# Cantor Paradoxes, Possible Worlds and Set Theory

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## Abstract

**:**

## 1. Introduction

## 2. Possible Worlds and Set Theory

**Definition**

**1.**

**maximally consistent**if for every well-formulated proposition p, $p\in S$ or $\neg p\in S$.

**Theorem**

**1.**

**Proof.**

- For definition 1 $\forall S\phantom{\rule{0.166667em}{0ex}}\left(S\phantom{\rule{4.pt}{0ex}}\mathrm{is}\phantom{\rule{4.pt}{0ex}}\mathrm{maximally}\phantom{\rule{4.pt}{0ex}}\mathrm{consistent}\right)\leftrightarrow \left(\right)open="("\; close=")">\forall q\left(\right)open="("\; close=")">q\in S\vee \neg q\in S\wedge \left(\right)open="("\; close=")">S\phantom{\rule{0.166667em}{0ex}}\mathrm{is}\phantom{\rule{4.pt}{0ex}}\mathrm{consistent}$.
- There is a set of propositions A that is maximally consistent.
- $CardP\left(A\right)>CardA$.
- For each set belonging to the power set $P\left(A\right)$ there will be a proposition of that set. $\forall S\left(\left(\right),S\in P\left(A\right)\to \exists q\phantom{\rule{4.pt}{0ex}}\left(\right)open="("\; close=")">q\phantom{\rule{4.pt}{0ex}}\mathrm{is}\phantom{\rule{4.pt}{0ex}}\mathrm{about}\phantom{\rule{4.pt}{0ex}}S\right)$.
- Let r be a similar proposition. By 1 it follows that: $\left(\right)open="("\; close=")">r\in A$.
- As 5 applies to any element of $P\left(A\right)$, there will be as many propositions A as sets belonging to the power set of A, so it follows that: $CardP\left(A\right)\le CardA$.

- It is not plausible to reject the definition of a maximally consistent set of Proposition 1.
- Nor is it reasonable to reject Cantor’s Theorem 3.

- Assumption 2 is that there is a set A of propositions that are maximally consistent.
- There is at least one proposition for each subset of the power set of A that is 4.

## 3. Seven Cantor Paradoxes

- C1
- There is no set of all truths.Suppose we have a set of all truths, let $A=\{{t}_{1},{t}_{2},\dots ,{t}_{n}\}$. There is a cardinality of the set of all truths $CardA=n$. However, it turns out that each subset of A may be assigned a true proposition. For example, if $B\subseteq A$, defined as the set $\{{p}_{i-1},{p}_{i},{p}_{i+1}\}$, then there exists at least one truth ${p}_{i}\in B$. This truth has to be part of the set A, because it is the set of all truths. However, the set of all truths then has as many elements as the power set of A, $P\left(A\right)$. Thus, $CardA\ge CardP\left(A\right)$, contrary to Cantor’s result. This form of reasoning shows that there is no set of all necessary truths and there is no set of all falsehoods, nor is there a set of all truths which are known by an omniscient being. In the case of necessary truths, the difference is that the set A is restricted. Each subset has assigned another as a trivially true and necessary proposition, because every true proposition belonging to a given subset is a necessary fact. In the case of the set of all falsehoods, set A will consist of all the denials of the set that have entered into the central argument about the truths. A false proposition may be assigned to each subset (e.g., if a subset of A is $\{{p}_{1},{p}_{2}\}$, then a falsehood can be made by stating that ${p}_{1}\notin \left(\right)open="\{"\; close="\}">{p}_{1},{p}_{2}$) and the argument works again in the same way. Finally, in the case of all true propositions known by an omniscient being, the argument works because the omniscient being—precisely because it is omniscient—must also know each subset of A, which is a subset of A.
- C2
- There is no set of all states of affairs or facts.This is a very similar argument to 1. This is because the states of affairs or facts are conceived as those which make a proposition ordinarily true (or false, depending on the case). If there is an argument for the truth, then it seems obvious that there will be an argument for those entities that are correlated with true propositions. Let A be the set of all states of affairs in the world. To each subset of A, we can assign a new state of affairs. For example, if there is a subset $B=\{{S}_{1},{S}_{2}\}$, then there is also a state of affairs ${S}_{3}$ constituted by the fact that ${S}_{1}$ belongs to the set B.
- C3
- There is no set of propositions or sentences that is maximally consistent.The reasoning is as detailed above. This is the argument that seems to directly attack actualistic conceptions, which work with “full stories.”
- C4
- There is no set of all possible states of affairs.This is a variation of 2 which is based on the total of all current states of affairs. Let A be the set of all possible states of affairs. To each subset of A we may assign a possible state of affairs. For example, if $B\subseteq A$ consists of $\{{S}_{1},{S}_{2}\}$, then there is a possible state of affairs and it is consistent, where ${S}_{1}\in B$. This argument seems to go directly against the way of conceiving all possible worlds as a maximum possible states of affairs [16].
- C5
- There is no set of all essences.This is an argument that has been directed at Plantinga’s modal conception, in which non-current possible entities are represented by a substance that is not found instantiated [2]. The essence is here a set of properties that are satisfied by one and only one individual in all possible worlds. It must be assumed that there are individual essences that are not only objects but also states of affairs. That is, each state of things, actual or merely possible, is assigned a property that is satisfied by this state of affairs, and nothing else in all possible worlds. We use special symbols called descriptions, and they are of the form $\left(\right)$ that we can read: the x that satisfies the function F. For example, if we want to specify the individual essence of the state of affairs in which the horse Rocinante is starving, we can define a consistent property to be something that is at the same time Rocinante and starving. That is, if R is the individual essence of horse Rocinante ($\left(\right)$, R fulfills the function H of being a horse) and S is the property of being starved ($\left(\right)$R fulfills the function S of being starved), then there is the property of $\left(\right)open="|"\; close>RH\left(R\right)$. It is trivial then that these individual essences or states of affairs can be defined, if there are individual essences for objects that constitute such states of affairs. Let A be the set of all individual essences. Each subset of A can be assigned an individual essence. In effect, let B be a set $B\subseteq A$ composed of $\{{E}_{1},{E}_{2}\}$, where ${E}_{1}$ and ${E}_{2}$ are individual essences. We consider that ${E}_{1}=\left(\right)open="|"\; close>RH\left(R\right)$ and ${E}_{2}=\left(\right)open="|"\; close>QM\left(Q\right)$ (for example, ${E}_{1}$ is the individual essence of the state of affairs for Rocinante to be starving and ${E}_{2}$ is the individual essence of being the man $\left(M\right)$ Don Quixote $\left(Q\right)$ a knight errant $\left(KE\right)$), then it is trivial that there is an individual essence ${E}_{3}={E}_{1}\cap {E}_{2}=\left(\right)open="["\; close="]">\left(\right)open="|"\; close>RH\left(R\right)\wedge \left(\right)open="|"\; close>RS\left(R\right)$ (i.e., the individual essence of the state of affairs for Rocinante to be starving, and Don Quixote a knight errant). Therefore, there are so many individual essences as subsets of the set of all individual essences. This creates the Cantor paradox immediately.
- C6
- There is no set of all entities.Simply assume that our ontology is based on a principle of mereology as follows: if there are two different objects x and y, then there is a mereological sum $(x+y)$. This is an intuitive plausible principle. Let A be the set of all entities of the world, such that $A=\left(\right)open="\{"\; close="\}">{x}_{1},{x}_{2},\dots ,{x}_{n}$. Each subset of $P\left(A\right)$ can be assigned a specific entity, constituted by the mereological sum of all entities that are members of that subset. According to the mereological principle indicated, these entities are also entities of the world and must be elements of the set A. It follows, then, that it appears that A has as many elements as its power set, contrary to what has been established by Cantor’s theorem.
- C7
- There is no set of all universals.Suppose there was a set of all universals $A=\left(\right)open="\{"\; close="\}">{U}_{1},{U}_{2},\dots ,{U}_{n}$. There is a power set of A, $P\left(A\right)$ whose cardinality must be greater than the cardinality of A. However, for each subset of A, we can define a universal complex set. For example, given $B\subseteq A$ such that $B=\left(\right)open="\{"\; close="\}">{U}_{1},{U}_{2}$, then there is a universal set which is the joint instantiation of ${U}_{1}$ and ${U}_{2}$. It happens that there are so many universal sets, as subsets of the set A, but all these universals belong to A, contrary to what has been established by Cantor’s theorem.

## 4. Cantor Paradoxes and Possible Worlds

- Linguistic (or propositional) theory.
- Combinatorial theory.
- Plantinga’s theory.
- Theory of possible worlds as properties.

#### 4.1. Linguistics (Propositional) Theories

**Definition**

**2.**

**A state of things is current**if the proposition expressing the giving of such a state of things is true.

**Definition**

**3.**

**The current world is determined**by the maximally consistent set of sentences and propositions, in which all elements are true.

**Definition**

**4.**

**A state of things is possible**if the proposition expressing this state of things belongs to at least one maximally consistent set.

**Definition**

**5.**

**A state of affairs is necessary**if the proposition expressing the given of this state of affairs belongs to all maximally consistent sets.

#### 4.2. Combinatorial Theories

**Definition**

**6.**

**The set of all possible states of affairs**is defined as the set of all ordered pairs (or n-tuples ordinate) of objects and properties $\left(\right)$, $<\phantom{\rule{3.33333pt}{0ex}}{P}_{2},{x}_{1}>\phantom{\rule{3.33333pt}{0ex}},<\phantom{\rule{3.33333pt}{0ex}}{P}_{2},{x}_{2}>\phantom{\rule{3.33333pt}{0ex}},\dots ,<\phantom{\rule{3.33333pt}{0ex}}{P}_{2},{x}_{n}\phantom{\rule{3.33333pt}{0ex}}>$, $\left(\right)$.

**Definition**

**7.**

**A given set of all states**, independent of other things, may be defined as all possible worlds that are the totality of all possible combinations of states of affairs.

#### 4.3. Plantinga’s Theory

- Are given by the characteristics of objects and properties.
- Are the entities that are true (or false) in relation to sentences or propositions.

- S includes ${S}^{*}$ iff, necessarily, if S is actual then ${S}^{*}$ is actual.
- S excludes ${S}^{*}$ iff S and ${S}^{*}$ cannot be together.

- First, Plantinga seems to require a set of all possible states of affairs (i.e., something that seems to be quantifying its definition of a “state of maximum things”, and the definitions of “inclusion” and “exclusion”), and this is what makes his conception sensitive to the argument C4.
- Second, even if Plantinga does not conceive of possible worlds as sets of propositions, he contends that there is a set of propositions associated with each possible world which must be maximally consistent. “Books” are susceptible to falling into the Cantor paradox of the C3 form, just as the modal linguistic conceptions.
- Third, Plantinga’s design disclaims possible objects using a set of individual essences. The postulation of a set of individual essences, however, seems to be affected by Cantor paradoxes of the C4 form.

#### 4.4. Theory of Possible Worlds as Properties

**Definition**

**8.**

**A structural universal**is a universal, that is, a certain determination that it is by its nature apt to be predicated by many things that arise from the complexion of other more basic universals.

**Example**

**1.**

**Definition**

**9.**

**A state of affairs S is possible**if there is at least one maximum structural universal U, such that if U was instantiated, then S would be given.

**Definition**

**10.**

**A state of affairs S is necessary**if for all maximum structural universals U, if U is instantiated, then S would be given.

- 1.
- It is a specification of a “maximum” individual—that is, an individual such that every individual is part of it.

**Definition**

**11.**

**maximum individual**as

- 2.
- The specification of the maximum individual is exhaustive, in the sense that, given a set of all universals $\left(\right)$ that are attributed to each party, either the universal ${P}_{i}$ will be attributed or not to that universal ${P}_{i}$ (for a universal ${P}_{i}\in \left(\right)open="\{"\; close="\}">{P}_{1},{P}_{2},\dots ,{P}_{n}$, of course). Then, the form by which a universal structural becomes a maximum, is because it will specify each determination held by each part of a possible world, through clauses in this way:

**Definition**

**12.**

**an exhaustive description**of all its parts. In turn, the parties from which a possible world is made can be fully encoded with the full specification of each of its parts, then with the full specification of each of the parts of those parties, etc.

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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Usó-Doménech, J.-L.; Nescolarde-Selva, J.-A.; Segura-Abad, L.; Gash, H.; Alonso-Stenberg, K.
Cantor Paradoxes, Possible Worlds and Set Theory. *Mathematics* **2019**, *7*, 628.
https://doi.org/10.3390/math7070628

**AMA Style**

Usó-Doménech J-L, Nescolarde-Selva J-A, Segura-Abad L, Gash H, Alonso-Stenberg K.
Cantor Paradoxes, Possible Worlds and Set Theory. *Mathematics*. 2019; 7(7):628.
https://doi.org/10.3390/math7070628

**Chicago/Turabian Style**

Usó-Doménech, José-Luis, Josué-Antonio Nescolarde-Selva, Lorena Segura-Abad, Hugh Gash, and Kristian Alonso-Stenberg.
2019. "Cantor Paradoxes, Possible Worlds and Set Theory" *Mathematics* 7, no. 7: 628.
https://doi.org/10.3390/math7070628