# A Finitely Axiomatized Non-Classical First-Order Theory Incorporating Category Theory and Axiomatic Set Theory

## Abstract

**:**

## 1. Introduction

#### 1.1. Motivation

- (i)
- The number of axioms of ZF(C) is infinite, as a result of which ZF(C) cannot be written down explicitly;
- (ii)

- (i)
- The natural numbers $0,1,2,\dots $ defined as sets;
- (ii)
- The countable set of natural numbers $\mathbb{N}=\{0,1,2,\dots \}$;
- (iii)
- Countably many subsets ${S}_{1},{S}_{2},{S}_{3},\dots $ of the set $\mathbb{N}$;
- (iv)
- A countable set K that contains the above subsets of $\mathbb{N}$, so $K=\{{S}_{1},{S}_{2},{S}_{3},\dots \}$.

#### 1.2. Related Works

“At present we can do no more than note that we have one more reason here to entertain reservations about set theory and that for the time being no way of rehabilitating this theory is known”.[6]

#### 1.3. Informal Overview of the Main Result

**Definition**

**1.**

**universe of discourse**is a category $\mathcal{C}$ consisting of:

- (i)
- A proper class of objects, each of which is a
**set**; - (ii)
- A proper class of arrows, each of which is a
**function**.

“Instead of having to say the entities to which the predicates might in principle apply, we can make things easier for ourselves by collectively calling these entities the universe of discourse”.[15]

**given**a set X and a function ${f}_{X}$, precisely one expression ${f}_{X}:y\mapsto z$ is true for each element y in the domain of ${f}_{X}$. For example, we have

**Top**,

**Mon**,

**Grp**, etc., which are subjects of study in category theory, can be viewed as subcategories of the category of sets and functions of Definition 1, thus providing a new approach to the foundational problem identified in [21]. While that latter point (ii) hardly needs elaboration, the former (i) does. The crux is that one does not need to apply the non-classical sum function axiom directly: what one uses is the theorem—or rather: the theorem schema—that given any set X, we can construct a new function on X by giving a function prescription. So, this is a philosophical nuance: in ZF one constructs a new object with the ∈-relation, but in the present framework one can construct a new function ${f}_{X}$ not with the ∈-relation but by simply defining which expressions ${f}_{X}:y\mapsto z$ are true for the elements y in the domain. On account of the sum function axiom, it is then a guarantee that the function ${f}_{X}$ exists. So, given any set X one can simply give a defining function prescription

#### 1.4. Point-by-Point Overview

- (i)
- Zermelo–Fraenkel set theory is the most widely accepted foundational theory for mathematics, but the problem is that it has two unwanted features:
- –
- It has infinitely many axioms;
- –
- It has a countable model in which the powerset of the naturals is countable;

- (ii)
- Since the 1950s it has been generally accepted that this problem has no standard solution;
- (iii)
- This paper presents a nonstandard solution, which opens up an entirely new research field that may be called “infinitary mathematical logic”;
- (iv)
- The main result, the non-classical theory $\mathfrak{T}$, incorporates category theory and axiomatic set theory in a single framework;
- (v)
- Category theory is incorporated by the ontological assumption that the universe of $\mathfrak{T}$ is a class of sets and a class of functions: this universe satisfies the axioms of category theory;
- (vi)
- Set theory is incorporated in the sense that the axioms of ZF can be derived from the axioms of $\mathfrak{T}$;
- (vii)
- $\mathfrak{T}$ can be considered stronger than ZF since it lacks the two unwanted features:
- –
- $\mathfrak{T}$ has finitely many axioms of finite length;
- –
- If $\mathfrak{T}$ has a model, it is uncountable;

- (viii)
- $\mathfrak{T}$ is introduced here as a candidate for a foundational theory for mathematics, the corresponding philosophy of mathematics being that mathematics—i.e., all of mathematics—can be viewed as the collection of valid inferences within the framework of $\mathfrak{T}$;
- (ix)
- The true merit of $\mathfrak{T}$ as a foundational theory for mathematics has yet to be established by a program for further research: negative results may invalidate the idea that $\mathfrak{T}$ is suitable as a foundational theory for mathematics.

## 2. Axiomatic Introduction

#### 2.1. Formal language

**Definition**

**2.**

**vocabulary**of the language ${L}_{\mathfrak{T}}$ of $\mathfrak{T}$ consists of the following:

- (i)
- The constants ∅ and $\mathbf{\omega}$, to be interpreted as the empty set and the first infinite ordinal;
- (ii)
- The constants ${1}_{\varnothing}$, to be interpreted as the inactive function;
- (iii)
- Simple variables $x,X,y,Y,\dots $ ranging over sets;
- (iv)
- For any constant $\widehat{\mathbf{X}}$ referring to an individual set, composite symbols ${f}_{\widehat{\mathbf{X}}},{g}_{\widehat{\mathbf{X}}},\dots $ with an occurrence of the constant $\widehat{\mathbf{X}}$ as the subscript are simple variables ranging over functions on that set $\widehat{\mathbf{X}}$;
- (v)
- For any simple variable X ranging over sets, composite symbols ${f}_{X},{g}_{X},\cdots $ with an occurrence of the variable X as the subscript are composite variables ranging over functions on a set X;
- (vi)
- Simple variables $\alpha ,\beta ,\dots $ ranging over all things (sets and functions);
- (vii)
- The binary predicates “∈” and “=”, and the ternary predicates “$(.):(.)\twoheadrightarrow (.)$” and “$(.):(.)\mapsto (.)$”;
- (viii)
- The logical connectives of first-order logic $\neg ,\wedge ,\vee ,\Rightarrow ,\iff $;
- (ix)
- The universal and existential quantification symbols $\forall ,\exists $;
- (x)
- The brackets “(’ and ‘)". □

**Remark**

**1.**

**Definition**

**3.**

**syntax**of the language ${L}_{\mathfrak{T}}$ is defined by the following clauses:

- (i)
- If t is a constant, or a simple or a composite variable, then t is a term;
- (ii)
- If ${t}_{1}$ and ${t}_{2}$ are terms, then ${t}_{1}={t}_{2}$ and ${t}_{1}\in {t}_{2}$ are atomic formulas;
- (iii)
- If ${t}_{1}$, ${t}_{2}$, and ${t}_{3}$ are terms, then ${t}_{1}:{t}_{2}\twoheadrightarrow {t}_{3}$ and ${t}_{1}:{t}_{2}\mapsto {t}_{3}$ are atomic formulas;
- (iv)
- If $\Phi $ and $\Psi $ are formulas, then $\neg \Phi $, $(\Phi \wedge \Psi )$, $(\Phi \vee \Psi )$, $(\Phi \Rightarrow \Psi )$, $(\Phi \iff \Psi )$ are formulas;
- (v)
- If $\Psi $ is a formula and t a simple variable ranging over sets, over all things, or over functions on a constant set, then $\forall t\Psi $ and $\exists t\Psi $ are formulas;
- (vi)
- If X and ${f}_{\widehat{\mathbf{X}}}$ are simple variables ranging respectively over sets and over functions on the set $\widehat{\mathbf{X}}$, ${f}_{X}$ a composite variable with an occurrence of X as the subscript, and $\Psi $ a formula with an occurrence of a quantifier $\forall {f}_{\widehat{\mathbf{X}}}$ or $\exists {f}_{\widehat{\mathbf{X}}}$ but with no occurrence of X, then $\forall X[X\backslash \widehat{\mathbf{X}}]\Psi $ and $\exists X[X\backslash \widehat{\mathbf{X}}]\Psi $ are formulas. □

**Remark**

**2.**

**Definition**

**4.**

- (i)
- If t is a simple variable ranging over sets, over all things, or over functions on a constant set, then $\forall t$ and $\exists t$ are
**quantifiers with a simple variable**; - (ii)
- If ${f}_{X}$ is a composite variable, then $\forall {f}_{X}$ and $\exists {f}_{X}$ are a
**quantifiers with a composite variable**.

**double quantifier**. □

**Definition**

**5.**

- (i)
- An occurrence of ${f}_{X}$ in a formula $\Psi $ is
**free**if that occurrence is neither in the scope of a quantifier with the composite variable ${f}_{X}$ nor in the scope of a quantifier with the simple variable X; - (ii)
- An occurrence of ${f}_{X}$ in a formula $\Psi $ is
**bounded**if that occurrence is in the scope of a quantifier with the composite variable ${f}_{X}$.

**sentence**is a formula with no free variables—simple or composite. A formula is

**open**in a variable if there is a free occurrence of that variable. A formula that is open in a composite variable ${f}_{X}$ is also open in the simple variable X. □

**Definition**

**6.**

**semantics**of any sentence without a quantifier with a composite variable is as usual. Furthermore,

- (i)
- A sentence $\forall X\Psi $ with an occurrence of a quantifier $\forall {f}_{X}$ or $\exists {f}_{X}$ with the composite variable ${f}_{X}$ is valid in a model $\mathcal{M}$ if and only if for every assignment g that assigns an individual set $g\left(X\right)=\underline{\mathbf{X}}$ in $\mathcal{M}$ as a value to the variable X, the sentence $[\underline{\mathbf{X}}\backslash X]\Psi $ is valid in $\mathcal{M}$;
- (ii)
- The sentence $[\underline{\mathbf{X}}\backslash X]\Psi $ obtained in clause (i) is a sentence without a quantifier with a composite variable, hence with usual semantics.

#### 2.2. Set-Theoretical Axioms

**Axiom**

**1.**

**Axiom**

**2.**

**Axiom**

**3.**

**Axiom**

**4.**

**Axiom**

**5.**

**Remark**

**3.**

**Remark**

**4.**

**Definition 7.**(Extension of vocabulary of ${L}_{\mathfrak{T}}$.)

**Notation**

**1.**

**Zermelo ordinals**at this point as a notation for these singletons:

**Remark**

**6.**

**Remark**

**5.**

**Remark**

**7.**

**Remark**

**6.**

**Remark**

**8.**

**Remark**

**7.**

**Axiom**

**9.**

**Definition**

**8.**

**two-tuple**$\langle \alpha ,\beta \rangle $ is the pair set of $\alpha $ and the pair set of $\alpha $ and $\beta $; using the iota-operator we get

**Remark**

**8.**

#### 2.3. Standard Function-Theoretical Axioms

**Axiom**

**10.**

**Remark**

**9.**

**set**that has this property: Axiom 2 guarantees, namely, that a function on a set X is not a set! □

**Axiom**

**11.**

**Axiom**

**12.**

**Axiom**

**13.**

**Remark**

**10.**

**Axiom**

**14.**

**Remark**

**11.**

**Remark**

**12.**

**Notation**

**2.**

**Axiom**

**15.**

**Remark**

**13.**

**Remark**

**14.**

**Axiom**

**16.**

**Axiom**

**17.**

**Axiom**

**18.**

**Remark**

**15.**

**Axiom**

**19.**

**Remark**

**16.**

#### 2.4. The Non-Classical Function-Theoretical Axiom and Inference Rules

**Definition**

**9.**

**vocabulary**of ${L}_{\mathfrak{T}}$ as given by Definition 2 is extended:

- (i)
- With symbols “ı”, the iota-operator, and “⋀”, the conjunctor;
- (ii)
- For any constant $\widehat{\mathbf{X}}$ denoting a set, with enough composite symbols ${\widehat{\mathbf{f}}}_{{\alpha}^{+}}$, ${\widehat{\mathbf{h}}}_{{\beta}^{+}}$, … such that each of these is a variable that ranges over a family of ur-functions indexed in $\widehat{\mathbf{X}}$.

**syntax**of ${L}_{\mathfrak{T}}$ as given by Definition 3 is extended with the following clauses:

- (iii)
- If ${\widehat{\mathbf{f}}}_{{\alpha}^{+}}$ is a variable as in clause (ii) of Definition 9, then ${\widehat{\mathbf{f}}}_{{\alpha}^{+}}$ is a term;
- (iv)
- If t is a term and ${u}_{{t}^{+}}$ is a composite term with an occurrence of t, and $\beta $ is a variable ranging over all things, then $\u0131\beta ({u}_{{t}^{+}}:t\mapsto \beta )$ is a iota-term denoting the image of t under the ur-function ${u}_{{t}^{+}}$;
- (v)
- If $\widehat{\mathbf{X}}$ is a constant designating a set, $\alpha $ a simple variable ranging over all things, and $\Psi \left(\alpha \right)$ an atomic formula of the type $t:{t}^{\prime}\mapsto {t}^{\prime \prime}$ that is open in $\alpha $, then ${\bigwedge}_{\alpha \in \widehat{\mathbf{X}}}\Psi \left(\alpha \right)$ is a formula;
- (vi)
- If $\Phi $ is a formula with a subformula ${\bigwedge}_{\alpha \in \widehat{\mathbf{X}}}\Psi \left(\alpha \right)$ as in (v) with an occurrence of a composite variable ${f}_{{\alpha}^{+}}$, then ${(\forall {f}_{{\alpha}^{+}})}_{\alpha \in \widehat{\mathbf{X}}}\Phi $ and ${(\exists {f}_{{\alpha}^{+}})}_{\alpha \in \widehat{\mathbf{X}}}\Phi $ are formulas;
- (vii)
- If X is a simple variable ranging over sets, and $\Psi $ a formula with no occurrence of X but with a subformula ${(\forall {f}_{{\alpha}^{+}})}_{\alpha \in \widehat{\mathbf{X}}}\Phi $ as in (vi), then $\forall X[X\backslash \widehat{\mathbf{X}}]\Psi $ and $\exists X[X\backslash \widehat{\mathbf{X}}]\Psi $ are formulas with a subformula ${(\forall {f}_{{\alpha}^{+}})}_{\alpha \in X}[X\backslash \widehat{\mathbf{X}}]\Phi $;
- (viii)
- If X is a simple variable ranging over sets, and $\Psi $ a formula with no occurrence of X but with a subformula ${(\exists {f}_{{\alpha}^{+}})}_{\alpha \in \widehat{\mathbf{X}}}\Phi $ as in (vi), then $\forall X[X\backslash \widehat{\mathbf{X}}]\Psi $ and $\exists X[X\backslash \widehat{\mathbf{X}}]\Psi $ are formulas with a subformula ${(\exists {f}_{{\alpha}^{+}})}_{\alpha \in X}[X\backslash \widehat{\mathbf{X}}]\Phi $. □

**Remark**

**17.**

**Definition**

**10.**

- (i)
- If $\widehat{\mathbf{X}}$ is a constant designating a set, X and $\alpha $ simple variables ranging over sets c.q. things, and ${f}_{{\alpha}^{+}}$ a composite variable ranging over ur-functions on ${\alpha}^{+}$, then
- ${(\forall {f}_{{\alpha}^{+}})}_{\alpha \in \widehat{\mathbf{X}}}$ is a
**multiple universal quantifier**; - ${(\exists {f}_{{\alpha}^{+}})}_{\alpha \in \widehat{\mathbf{X}}}$ is a
**multiple existential quantifier**; - ${(\forall {f}_{{\alpha}^{+}})}_{\alpha \in X}$ in the scope of a quantifier $\forall X$ is a
**universally generalized multiple universal quantifier**; - ${(\forall {f}_{{\alpha}^{+}})}_{\alpha \in X}$ in the scope of a quantifier $\exists X$ is an
**existentially generalized multiple universal quantifier**; - ${(\exists {f}_{{\alpha}^{+}})}_{\alpha \in X}$ in the scope of a quantifier $\forall X$ is a
**universally generalized multiple existential quantifier**; - ${(\exists {f}_{{\alpha}^{+}})}_{\alpha \in X}$ in the scope of a quantifier $\exists X$ is an
**existentially generalized multiple existential quantifier**;

- (ii)
- If $\widehat{\mathbf{X}}$ is a constant designating a set, X a simple variable ranging over sets, and $\alpha $ a simple variable ranging over all things, then
- ${\bigwedge}_{\alpha \in \widehat{\mathbf{X}}}$ is a
**conjunctive operator with constant range**; - ${\bigwedge}_{\alpha \in X}$ is a
**conjunctive operator with variable range**. □

**multiple quantifier**and that ${(\u029e{f}_{{\alpha}^{+}})}_{\alpha \in X}$ in the scope of a quantifier ${\u029e}^{\prime}X$ is a

**generalized multiple quantifier**.

**Definition**

**11.**

**scope of the conjunctive operator**; furthermore,

- (i)
- If there is an occurrence of a variable $\alpha $ and/or ${\widehat{\mathbf{f}}}_{{\alpha}^{+}}$ in the scope of the conjunctive operator, then a formula ${\bigwedge}_{\alpha \in \widehat{\mathbf{X}}}\Psi $ has a
**semantic occurrence**of each of the constants $\widehat{\alpha}$ referring to a thing in the range of the variable $\alpha $, and/or of each of the constant ur-functions ${\widehat{\mathbf{u}}}_{{\widehat{\alpha}}^{+}}$ over which the variable ${\widehat{\mathbf{f}}}_{{\alpha}^{+}}$ ranges—a subformula ${\bigwedge}_{\alpha \in \widehat{\mathbf{X}}}\Psi $ has thus to be viewed as the conjunction of all the formulas $[\widehat{\alpha}\backslash \alpha ]{[{\widehat{\mathbf{u}}}_{{\widehat{\alpha}}^{+}}\backslash \widehat{\mathbf{f}}]}_{{\alpha}^{+}}]\Psi $ with $\widehat{\alpha}\in \widehat{\mathbf{X}}$. - (ii)
- If there is an occurrence of a composite variable ${f}_{{\alpha}^{+}}$ in the scope of the conjunctive operator, then the subformula ${\bigwedge}_{\alpha \in \widehat{\mathbf{X}}}\Psi \left({f}_{{\alpha}^{+}}\right)$ has a
**free semantic occurrence**of each of the simple variables ${f}_{{\widehat{\alpha}}^{+}}$ ranging over ur-functions on the singleton of $\widehat{\alpha}$ with $\widehat{\alpha}\in \widehat{\mathbf{X}}$—the formula ${\bigwedge}_{\alpha \in \widehat{\mathbf{X}}}\Psi \left({f}_{{\alpha}^{+}}\right)$ has thus to be viewed as the conjunction of all the formulas $[\widehat{\alpha}\backslash \alpha ][{f}_{{\widehat{\alpha}}^{+}}\backslash {f}_{{\alpha}^{+}}]\Psi $. □

**Definition**

**12.**

**scope of the multiple quantifier**; likewise for the scope of the generalized multiple quantifiers of Definition 10. If a formula $\Psi $ has a free semantic occurrence of each of the simple variables ${f}_{{\widehat{\alpha}}^{+}}$ with a constant $\widehat{\alpha}\in \widehat{\mathbf{X}}$, then a formula ${(\u029e{f}_{{\alpha}^{+}})}_{\alpha \in \widehat{\mathbf{X}}}\Psi $ has a

**bounded semantic occurrence**of each of the simple variables ${f}_{{\widehat{\alpha}}^{+}}$ with a constant $\widehat{\alpha}\in \widehat{\mathbf{X}}$. A non-classical formula $\Psi $ without free occurrences of variables is a

**sentence**. If X is a simple variable ranging over sets and $\Psi $ is a sentence with an occurrence of a multiple quantifier ${(\u029e{f}_{{\alpha}^{+}})}_{\alpha \in \widehat{\mathbf{X}}}$ and with no occurrence of X, then $\forall X[X\backslash \widehat{\mathbf{X}}]\Psi $ and $\exists X[X\backslash \widehat{\mathbf{X}}]\Psi $ are sentences with an occurrence of a generalized multiple quantifier ${(\u029e{f}_{{\alpha}^{+}})}_{\alpha \in X}$. □

**Axiom**

**20.**

**Inference Rule**

**1.**

**Inference Rule**

**2.**

**Inference Rule**

**3.**

**Inference Rule**

**4.**

**Remark**

**18.**

**Example**

**1.**

**Remark**

**19.**

- (i)
- That SUM-F is a typographically finite sentence;
- (ii)
- That an instance (46) of SUM-F, deduced by applying Nonstandard Universal Elimination, is a typographically finite sentence;
- (iii)
- That a Formula (48), deduced from an instance of SUM-F by applying Multiple Universal Elimination, is a typographically finite sentence;
- (iv)
- That a conjunction (50), deduced by applying Rule-C to a sentence deduced from SUM-F by successively applying Nonstandard Universal Elimination and Multiple Universal Elimination, is a typographically finite sentence.

**Inference Rule**

**5.**

**collection**of formulas ${\left\{\right[I\left(\alpha \right)\backslash \alpha ]\Psi (\alpha \left)\right\}}_{I\left(\alpha \right)\in \widehat{\mathbf{X}}}$ in Equation (60) is itself not a well-formed formula of the language ${L}_{\mathfrak{T}}$, but each of the formulas in the collection is.

**Inference Rule**

**6.**

**arbitrary**family of ur-functions indexed in $\widehat{\mathbf{X}}$. □

**Inference Rule**

**7.**

**specific**family of ur-functions indexed in $\widehat{\mathbf{X}}$. □

**Remark**

**20.**

**arbitrary**constant $\widehat{\mathbf{X}}$, and Nonstandard Existential Quantification, i.e., the rule

**specific**constant $\widehat{\mathbf{X}}$, are the same as in the standard case, but with the understanding that upon quantification a multiple quantifier ${(\u029e{f}_{{\alpha}^{+}})}_{\alpha \in \widehat{\mathbf{X}}}$ in $\Psi $ becomes a generalized multiple quantifier ${(\u029e{f}_{{\alpha}^{+}})}_{\alpha \in X}$ in $[X\backslash \widehat{\mathbf{X}}]\Psi $, and a conjunctive operator ${\bigwedge}_{\alpha \in \widehat{\mathbf{X}}}$ with constant range in $\Psi $ becomes a conjunctive operator ${\bigwedge}_{\alpha \in X}$ with variable range in $[X\backslash \widehat{\mathbf{X}}]\Psi $. □

## 3. Discussion

#### 3.1. Main theorems

**Theorem**

**1.**

**Proof.**

**Theorem**

**2.**

**Proof.**

**Remark**

**21.**

**construct**a function ${f}_{X}$ by giving a function prescription—what we then actually do is define an ur-function for every $\alpha \in X$; the function ${f}_{X}$ then exists on account of SUM-F. Furthermore, by constructing the function we construct its graph, which exists on account of Theorem 1. Generally speaking, if we define an ur-function for each singleton ${\alpha}^{+}\subset X$, then we do not yet have the graphs of these ur-functions in a set. However, in the present framework, the set of these graphs is guaranteed to exist. Ergo, giving a function prescription is constructing a set! □

#### 3.2. Derivation of SEP and REP of ZF

**Theorem**

**3.**

**Proof.**

**Theorem**

**4.**

**Proof.**

**Remark**

**22.**

#### 3.3. Model theory

**Definition**

**13.**

**model**$\mathcal{M}$ of the present theory $\mathfrak{T}$ consists of the

**universe**$\left|\mathcal{M}\right|$ of $\mathcal{M}$, which is a concrete category made up of a nonempty collection of objects (sets) and a nonempty collection of arrows (functions on sets), and the

**language**${L}_{\mathcal{M}}$ of $\mathcal{M}$, which is the language ${L}_{\mathfrak{T}}$ of $\mathfrak{T}$ extended with a constant for every object and for every arrow in $\left|\mathcal{M}\right|$, such that the axioms of $\mathfrak{T}$ are valid in $\mathcal{M}$. □

**Definition 14.**(Validity of non-classical formulas):

- (i)
- A sentence $\forall X\ne \varnothing \Psi $ with a non-classical subformula $\Psi $, such as the sum function axiom, is
**valid**in a model $\mathcal{M}$ of $\mathfrak{T}$ if and only if for every assignment g that assigns an individual nonempty set $g\left(X\right)=\underline{\mathbf{X}}$ in $\left|\mathcal{M}\right|$ as a value to the variable X, $[\underline{\mathbf{X}}\backslash X]\Psi $ is valid in $\mathcal{M}$; - (ii)
- A sentence ${(\forall {f}_{{\alpha}^{+}})}_{\alpha \in \underline{\mathbf{X}}}\Phi $ with an occurrence of an individual nonempty set $\underline{\mathbf{X}}$ of $\left|\mathcal{M}\right|$, such as an instance of SUM-F, is
**valid**in a model $\mathcal{M}$ of $\mathfrak{T}$ if and only if for every “team assignment” g that assigns an individual ur-function $g\left({f}_{{\underline{\mathbf{\alpha}}}^{+}}\right)={\underline{\mathbf{u}}}_{{\underline{\mathbf{\alpha}}}^{+}}$ in $\left|\mathcal{M}\right|$ as a value to each variable ${f}_{{\underline{\mathbf{\alpha}}}^{+}}$ semantically occurring in $\Phi $, the sentence $[{\underline{\mathbf{f}}}_{{\alpha}^{+}}^{g}\backslash {f}_{{\alpha}^{+}}]\Phi $ with the variable ${\underline{\mathbf{f}}}_{{\alpha}^{+}}^{g}$ ranging over the family of ur-functions ${\left(\right)}_{{\underline{\mathbf{u}}}_{{\alpha}^{+}}}$ is valid in $\mathcal{M}$; - (iii)
- A sentence $\exists t{\rm Y}$ with an occurrence of a simple variable t ranging over sets or over functions on a set $\underline{\mathbf{X}}$ and with ${\rm Y}$ being a non-classical formula, such as the sentences that can be obtained by successively applying Nonstandard Universal Elimination and Multiple Universal Elimination to SUM-F, is
**valid**in a model $\mathcal{M}$ of $\mathfrak{T}$ if and only if for at least one assignment g that assigns an individual function $g\left(t\right)={\underline{\mathbf{F}}}_{\underline{\mathbf{X}}}$ or an individual nonempty set $g\left(t\right)=\underline{\mathbf{Y}}$ as value to the variable t, the sentence $\left[g\right(t)\backslash t]{\rm Y}$ is valid in $\mathcal{M}$; - (iv)
- A sentence ${\bigwedge}_{\alpha \in \underline{\mathbf{X}}}\Psi ({\underline{\mathbf{f}}}_{{\alpha}^{+}},\alpha )$ is
**valid**in a model $\mathcal{M}$ of $\mathfrak{T}$ if and only if for every assignment g that assigns an individual ur-function $g\left({\underline{\mathbf{f}}}_{{\alpha}^{+}}\right)={\underline{\mathbf{u}}}_{{\underline{\mathbf{\alpha}}}^{+}}$ from the range ${\left(\right)}_{{\underline{\mathbf{u}}}_{{\alpha}^{+}}}$ of the variable ${\underline{\mathbf{f}}}_{{\alpha}^{+}}$ and an individual $\underline{\mathbf{\alpha}}$ as values to the variables ${\underline{\mathbf{f}}}_{{\alpha}^{+}}$ and $\alpha $ respectively, the sentence $[\underline{\mathbf{\alpha}}\backslash \alpha ][{\underline{\mathbf{u}}}_{{\underline{\mathbf{\alpha}}}^{+}}\backslash {\underline{\mathbf{f}}}_{{\alpha}^{+}}]\Psi $ is valid in $\mathcal{M}$.

**Proposition**

**1.**

**Proof.**

**not**in $\mathcal{M}$, and let $\underline{\mathbf{h}}\in \underline{\mathbf{A}}$. All numbers $0,1,2,\dots $ are in $\mathcal{M}$ (including $\underline{\mathbf{h}}$), so for an arbitrary number $\underline{\mathbf{n}}\in \mathbb{N}$ there is thus on account of the ur-function axiom (Axiom 18) an ur-function on $\left\{\underline{\mathbf{n}}\right\}$ that maps $\underline{\mathbf{n}}$ to $\underline{\mathbf{n}}$ and an ur-function on $\left\{\underline{\mathbf{n}}\right\}$ that maps $\underline{\mathbf{n}}$ to $\underline{\mathbf{h}}$. Since $\mathbb{N}$ is in $\mathcal{M}$, we get on account of SUM-F and Nonstandard Universal Elimination that

**Remark**

**23.**

**stronger**than ZF. The crux here is that the non-classical sentence (76) has to be valid in $\mathcal{M}$: the notion of validity of Definition 14 entails that there are uncountably many variables ${\underline{\mathbf{f}}}_{{p}^{+}}^{g}$, ranging over a family of individual ur-functions indexed in $\mathbb{N}$, in the language of the model. As a result, the subsets of $\mathbb{N}$ that can be constructed within the model are non-denumerable—a model of $\mathfrak{T}$ in which Multiple Universal Elimination, inference rule 2, applies for at most countably many variables ${\underline{\mathbf{f}}}_{{p}^{+}}^{g}$ is thus nonexistent. Thus speaking, the Löwenheim–Skolem theorem does not apply because $\mathfrak{T}$ is a

**non-classical**first-order theory, meaning that $\mathfrak{T}$ cannot be reformulated as a standard first-order theory. □

**Proposition**

**2.**

**Proof.**

**proper class**of simple quantifiers ${f}_{{\widehat{\alpha}}^{+}}$ ranging over ur-functions on the singleton of a thing $\widehat{\alpha}$. The universe of $\mathfrak{T}$, however, does not contain a set $\widehat{\mathbf{U}}$ of all things, so there is no multiple quantifier ${(\forall {f}_{{\alpha}^{+}})}_{\alpha \in \widehat{\mathbf{U}}}$ which would be equivalent to quantifier $\forall \Phi $: a multiple quantifier ${(\forall {f}_{{\alpha}^{+}})}_{\alpha \in \widehat{\mathbf{X}}}$ is

**at most**equivalent to an infinite set of simple quantifiers ${f}_{{\widehat{\alpha}}^{+}}$ and the degree of infinity is then bounded by the notion of a set. Thus, since a set does not amount to a proper class, a multiple quantifier ${(\forall {f}_{{\alpha}^{+}})}_{\alpha \in \widehat{\mathbf{X}}}$ does not amount to second-order quantification. See Figure 2 for an illustration.

**cannot**construct any image set of $\widehat{\mathbf{U}}$. That shows that our non-classical first-order quantification is not the same as second-order quantification.

#### 3.4. The Axioms of Category Theory

- (i)
- That domain and codomain of any function on any set are unique;
- (ii)
- That, given sets X and Y and functions ${f}_{X}$ and ${g}_{Y}$ with $Y={f}_{X}\left[X\right]$, there is a function ${h}_{X}={g}_{Y}\circ {f}_{X}$ such that ${h}_{X}$ maps every $\alpha \in X$ to the image under ${g}_{Y}$ of its image under ${f}_{X}$;
- (iii)
- That for any set X there is a function ${1}_{X}$ such that ${f}_{X}\circ {1}_{X}={f}_{X}$ and ${1}_{{f}_{X}\left[X\right]}\circ {f}_{X}={f}_{X}$ for any function ${f}_{X}$ on X.

#### 3.5. Concerns Regarding Inconsistency

**Conjecture**

**1.**

**Conjecture**

**2.**

## 4. Concluding Remarks

#### 4.1. Limitations of the Present Study

#### 4.2. Aesthetic Counterarguments

#### 4.3. Main Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Venn diagram of the Siamese twin functions ${f}_{X}$ and ${h}_{Y}$. The left oval together with the black point inside it is a Venn diagram representing the singleton $\left\{{f}_{X}\right\}$; the right oval together with the black point inside it is a Venn diagram representing the singleton $\left\{{h}_{y}\right\}$. The upper arrow represents the mapping of ${f}_{X}$ to ${h}_{Y}$ by ${h}_{Y}$, the lower arrow the mapping of ${h}_{Y}$ to ${f}_{X}$ by ${f}_{X}$.

**Figure 2.**Illustration of the heuristic argument. In both diagrams (

**a**,

**b**), all things in the universe of T are for illustrative purposes represented on the horizontal and vertical axes. In diagram (

**a**), the dotted black line represents an arbitrary functional relation $\widehat{\Phi}$: each dot corresponds to a constant ur-function as indicated, so the dotted line is equivalent to a proper class of ur-functions. In diagram (

**b**) it is indicated of which things on the horizontal axis the set $\widehat{\mathbf{X}}$ is made up, and each of the black dots within the red oval corresponds to a constant ur-functions: the dotted line segment is thus equivalent to a set of ur-functions. So, a multiple quantifier ${(\forall {f}_{{\alpha}^{+}})}_{\alpha \in \widehat{\mathbf{X}}}$ cannot be equivalent to a quantifier $\forall \Phi $.

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Cabbolet, M.J.T.F.
A Finitely Axiomatized Non-Classical First-Order Theory Incorporating Category Theory and Axiomatic Set Theory. *Axioms* **2021**, *10*, 119.
https://doi.org/10.3390/axioms10020119

**AMA Style**

Cabbolet MJTF.
A Finitely Axiomatized Non-Classical First-Order Theory Incorporating Category Theory and Axiomatic Set Theory. *Axioms*. 2021; 10(2):119.
https://doi.org/10.3390/axioms10020119

**Chicago/Turabian Style**

Cabbolet, Marcoen J. T. F.
2021. "A Finitely Axiomatized Non-Classical First-Order Theory Incorporating Category Theory and Axiomatic Set Theory" *Axioms* 10, no. 2: 119.
https://doi.org/10.3390/axioms10020119