d_E$, unless otherwise stated. We denote the n-dimensional Lebesgue measure, Hausdorff dimension, and small inductive topological dimension by ${\lambda }_n$, ${dim}_H$, and ${dim}_{ind}$, respectively. Throughout the cardinality calculations, we use standard notation from set theory [27]; in particular, under the Continuum Hypothesis, $\mathfrak{c}={\aleph }_1$, and under the Generalized Continuum Hypothesis, $2^{\mathfrak{c}}={\aleph }_2$.

2.1. Basic Definitions and Remarks

Definition 2.1

(Fuzzy set and $\alpha -$ cuts). A fuzzy set on ${\mathbb{R}}^n$ is a function $u:{\mathbb{R}}^n\to [0,1].$ For the level parameter $0<\alpha \le 1,$ the $\alpha -$cut (crisp set level) of u is ${\left[u\right]}_{\alpha }=\{x\in {\mathbb{R}}^n:u\left(x\right)\ge \alpha \}.$ The core of u is $\mathit{core}\left(u\right)=\left[u\right]=\{x\in {\mathbb{R}}^n:u\left(x\right)=1\},$ and the support of u is $\mathit{supp}\left(u\right):={cl}_{{\mathbb{R}}^n}\{x\in {\mathbb{R}}^n:u\left(x\right)>0\}.$

Definition 2.2

(Special Fuzzy Sets). A fuzzy set $u:{\mathbb{R}}^n\to [0,1]$ is upper semicontinuous(usc) if, for every $0<\alpha \le 1,$ the cut ${\left[u\right]}_{\alpha }$ is closed. We call u normal if ${\left[u\right]}_1\ne \varnothing .$

Remark 2.3.

The geometry of a fuzzy set $u:{\mathbb{R}}^n\to [0,1]$ is naturally encoded in its family of nested $\alpha -$cuts as ${\left[u\right]}_{\beta }\subseteq {\left[u\right]}_{\alpha }$ whenever $0<\alpha <\beta \le 1.$

Definition 2.4

(The fuzzy fractal profile). Let $u:{\mathbb{R}}^n\to [0,1]$ be a fuzzy set and given level parameter $-\alpha .$ The fuzzy fractal profile of u at level $0<\alpha \le 1$ denoted by $Profil{e}_u\left(\alpha \right)$ is the pair $(D_u\left(\alpha \right),m_u\left(\alpha \right))$ where $D_u$ is the Hausdorff dimension profile defined by $D_u:(0,1]\to [0,n],D_u\left(\alpha \right)={dim}_H\left({\left[u\right]}_{\alpha }\right);$ and, the Lebesgue measure profile defined by $m_u:(0,1]\to [0,\infty ],m_u\left(\alpha \right)={\lambda }_n\left({\left[u\right]}_{\alpha }\right).$ The total fuzzy fractal profile of measurable u denoted by $Profil{e}_u$ is the pair $(D,m)$ where $D=\underset{\alpha >0}{sup}{D}_u\left(\alpha \right);$ and, $m={\int }_0^1{m}_u\left(\alpha \right)d\alpha ,$ respectively.

Remark 2.5.

The fuzzy fractal profile $Profil{e}_u\left(\alpha \right)$ and its components are non-increasing in terms of the level parameter $0<\alpha \le 1.$

Definition 2.6

(Fuzzy Fractal: Mandelbrot Context). Let $u:{\mathbb{R}}^n\to [0,1]$ be a fuzzy set. Then:

(a)

u is a fuzzy fractal if there exists $\alpha \in (0,1]$ such that ${\left[u\right]}_{\alpha }\ne \varnothing$ and ${dim}_H\left({\left[u\right]}_{\alpha }\right)>{dim}_{ind}\left({\left[u\right]}_{\alpha }\right)$.

(b)

u is a strong fuzzy fractal if for all $\alpha \in (0,1]$, ${\left[u\right]}_{\alpha }\ne \varnothing$ and ${dim}_H\left({\left[u\right]}_{\alpha }\right)>{dim}_{ind}\left({\left[u\right]}_{\alpha }\right)$.

We denote the set of all fuzzy fractals and strong fuzzy fractals by ${\mathcal{F}}_1\left({\mathbb{R}}^n\right)$ and ${\mathcal{F}}_2\left({\mathbb{R}}^n\right)$, respectively(${\mathcal{F}}_2\left({\mathbb{R}}^n\right)\subseteq {\mathcal{F}}_1\left({\mathbb{R}}^n\right)$).

Example 2.7

(Indicator Fuzzy Fractals). Let $u:{\mathbb{R}}^n\to [0,1]$ be given by $u\left(x\right)={\mathbf{1}}_F\left(x\right)$ where $F\subseteq {\mathbb{R}}^n$ is a non-empty deterministic fractal. Then, for all $\alpha \in (0,1]$, ${\left[u\right]}_{\alpha }=F$ yielding u to be a strong fuzzy fractal. Furthermore, u is upper semicontinuous(usc) if and only if F is closed. In particular, $u:\mathbb{R}\to [0,1],u\left(x\right)={\mathbf{1}}_C\left(x\right)$ where C is the middle-third Cantor set is a strong usc fuzzy fractal. We denote the family of all indicator usc fuzzy fractals by $\mathcal{I}.$

Remark 2.8.

$Card\left(\mathcal{I}\right)=\mathfrak{c}.$

Definition 2.9

(Weak Admissible Distance Kernels). A function $g:[0,\infty )\to (0,1]$ is called a weak admissible distance kernel if it is bijective, $g\left(0\right)=1$, and g is continuous at 0. We denote the family of all weak admissible distance kernels by ${\mathcal{G}}_w$.

Remark 2.10.

$Card\left({\mathcal{G}}_w\right)=2^{\mathfrak{c}}.$

Definition 2.11

(Admissible Distance Kernels). A function $g:[0,\infty )\to (0,1]$ is called an admissible distance kernel if it is bijective, $g\left(0\right)=1,$ and g is continuous everywhere. We denote the family of all admissible distance kernels by $\mathcal{G}.$

Remark 2.12.

$Card\left(\mathcal{G}\right)=\mathfrak{c}.$

Example 2.13.

Three examples of the family $\mathcal{G}$ are as follow:

(a)

The exponential kernel $g_1(t,\lambda )=e^{-\lambda t}(\lambda >0),t\ge 0$ has the inverse $g_1^{-1}(\alpha ,\lambda )=-{\lambda }^{-1}log\left(\alpha \right).$

(b)

The Gaussian tail kernel $g_2\left(t\right)=2(1-{\Phi }_Z\left(t\right)),t\ge 0$ has the inverse $g_2^{-1}\left(\alpha \right)={\Phi }_Z^{-1}(1-\alpha /2).$

(b)

The rational kernel $g_3\left(t\right)=1/(1+t),t\ge 0$ has the inverse $g_3^{-1}\left(\alpha \right)=(1-\alpha )/\alpha .$

Definition 2.14

(Family of Kernel-generated Fuzzy Extensions). Let $F\subseteq {\mathbb{R}}^n$ be non-empty, and let $g\in \mathcal{G}.$ The $g-$generated fuzzy extension of F is the fuzzy set $u_{F,g}\left(x\right)=g\left(d_E(x,F)\right),x\in {\mathbb{R}}^n,$ where $d_E(x,F)=\underset{y\in F}{inf}\left(d_E(x,y)\right).$ We denote the family of all kernel-generated fuzzy extensions via F by ${\mathcal{K}}_F.$ The family of all kernel-generated fuzzy extensions is denoted by $\mathcal{K}.$

Remark 2.15.

$Card\left({\mathcal{K}}_F\right)=\mathfrak{c},\varnothing \ne F\subseteq {\mathbb{R}}^n.$

Remark 2.16.

$Card\left(\mathcal{K}\right)=2^{\mathfrak{c}}.$

Remark 2.17.

The kernel-generated fuzzy extension $u_{F,g}$ is continuous on ${\mathbb{R}}^n$ and hence usc.

Remark 2.18.

The kernel-generated fuzzy extension $u_{F,g}$ has the $\alpha -$cut given by ${\left[u_{F,g}\right]}_{\alpha }=\{x\in {\mathbb{R}}^n:d_E(x,F)\le g^{-1}\left(\alpha \right)\}.$ In particular, ${\left[u_{F,g}\right]}_1=\overline{F}.$

2.2. Key Established Existence Theorems

Theorem 2.19

(Hausdorff Dimension Theorem). For any real $r>0$, there is aleph-one thin deterministic fractals with Hausdorff dimension r in ${\mathbb{R}}^n$, where $\lceil r\rceil \le n$.

Theorem 2.20

(Generalized Hausdorff Dimension Theorem). For any real $r>0$ and $l\ge 0$, there are aleph-two deterministic fractals-denoted by the set ${\mathcal{D}}_n(r,l)$- with Hausdorff dimension $\rho =r{\mathbf{1}}_{\left\{0\right\}}\left(l\right)+n{\mathbf{1}}_{(0,\infty )}\left(l\right)$ and Lebesgue measure l in ${\mathbb{R}}^n$, where $\lceil r\rceil \le n$.

Remark 2.21.

Under the additional compactness assumption, the cardinality of the associated available set -denoted by ${\mathcal{D}}_n^{comp}(r,l)$- in the Theorem 2.20 drops to aleph-one.

Theorem 2.22

(Second Generalized Hausdorff Dimension Theorem). For any real $r>0$ and $l\ge 0$, there are aleph-two random fractals -denoted by the set ${\mathcal{R}}_n(r,l)$-with Hausdorff dimension $\rho =r{\mathbf{1}}_{\left\{0\right\}}\left(l\right)+n{\mathbf{1}}_{(0,\infty )}\left(l\right)$ almost surely, and expected Lebesgue measure l in ${\mathbb{R}}^n$, where $\lceil r\rceil \le n$.

Remark 2.23.

An analogous compactness restriction applies to Theorem 2.22: if the random fractals are required to be compact-valued, the cardinality of the available associated set -denoted by ${\mathcal{R}}_n^{comp}(r,l)$- drops to aleph-one.

3. Auxiliary Results

Proposition 3.1

(The fuzzy fractal profile characterization). Let $F\subseteq {\mathbb{R}}^n$ be a non-empty compact set, $g\in \mathcal{G},$ and given level parameter $-\alpha .$ Then, the fuzzy fractal profile of kernel-generated fuzzy extension $Profil{e}_{u_{F,g}}\left(\alpha \right)=(D_{u_{F,g}}\left(\alpha \right),m_{u_{F,g}}\left(\alpha \right))$ has the following features:

(a)

$D_{u_{F,g}}\left(\alpha \right)=n.{\mathbf{1}}_{(0,1)}\left(\alpha \right)+{dim}_H\left(F\right).{\mathbf{1}}_{\left\{1\right\}}\left(\alpha \right),\mathrm{for}\mathrm{all}\alpha \in (0,1]$

(b)

$m_{u_{F,g}}\left(\alpha \right)>0,\mathrm{for}\mathrm{all}\alpha \in (0,1).$

Proof. 

(a) First, let $0<\alpha <1.$ Then, $g^{-1}\left(\alpha \right)>0,$ and ${\left[u_{F,g}\right]}_{\alpha }=\{x\in {\mathbb{R}}^n:d_E(x,F)\le g^{-1}\left(\alpha \right)\}.$ Choose $y\in F.$ then, $\overline{B}(y,g^{-1}\left(\alpha \right))\subseteq {\left[u_{F,g}\right]}_{\alpha }.$ Hence, ${\left[u_{F,g}\right]}_{\alpha }$ has nonempty interior, and hence Hausdorff dimension $n.$ Thus, $D_{u_{F,g}}\left(\alpha \right)={dim}_H\left({\left[u_{F,g}\right]}_{\alpha }\right)=n.$ Second, let $\alpha =1.$ Then, by Remark 2.18 and closeness of $F,$${\left[u_{F,g}\right]}_1=\overline{F}=F.$ Thus, $D_{u_{F,g}}\left(1\right)={dim}_H\left({\left[u_{F,g}\right]}_1\right)={dim}_H\left(F\right).$

(b) This follows from proof in part (a) in which ${\left[u_{F,g}\right]}_{\alpha }$ has nonempty interior and hence positive Lebesgue measure. □

Proposition 3.2

(The weak fuzzy fractal profile characterization). Let $F\subset {\mathbb{R}}^n$ be a non-empty bounded closed set, $g\in {\mathcal{G}}_w$, and given level parameter α. Then, the fuzzy fractal profile of the weak kernel-generated fuzzy extension ${\mathrm{Profile}}_{u_{F,g}}\left(\alpha \right)=\left(D_{u_{F,g}}\left(\alpha \right),m_{u_{F,g}}\left(\alpha \right)\right)$ has the following features:

(a)

$D_{u_{F,g}}\left(\alpha \right)=n·{\mathbf{1}}_{(0,1)}\left(\alpha \right)+{dim}_{\mathrm{H}}\left(F\right)·{\mathbf{1}}_{\left\{1\right\}}\left(\alpha \right)$, for all $\alpha \in (0,1]$;

(b)

$m_{u_{F,g}}\left(\alpha \right)>0$, for all $\alpha \in (0,1)$.

Proof. 

(a) First, let $0<\alpha <1$. Since g is continuous at 0 and $g\left(0\right)=1$, there exists ${\delta }_{\alpha }>0$ such that $0\le t<{\delta }_{\alpha }\Rightarrow g\left(t\right)>\alpha$. Then $\{x:d_E(x,F)<{\delta }_{\alpha }\}\subseteq {\left[u_{F,g}\right]}_{\alpha }$. Because $F\ne \varnothing$, this set contains a nonempty open ball. Hence ${\left[u_{F,g}\right]}_{\alpha }$ has nonempty interior, and therefore $D_{u_{F,g}}\left(\alpha \right)={dim}_{\mathrm{H}}\left({\left[u_{F,g}\right]}_{\alpha }\right)=n$. Second, let $\alpha =1$. Since g is bijective and $g\left(0\right)=1$, the value 1 is attained only at 0, so ${\left[u_{F,g}\right]}_1=\{x:d_E(x,F)=0\}=\overline{F}=F$ by closeness of F. Thus $D_{u_{F,g}}\left(1\right)={dim}_{\mathrm{H}}\left({\left[u_{F,g}\right]}_1\right)={dim}_{\mathrm{H}}\left(F\right)$.

(b) This follows from the proof of part (a) in which ${\left[u_{F,g}\right]}_{\alpha }$ has nonempty interior and hence positive Lebesgue measure. □

Proposition 3.3

(The Kernel-generated Fuzzy Fractal). Let $F\subseteq {\mathbb{R}}^n$ be a non-empty compact deterministic fractal and let $g\in \mathcal{G}$. Then the kernel-generated fuzzy extension

$$u_{F,g}\left(x\right)=g\left(d_E(x,F)\right),x\in {\mathbb{R}}^n,$$

(1)

is a usc fuzzy fractal.

Proof. 

At the core level $\alpha =1$, we have $g^{-1}\left(1\right)=0$, and hence ${\left[u_{F,g}\right]}_1=F.$ Therefore, ${dim}_H\left({\left[u_{F,g}\right]}_1\right)={dim}_H\left(F\right)>{dim}_{\mathrm{ind}}\left(F\right)={dim}_{\mathrm{ind}}\left({\left[u_{F,g}\right]}_1\right),$ so $u_{F,g}$ is a fuzzy fractal. Finally, the usc property follows from Remark 2.17. □

Remark 3.4

(Dimension Jump). Given a kernel-generated fuzzy fractal $u_{F,g}$. Then, the Hausdorff dimension profile $D_{u_{F,g}}$ has a sharp jump from its value of ${dim}_H\left(F\right)$ at the core level $\alpha =1$ to its value of n at the lower positive levels $0<\alpha <1.$ Hence, the Hausdorff dimension profile involves more information than then the single deterministic Hausdorff dimension value.

Remark 3.5

(Fractal Categorization). Given a kernel-generated fuzzy fractal $u_{F,g}$. Then, $u_{F,g}$ is not strong fuzzy fractal(Indeed, for any $0<\alpha <1,{dim}_H\left({\left[u_{F,g}\right]}_{\alpha }\right)=n={dim}_{\mathrm{ind}}\left({\left[u_{F,g}\right]}_{\alpha }\right)$).

Example 3.6

(Fuzzy middle-third Cantor Fractals). Let C denote the deterministic middle-third Cantor set with ${dim}_H\left(C\right)={\displaystyle \frac{log\left(2\right)}{log\left(3\right)}}.$ Then, for $g_i(1\le i\le 3)$ given in Example 2.13, the three distinct kernel-generated fuzzy fractals $u_{C,g_i}(1\le i\le 3)$ have the fuzzy Hausdorff dimension profile given by $D_{u_{F,g}}\left(\alpha \right)=n{\mathbf{1}}_{(0,1)}\left(\alpha \right)+\left({\displaystyle \frac{log\left(2\right)}{log\left(3\right)}}\right){\mathbf{1}}_{\left\{1\right\}}\left(\alpha \right),\mathrm{for}\mathrm{all}\alpha \in (0,1].$

4. Main Results

The inverse problem is considerably more complex in the fuzzy category than in its deterministic and random predecessors. In those two settings one prescribes a single numerical pair $(\rho ,l)$: a Hausdorff dimension together with a Lebesgue measure. In the fuzzy setting the prescribed object is an entire fuzzy fractal profile ${\mathrm{Profile}}_u\left(\alpha \right)=(D_u\left(\alpha \right),m_u\left(\alpha \right))$, $0<\alpha \le 1$, that is, a whole family of dimension–measure pairs indexed by the level parameter; the existence question therefore admits several formulations, according to which part of the profile is fixed. In order to (i) keep our results aligned with the established deterministic and random theorems and (ii) bring those theorems directly to bear on the construction, we restrict attention throughout to profiles whose Hausdorff dimension component $D_u$ is a two-valued jump function, prescribed at the core level $\alpha =1$.

This section discuss the existence of fractals with prescribed fuzzy fractal profile. Section 4.1 establishes the result for the case of fuzzy fractals in two routes: (i) the weak admissible kernel method, (ii) the decorated membership method. Section 4.2 establishes the similar result for the strong fuzzy fractals counterparts.

4.1. Existence of Fuzzy Fractals

We begin with a baseline existence result that secures the regularity property of upper semicontinuity at the cost of cardinality. The main theorem of this section will then trade this regularity for a strictly larger family.

Proposition 4.1

(Existence of usc Fuzzy Fractals). For any real $r>0$, $l\ge 0$, and $n\in \mathbb{N}$ with $\lceil r\rceil \le n$, put $\rho :=r{\mathbf{1}}_{\left\{0\right\}}\left(l\right)+n{\mathbf{1}}_{(0,\infty )}\left(l\right).$ Then, there exist aleph-one usc fuzzy fractals with fuzzy profile $(\rho ,l)$ at the core level $\alpha =1.$

Proof. 

We consider the map ${\mathrm{\Psi }}_0:{\mathcal{D}}_n^{comp}(r,l)\times \mathcal{G}\to {\mathcal{F}}_1\left({\mathbb{R}}^n\right)$ given by ${\mathrm{\Psi }}_0(F,g)=u_{F,g}.$ Then, for $S_0={\mathrm{\Psi }}_0({\mathcal{D}}_n^{comp}(r,l)\times \mathcal{G})$ we have $Card\left(S_0\right)=c$ given straightforward two -sided lower bounds inference from Remark 2.21, Proposition 3.1 and Proposition 3.3. □

The aleph-one ceiling above is an artifact of the compactness restriction (Remark 2.21) inherited through the kernel construction. Relaxing upper semicontinuity lifts it: the following theorem produces aleph-two fuzzy fractals with the same profile, which are necessarily non-usc (Remark 4.3).

Theorem 4.2

(Third Generalized Hausdorff Dimension Theorem). For any real $r>0$, $l\ge 0$, and $n\in \mathbb{N}$ with $\lceil r\rceil \le n$, put $\rho :=r{\mathbf{1}}_{\left\{0\right\}}\left(l\right)+n{\mathbf{1}}_{(0,\infty )}\left(l\right).$ Then, there exist aleph-two fuzzy fractals with fuzzy profile $(\rho ,l)$ at the core level $\alpha =1.$

Proof. 

We prove the claim in two routes as follows:

The First Proof: A Weak Admissible Kernel Route

Step (i): Core and Kernel. By Remark 2.21, choose one nonempty compact deterministic fractal $F\subseteq {\mathbb{R}}^n$ such that $({dim}_H\left(F\right),{\lambda }_n\left(F\right),{dim}_{ind}\left(F\right))=(\rho ,l,0).$ Then, for each $g\in {\mathcal{G}}_w,$ define $u_{F,g}\left(x\right)=g\left(d_E(x,F)\right).$

Step (ii): Fuzzy Fractality. Proposition 3.2 gives ${dim}_H\left({\left[u_{F,g}\right]}_{\alpha }\right)=n·{\mathbf{1}}_{(0,1)}\left(\alpha \right)+{dim}_{\mathrm{H}}\left(F\right)·{\mathbf{1}}_{\left\{1\right\}}\left(\alpha \right).$ Since ${\left[u_{F,g}\right]}_1=F,$ each $u_{F,g}$ has the same core $F,$ and the top cut witness fuzzy fractality, i.e., ${dim}_H\left({\left[u_{F,g}\right]}_1\right)=\rho >0={dim}_{ind}\left({\left[u_{F,g}\right]}_1\right).$

Step (iii): The Map. For the given bounded nonempty $F\subseteq {\mathbb{R}}^n,$ the distance function $h:{\mathbb{R}}^n\to [0,\infty )$ given by $h\left(x\right)=d_E(x,F)$ is onto. Hence, if $g_1\ne g_2,$ then there exists $t\in [0,\infty )$ such that $g_1\left(t\right)\ne g_2\left(t\right),$ and there exists $x\in {\mathbb{R}}^n$ such that $h\left(x\right)=d_E(x,F)=t.$ Therefore, $u_{F,g_1}\left(x\right)=g_1\left(d_E(x,F)\right)\ne g_2\left(d_E(x,F)\right)=u_{F,g_2}\left(x\right).$ Accordingly, $u_{F,g_1}\ne u_{F,g_2}.$ Thus, the map ${\mathrm{\Psi }}_{11}:{\mathcal{G}}_w\to {\mathcal{F}}_1\left({\mathbb{R}}^n\right)$ given by ${\mathrm{\Psi }}_{11}\left(g\right)=u_{F,g}$ is injective.

Step (iv): The Count. Finally, from step(iii), setting ${\mathcal{S}}_{11}={\mathrm{\Psi }}_{11}\left({\mathcal{G}}_w\right)$ and Remark 2.10 it follows that $2^c=Card\left({\mathcal{G}}_w\right)=Card\left({\mathcal{S}}_{11}\right)\le Card\left({\mathcal{F}}_1\left({\mathbb{R}}^n\right)\right).$ Hence, $Card\left({\mathcal{F}}_1\left({\mathbb{R}}^n\right)\right)=2^c.$

The Second Proof: A Decorated Membership Route

Step (i): Core and Kernel. Again, by Remark 2.21, choose one nonempty compact deterministic fractal $F\subseteq {\mathbb{R}}^n$ such that $({dim}_H\left(F\right),{\lambda }_n\left(F\right),{dim}_{ind}\left(F\right))=(\rho ,l,0).$ Next, fix one admissible kernel $g\in \mathcal{G},$ for instance $g\left(t\right)=exp(-t),$ set $c_0={\displaystyle \frac{1}{2}},$ and put $R=g^{-1}\left(c_0\right)=log2>0$.

Step (ii): Decoration Site. Define the far field

$$M:=\{x\in {\mathbb{R}}^n:d_E(x,F)\ge R+1\}.$$

(2)

The following observations hold:

1.

Since F is bounded, M contains a Euclidean ball; hence $Card\left(M\right)=c.$

2.

The far field M lies at distance at least 1 from the closed parallel body ${\left[u_{F,g}\right]}_{c_0}=\{x\in {\mathbb{R}}^n:d_E(x,F)\le R\}.$ By construction, $F\subseteq {\left[u_{F,g}\right]}_{c_0}\subseteq M^c,$ implying $M\cap {\left[u_{F,g}\right]}_{c_0}=\varnothing .$

3.

For $x\in M,$ we have $d_E(x,F)\ge R+1>R,$ so $u_{F,g}\left(x\right)=g\left(d_E(x,F)\right)<g\left(R\right)=c_0.$

Step (iii): The Family. Let $\mathcal{N}=\{S\subseteq M:S\mathrm{non}-\mathrm{closed}\mathrm{in}{d}_E\}.$ Then, by step(ii) item (1), $Card\left(\mathcal{N}\right)=2^c.$ For each $S\in \mathcal{N},$ define the fuzzy set:

$$u_S\left(x\right):=max\{u_{F,g}\left(x\right),c_0{\mathbf{1}}_S\left(x\right)\},x\in {\mathbb{R}}^n.$$

(3)

Step (iv): Cuts and Profile. For given $S\in \mathcal{N}$ and every $0<\alpha \le 1$ by equation (3):

$${\left[u_S\right]}_{\alpha }={\left[u_{F,g}\right]}_{\alpha }\cup S_{c_0,\alpha }:S_{c_0,\alpha }=S\cap \{x\in {\mathbb{R}}^n:c_0\ge \alpha \}).$$

(4)

We investigate the fuzzy profile as follows:

1.

Case $\alpha =1:$ Here, by equation (4) $S_{c_0,\alpha }=\varnothing ,$ and ${\left[u_S\right]}_1={\left[u_{F,g}\right]}_1=\overline{F}=F.$ Hence, ${dim}_H\left({\left[u_S\right]}_1\right)=\rho >0.$

2.

Case $0<\alpha <1:$ Here, as in proof (part (a)) of Proposition 3.1, ${dim}_H\left({\left[u_{F,g}\right]}_{\alpha }\right)=n.$ In addition, ${dim}_H\left(S_{c_0,\alpha }\right)\le n.$ Hence, by equation (4) it follows that: ${dim}_H\left({\left[u_S\right]}_{\alpha }\right)=max\{n,{dim}_H\left(S_{c_0,\alpha }\right)\}=n.$

Accordingly, both cases can be summarized as ${dim}_H\left({\left[u_S\right]}_{\alpha }\right)=n·{\mathbf{1}}_{(0,1)}\left(\alpha \right)+{dim}_{\mathrm{H}}\left(F\right)·{\mathbf{1}}_{\left\{1\right\}}\left(\alpha \right).$ Finally, it is straightforward to check that $m_{u_S}\left(\alpha \right)>0,\mathrm{for}\mathrm{all}\alpha \in (0,1).$

Step (v): Fuzzy Fractality. For given $S\in \mathcal{N},$ by step(iv) item(1), we have ${dim}_H\left({\left[u_S\right]}_1\right)=\rho >0={dim}_{ind}\left({\left[u_S\right]}_1\right)$ where in which ${\left[u_S\right]}_1=F\ne \varnothing .$ Thus, $u_S$ is a fuzzy fractal.

Step (vi): The Map . First, for given $S\in \mathcal{N},$ using step(ii) item(2), we have:

$$M\cap {\left[u_S\right]}_{c_0}=M\cap ({\left[u_{F,g}\right]}_{c_0}\cup S)=(M\cap {\left[u_{F,g}\right]}_{c_0})\cup (M\cap S)=\varnothing \cup S=S.$$

(5)

Second, let $u_{S_1}=u_{S_2}.$ Then, by equation (5) we have $S_1=S_2.$ Thus, the map ${\mathrm{\Psi }}_{12}:\mathcal{N}\to {\mathcal{F}}_1\left({\mathbb{R}}^n\right)$ given by ${\mathrm{\Psi }}_{12}\left(S\right)=u_S$ is injective.

Step (vii): The Count . Finally, from step(vi), setting ${\mathcal{S}}_{12}={\mathrm{\Psi }}_{12}\left(\mathcal{N}\right)$ and Step(iii) it follows that $2^c=Card\left(\mathcal{N}\right)=Card\left({\mathcal{S}}_{12}\right)\le Card\left({\mathcal{F}}_1\left({\mathbb{R}}^n\right)\right).$ Hence, $Card\left({\mathcal{F}}_1\left({\mathbb{R}}^n\right)\right)=2^c.$

□

Remark 4.3.

The available fuzzy fractals in Theorem 4.2 are indeed non-usc. To see this fact, it is sufficient in the second proof of the Theorem to consider the non-closed set ${\left[u_S\right]}_{c_0}={\left[u_{F,g}\right]}_{c_0}\cup S$ in which its components are separated with positive distance; the first component is closed; and, the second component is non-closed.

Remark 4.4.

The two routes in the proof of Theorem 4.2 have the following similarities: First, both routes provide aleph-two fuzzy fractals. Second, both routes provide similar fuzzy fractal profiles where the Hausdorff dimension profile is a two-valued jump function. Finally, both routes use kernels in their provisions.

Remark 4.5.

The two routes in the proof of Theorem 4.2 have the following differences: First, the weak admissible kernel route uses the full force of cardinality of such kernels while the decorated membership route retains a smooth kernel but utilizes perturbing a single far field set. Second, while the weak admissible route is silent on additional non-usc feature of the provided fuzzy fractals, the decorated membership route proves their existence.

4.2. Existence of Strong Fuzzy Fractals

We now turn to the strong counterpart, where fractality is required at every level $0<\alpha \le 1$. The kernel-generated construction of Section 4.1 is unavailable here, since it is never strong (Remark 3.5); we instead employ the indicator fuzzy fractals of Example 2.7. As before, we first record a usc baseline at the cost of cardinality, and then relax upper semicontinuity to reach the maximal family.

Proposition 4.6

(Existence of usc Strong Fuzzy Fractals). For any real $r>0$, $l\ge 0$, and $n\in \mathbb{N}$ with $\lceil r\rceil \le n$, put $\rho :=r{\mathbf{1}}_{\left\{0\right\}}\left(l\right)+n{\mathbf{1}}_{(0,\infty )}\left(l\right)$. Then, there exist aleph-one usc strong fuzzy fractals with fuzzy profile $(\rho ,l)$ at the core level $\alpha =1$.

Proof. 

We consider the map ${\mathrm{\Psi }}_{21}:{\mathcal{D}}_n^{\mathrm{comp}}(r,l)\to {\mathcal{F}}_2\left({\mathbb{R}}^n\right)$ given by ${\mathrm{\Psi }}_{21}\left(F\right)={\mathbf{1}}_F$. By Example 2.7, each ${\mathbf{1}}_F$ is a usc strong fuzzy fractal whose profile equals $(\rho ,l)$ at every level $0<\alpha \le 1$, in particular at the core level $\alpha =1$; and the map is injective, since $F={\left[{\mathbf{1}}_F\right]}_1$. Then, for $S_{21}={\mathrm{\Psi }}_{21}\left({\mathcal{D}}_n^{\mathrm{comp}}(r,l)\right)$ we have $\mathrm{Card}\left(S_{21}\right)=\mathfrak{c}$ by Remark 2.21. □

Remark 4.7

(The indicator is inessential). The map ${\mathrm{\Psi }}_{21}\left(F\right)={\mathbf{1}}_F$ is deliberately minimal, but the usc baseline of Proposition 4.6 does not depend on it. To see this feature, given a compact fractal core F, choose pairwise disjoint closed balls $\overline{B}(p_n,{\rho }_n)$, $n\ge 2$, in its complement $F^c$, and place in each ball a compact, totally disconnected fractal $F_n$ of positive Hausdorff dimension via a similarity transformation, with $F_1:=F$. Then ${\left\{F_n\right\}}_{n=1}^{\infty }$ is a sequence of pairwise disjoint closed fractals with $F_1=F$. Set $u_F\left(x\right):={\sum }_{n=1}^{\infty }{a}_n{\mathbf{1}}_{F_n}\left(x\right)$ where ${\left\{a_n\right\}}_{n=1}^{\infty }$ is strictly decreasing sequence in $(0,1]$. Disjointness makes $u_F$ well-defined with cuts ${\left[u_F\right]}_{\alpha }={\bigcup }_{n\le a_n^{-1}\left(\alpha \right)}{F}_n$ and core ${\left[u_F\right]}_1=F$; each cut is a finite union of totally disconnected fractals, so ${dim}_{\mathrm{ind}}\left({\left[u_F\right]}_{\alpha }\right)=0<{dim}_H\left({\left[u_F\right]}_{\alpha }\right)$ and $u_F$ is a usc strong fuzzy fractal. Next, the assignment ${\mathrm{\Psi }}_{21}^{*}:F\mapsto u_F$ is again injective and reproduces the cardinality ${\aleph }_1$. Thus, the result is carried again by genuinely multi-valued fuzzy fractals and not by the indicator alone; what the indicator route fixes is the dimension profile, held here at the two-valued jump.

The aleph-one ceiling above is again an artifact of the compactness restriction (Remark 2.21), here entering through the closedness of the core required for upper semicontinuity. Dropping closedness admits the full available set ${\mathcal{D}}_n(r,l)$ of cores and lifts the family to aleph-two, as the following theorem shows; the resulting strong fuzzy fractals are necessarily non-usc.

Theorem 4.8

(Third Generalized Hausdorff Dimension Theorem: Strong). For any real $r>0$, $l\ge 0$, and $n\in \mathbb{N}$ with $\lceil r\rceil \le n$, put $\rho :=r{\mathbf{1}}_{\left\{0\right\}}\left(l\right)+n{\mathbf{1}}_{(0,\infty )}\left(l\right).$ Then, there exist aleph-two strong fuzzy fractals with fuzzy profile $(\rho ,l)$ at the core level $\alpha =1.$

Proof. 

Step (i). We consider the injective map ${\mathrm{\Psi }}_{22}:{\mathcal{D}}_n(r,l)\to {\mathcal{F}}_2\left({\mathbb{R}}^n\right)$ given by ${\mathrm{\Psi }}_{22}\left(F\right)={\mathbf{1}}_F.$ Step(ii). We define ${\mathcal{S}}_{22}={\mathrm{\Psi }}_{22}\left({\mathcal{D}}_n(r,l)\right).$ Then, from step (i) and Theorem 2.20, it follows that $2^c=Card\left({\mathcal{D}}_n(r,l)\right)=Card\left({\mathcal{S}}_{22}\right)\le Card\left({\mathcal{F}}_2\left({\mathbb{R}}^n\right)\right).$ Hence, $Card\left({\mathcal{F}}_2\left({\mathbb{R}}^n\right)\right)=2^c.$ □

Remark 4.9.

The available strong fuzzy fractals in Theorem 4.8 are indeed generally non-usc; to see this, it is sufficient to note that the aleph-two cardinality of ${\mathcal{D}}_n(r,l)$ is carried by non-closed cores F, for which the indicator ${\mathbf{1}}_F$ has the non-closed cut ${\left[{\mathbf{1}}_F\right]}_1=F$ (Example 2.7).

Remark 4.10.

Theorem 4.8 realizes the strong family through the canonical embedding of deterministic fractals into the fuzzy category via indicator membership functions. This strong case should therefore be read as a sharp crisp-core counterpart to Theorem 4.2. Moreover, the genuinely multi-valued enrichment of Remark 4.7 applies verbatim whenever $\overline{F}\ne {\mathbb{R}}^n$, since it places its auxiliary fractals in the nonempty interior of $F^c$ and leaves F in place; the dense case $\overline{F}={\mathbb{R}}^n$, which carries the aleph-two cardinality, admits no such complementary room and is left to future work.

5. Discussion

5.1. Summary of Contributions

This work established an existence theorem for fuzzy fractals of a prescribed fuzzy fractal profile with the highest possible cardinal number of aleph-two (Theorem 4.2). In doing so, it extended to the fuzzy setting the former existence theorems of the deterministic (Theorem 2.20) and random (Theorem 2.22) settings, completing the third stage of the inverse Hausdorff-dimension program in the fuzzy category.

The distinctive feature of the fuzzy setting is that the single Hausdorff dimension of the deterministic and random predecessors is replaced by the fuzzy fractal profile ${\mathrm{Profile}}_u\left(\alpha \right)=(D_u\left(\alpha \right),m_u\left(\alpha \right))$. The kernel-generated construction exhibits a sharp dimension jump (Remark 3.4): the deterministic value ${dim}_H\left(F\right)$ is retained at the core level $\alpha =1$, while the full ambient dimension n is attained at every lower level $0<\alpha <1$. The profile therefore carries strictly more information than the single dimension value.

The provision of the aleph-two family is governed by a regularity–cardinality dichotomy that parallels the deterministic and random cases. Imposing upper semicontinuity restricts the family to aleph-one (Proposition 4.1), whereas relaxing it recovers the maximal aleph-two (Theorem 4.2); this is exactly the mechanism by which the compactness (Remark 2.21) and compact-valued (Remark 2.23) restrictions collapse the deterministic and random families to aleph-one.

The key contributions to the fractal geometry literature align with those of the deterministic and random cases. First, the work highlights the advantage of Cantor-type cores over other classical fractals in securing the existence of fuzzy fractals. Second, the existence proof is constructive. Third, it furnishes a further example of a family of sets with cardinal number aleph-two. Fourth, it demonstrates that regardless of settings, the regularity has inverse relationship with cardinality. Finally, the cardinality of the available fractals is independent of the construction setting—deterministic, random, or fuzzy—and equals aleph-two.

Table 1 summarizes the inverse problem program across the former published literature and the present paper.

5.2. Future Work

Future work may extend and assess the present results in several directions.

First, the existence theorems established here concern fuzzy fractal profiles whose first component is a two-valued jump function, with the top slice at the core level $\alpha =1$ carrying the existence properties inherited from the deterministic setting. A natural next step is to generalize them to profiles whose first component is simultaneously multi-valued and fractal at every level, so that fractality is distributed across all cuts rather than concentrated at the core. Naturally, since the prescribed parameters differ from those of the deterministic and random settings, the resulting existence results need not align with those two former settings.

Second, an alternative direction concerns the regularity–cardinality dichotomy left open by the present work: to determine the maximal cardinality of the existence families under the upper-semicontinuity condition, for both fuzzy fractals and strong fuzzy fractals.

Third, only the Hausdorff fractal dimension is treated here. It is plausible to evaluate the counterpart results for other fractal dimensions, such as the box-counting, packing, or Assouad dimensions.

Fourth, the present setting is Euclidean throughout. It is plausible to establish and assess analogous results in more abstract, non-Euclidean fuzzy fractal settings.

Finally, it is plausible to investigate whether the fuzzy fractal profiles and existence results obtained here also arise as attractors of contractive iterated fuzzy systems.

5.3. Conclusion

This work has provided a further partial solution to the inverse existence problem—constructing families of fractals with a prescribed fractal dimension and Lebesgue measure—now in the fuzzy category. It secured the fuzzy case at cardinality aleph-two and extended the earlier deterministic and random results across both setting and regularity, exhibiting a single regularity–cardinality dichotomy. In doing so, it opened the way toward realizing non-constant profiles that are fractal at every level, and toward the maximal-cardinality question under upper semicontinuity.

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