Investigating the Concept of Infinity Through Logical Axioms

In an act of prolonged inquisitiveness, a plain expression could be dormant curiosity; I’ve been trying to understand the concept of infinity and attempt to find meaning in this phrase on and off for about 2 to 2 ½ years. Infinity is a word commonly found in the English language, and most people can describe it in some detail. But what does this word truly mean in the context of our lives? I would ask people from time to time to describe what infinity means to them and how they would describe it. Here is some of what they have said:

• “Infinity means everything.”
• “In math, it’s all of the numbers.”
• “Expansive, never-ending, everything.”
• “Everything in the universe, past, present, and future.”
• “The limit of what we know, and beyond that.” (I really like this one)
• “All things in time and space”

Some excellent explanations on what infinity means. It was a big help to give me a jump start and power charge this dormant curiosity. My understanding of infinity began with my blog post regarding the impact and flaws of mathematical systems titled: Deeper Scrutiny of Mathematical Systems Through Hofstadter’s (1979) Gödel, Escher, Bach: An Eternal Golden Braid. Hofstadter’s work¹ is impactful for many reasons; for me, it was his analysis of Gödel’s Vortex, which relates to infinity and ultimately sees the practice as recursive and self-referential in nature. Even before writing my Hofstadter piece, I had always been struck by this concept of what infinity actually means. I always question the problem with infinity from one of Hofstadter’s rules related to mathematical systems: Within infinite sets, there is always a contradiction. With this piece, I hope to challenge that notion of Hofstadter, to find a pearl of tautological design without contradiction.

The summary of infinity within my Hofstadter blog explores four dimensions of infinity: Aristotle’s first dimension (potential/actual infinities), a two-dimensional infinite (endless planes), a three-dimensional infinite (infinite spatial arrangements, like Escher or Bach’s work), and a fourth-dimensional infinite (infinite time). The fourth dimension, encompassing time, raises questions about origin and continuity, suggesting true infinity requires a temporal dimension, beyond what’s captured in Gödel, Escher, Bach (GEB), or current human understanding.

Three additional texts will help us in our observation of infinity through logic. The first comes from David Hilbert, who gave a talk at the Westphalian Mathematical Society in Munster, Germany, on June 4, 1925². He critiques the famous German Mathematician Karl Weierstrass’ concept of the infinite, problems with paradoxes related to Cantor’s Set Theory, and the discussion of transfinite and finite numbers. Now, I have been critical of Cantor in the past, notably due to Bertrand Russell’s Set Theory paradox; however, Hilbert does provide some levity to Cantor’s transfinite and finite numbers in sets. Although Hilbert’s work may seem antiquated, it is a significant starting point for understanding the infinite. Ultimately, Hilbert’s final position on the infinite is that the actual infinite is valid in mathematics as long as it is treated formally to avoid paradoxes. Even though Hilbert criticizes logic, “Mathematics, therefore, can never be grounded solely on logic” (p. 192), he is almost calling for logic to be the formal tool to avoid paradoxes.

The second text is a 1748 work called Introduction to the Analysis of Infinite: Book I by Leonard Euler³. Euler ensures that infinity can be a useful and rigorous mathematical concept rather than a mystical, or even contradictory one. A large part of Book I involves the understanding of geometric shapes and curves and their infinite process. He sees infinity as something that can be systematically analyzed and used to expand human understanding of functions, curves, and the nature of mathematical continuity.

The third text is a more modern (objectively) text titled Complexity, Existence and Infinite Analysis by Giovanni Merlo in 2012⁴. He proposes a stunning loop of infinity through a bio-conditional term of “first, that a derivative truth is contingent if and only if it contains infinitely complex concepts and, second, that a derivative truth contains infinitely complex concepts if and only if it does not admit of a finite proof” (p. 9) through Leibnizian analysis⁵. He breaks down truths between:

1. Derivative Truth (Truth)
2. Non-Derivative Truth (Identities)

For example, in science, water boils at 100°C at 1 atm pressure is considered a derivative truth under specific conditions, derived through observation and experimentation. A non-derivative truth does not rely on external justification, such as the statement “It will rain or it won’t rain”. This statement is true in all cases, as your choices are rain or no rain. Merlo’s analysis of infinity suggests that the concept of infinity is done through existence, and existence is the best out of several possible and unrefined existences. He attempts to make a temporal argument relating to the fourth-dimensional infinite.

Logical Axioms for Infinity

I seem outmatched given the literature on how logic may not be the best form to develop a concept of infinity. I am willing to acquiesce, but not before I present something that can be debated. First, let’s use a base for how we will find a logical axiom. In my previous work: Analytic Epistemology: What is It and Why is it Needed? I developed an inductive taxonomy to show the process of axioms for truth claims.

The sea of ideas needs to be focused to create concepts, then from concepts to applicable and logical propositions. If the propositions are logical, they are tested further to create clear axioms through formal logic. Once the logical axioms are developed, we determine truth claims in the form of tautological or contradictions. Furthermore, we must ensure our logical axioms follow a formal concept based on what we know about infinity, based on the research in the introduction:

1. Infinity Must Be Multidimensional
2. Infinity Is Valid When Formally Managed
3. Existence and Truth Are Tied to Infinite Complexity

Despite my criticism of set theory, I still believe it is valuable to attempt to define infinity through logical notation. For this instance, I will use the Zermelo-Fraenkel (ZF) set theory, which allows for more restrictions in axioms to avoid contradictions. In ZF set theory, infinity is introduced via the Standard Set Theoretic Axiom of Infinity, which asserts the existence of an infinite set. Informally, it says there’s a set that contains “enough” elements to never run out—something we can keep counting in forever. Formally, it’s expressed in first-order logic with a predicate for set membership (∈). Here’s how it looks:

Standard Set Theoretic:

∃S ( ∅ ∈ S ∧ ∀ x (x ∈ S → {x} ∈ S ))

Breakdown:

• ∃S: “There exists a set (S) such that…”
  • This declares the existence of some set we’ll define as infinite.
• ∅ ∈ S: “The empty set ∅ is a member of (S).”
  • This gives us a starting point, a “base element.”
• ∀ x (x ∈ S → {x} ∈ S )): “For all (x), if (x) is in (S), then the singleton set {x} is also in (S).”
  • This is the inductive step: if an element is in (S), the “next” element (represented as {x}) is too. It ensures the set ‘keeps growing’ without end.

This encompasses a standard set for infinity that satisfies a definition of infinity. Infinity, intuitively, is the idea of something unbounded or unending. This axiom formalizes that by ensuring (S) can’t be finite, any finite set would eventually “stop” having new successors. However, this requires deeper scrutiny. The concept of infinity suggests that it encompasses everything we know and everything yet to be discovered. Therefore, to imply “everything known and yet to be discovered” we need a set that represents both what exists (the “known”) and what could exist (the “yet to be discovered”), connected logically with “and,” while ensuring it’s distinct from “nothing” (perhaps ∅) and tied to an unbounded potential.

Modified Set Theoretic:

∃U (∅ ∈ U ∧ ∀ x (x ∈ U → (x ∧ ¬ ∅) ∈ U) ∧ ∀y (¬ (y ∈ U) → ∃ z (z ∈ U ∧z ⊈ y)))

Breakdown:

• ∃U: “There exists a set (U) such that…”
  • (U) will represent the totality of “everything known and yet to be discovered.”
• ∅ ∈ U: “The empty set ∅ is in (U).”
  • We start with “nothing” as a foundation, mirroring the Axiom of Infinity, but here it’s the baseline of existence.
• ∀ x (x ∈ U → (x ∧ ¬ ∅) ∈ U): “For all (x), if (x) is in (U), then (x) combined with ‘not nothing’ is also in (U).
  • Here, ∧ (logical “and”) connects (x) (something known) with ¬ ∅ (the negation of nothing, i.e., something). This suggests each element in (U) builds on what exists and excludes pure nothingness, growing the set. PLEASE NOTE: We typically wouldn’t use the logical “and” in set theory, but this is to create a constructive operation.
• ∀y (¬ (y ∈ U) → ∃ z (z ∈ U ∧z ⊈ y))): “For all (y), if (y) is not in (U), there exists some (z) in (U) that is not a subset of (y).
  • This captures “everything yet to be discovered.” If something (y) isn’t in (U) (not yet known), (U) still contains something (z) beyond it, ensuring (U) is unbounded and always exceeds any finite or fixed boundary of knowledge.

The standard set theoretic (∃S) guarantees the existence of a set that’s infinite because it starts with ∅ and includes an unending succession. The modified set theoretic is essentially a model of the natural numbers N under a certain interpretation (via von Neumann ordinals), where each element has a successor, and there’s no “last” element, capturing the essence of infinity. Infinity, intuitively, is the idea of something unbounded or unending. This axiom formalizes that by ensuring (S) can’t be finite, any finite set would eventually “stop” having new successors. Furthermore, adding the concept of everything known and yet to be discovered, this axiom (∃U) describes a set (U) that starts with nothing, grows by including all that exists (“known”) conjoined with the condition of existence (“not nothing”), and remains open-ended to encompass all that’s yet to be discovered. It’s infinite in a way reminiscent of the Axiom of Infinity, but framed epistemically – (U) is a set that can’t be pinned down fully because it always exceeds any attempt to limit it. A caveat to this axiom is that standard set theory doesn’t use ∧ between sets directly; I’ve treated it as a conceptual stand-in. This is more of a philosophical axiom inspired by infinity than a strict ZF extension, as “known” and “yet to be discovered” are epistemic, not purely set-theoretic, notions.

With set theory axioms established, it describes a first-order mathematical understanding of infinity, but it fails to encompass the meaning of infinity through propositional logic notations. Can infinity be true, fulfilling our third rule established: Existence and Truth Are Tied to Infinite Complexity? For this, we will use propositional logic and are not bound by ZF frameworks for using “and” operators (∧). Propositional logic deals with statements (propositions) that are true or false, connected by operators like “and” (∧), “not” (¬), “or” (∨), and “implies” (→). It’s less about sets and quantifiers and more about abstract relationships between ideas, so we’ll need to adapt the infinite, growing nature of the axiom into a propositional form.

First, we define the following propositions:

• (K): “Everything known is true.” (Represents the body of current knowledge.)
• (N): “Nothing is true.” (Represents the concept of “nothing,” akin to ∅.)
• (D): “Everything yet to be discovered is true.” (Represents the unbounded potential beyond current knowledge.)

Propositional Axiom:

(K ∧ ¬N) ∧ D

Breakdown of the propositional logic axiom:

• K: “Everything known is true.”
  • This stands for the finite, grasped portion of reality or knowledge at any given point – like the starting elements in the Standard Set Theoretic Axiom of Infinity.
• ¬N: “Not nothing is true.”
  • This negates “nothing”, asserting that there’s always something, aligning with existence over absence.
• K ∧ ¬N: “Everything known and not nothing is true.”
  • The “and” ties the known to the rejection of nothingness, suggesting that what we know is grounded in something real and non-empty.
• D: “Everything yet to be discovered is true.”
  • This captures the infinite, open-ended aspect — there’s always more beyond (K).
• (K ∧ ¬N) ∧ D:  “Everything known and not nothing, and everything yet to be discovered, is true.”
  • This combines the known with the condition of existence (“not nothing”) and extends it to the undiscovered, implying a totality that’s both concrete and limitless.

To mimic the Axiom of Infinity’s sense of unending growth, we can add an implication that suggests this statement is never “finished.” For example:

Extended Propositional Axiom:

(K ∧ ¬N) ∧ D ∧ (D → (K′ ∧ D′))

We can imply inexhaustibility by asserting that (D) always holds alongside (K), no matter how (K) expands. In set theory, we would substitute (∧(D→(K′∧D′)) with (∃more), but we will keep it consistent with propositional logic.

In terms of truth, the entire statement holds a tautological truth value. If (K) is true (we know something), (N) is false (there’s not nothing), and (D) is true (there’s always more to discover), the whole statement is true. Unlike a finite system where (D) might eventually become false (nothing left to discover), this proposition assumes (D) remains true indefinitely, echoing the Axiom of Infinity’s unending succession. This proposition says: “What we know exists (not nothing), and there’s always more to know.” It’s a snapshot of an infinite process—everything known conjoined with existence, plus an unbounded horizon. It’s less about building an infinite set and more about asserting an infinite state.

Conclusion and Implications

In summation, we ask if our axioms meet the standards set by the three rules determined at the beginning of the blog:

1. Infinity Must Be Multidimensional
2. Infinity Is Valid When Formally Managed
3. Existence and Truth Are Tied to Infinite Complexity

The standard set theoretic axiom (∃S ( ∅ ∈ S ∧ ∀ x (x ∈ S → {x} ∈ S )) partially satisfies the multidimensional rule. The axiom defines an infinite set via a single construction (successor operation), which is somewhat linear (akin to the natural numbers). However, in set theory, this set can be used to construct multidimensional structures (e.g., higher cardinals, ordinals, or Cartesian products), indirectly supporting multidimensionality. This axiom is part of Zermelo-Fraenkel set theory, a formal system designed to rigorously manage infinity. It defines infinity via a precise logical structure, ensuring consistency and avoiding paradoxes (e.g., Russell’s paradox) through restricted comprehension – meeting the second rule. Lastly, the axiom ensures the existence of an infinite set, which underpins complex mathematical structures (e.g., real numbers, functions). Truth in mathematics often relies on this infinite foundation, as many theorems assume infinite sets. The axiom ties existence (of the set (S)) to the infinite complexity of set theory.

The modified set theoretic axiom (∃U (∅ ∈ U ∧ ∀ x (x ∈ U → (x ∧ ¬ ∅) ∈ U) ∧ ∀y (¬ (y ∈ U) → ∃ z (z ∈ U ∧ z ⊈ y))) meets the multidimensional criteria by establishing a baseline of nothingness as a starting point; combines each element with “not nothing,” suggesting a dual nature (known elements and existence); and ensures (U) exceeds any external boundary, introducing an unbounded potential dimension. These components reflect multiple perspectives: a starting point, an iterative growth tied to existence, and an infinite horizon beyond current knowledge. This axiom partially meets the standard for validity with formal management. The axiom uses set-theoretic notation, borrowing from the formal structure of Zermelo-Fraenkel (ZF) set theory, which is rigorous and designed to manage infinity. However, the logical “and” (x ∧ ¬ ∅) is problematic, and not nothing (¬ ∅) is not a well-defined set operation. An additional question to ask is: could this axiom be best served with a substitution for the logical connectors? What could that be? Lastly, the axiom ties existence and truth to infinite complexity effectively by establishing a base for existence, linking each element to a condition of “not nothing,” suggesting that truth (elements in (U)) is grounded in existence beyond emptiness, and ensuring (U) is infinitely expansive, always containing elements beyond any finite or fixed boundary, implying a complex, unending system.

The third axiom – the propositional axiom (K ∧ ¬N) ∧ D) meets the multidimensional rule for infinity. The axiom combines multiple dimensions of infinity: (K) (known knowledge), ¬N (existence over nothingness), and (D) (unbounded future discovery). These represent epistemic, existential, and potential aspects, making it inherently multidimensional. It partially meets the validity rule by using a formal system; however, its terms ((K), (N), (D)) are abstract and less rigorously defined than set-theoretic constructs. Without precise definitions (e.g., what counts as “yet to be discovered”), it risks ambiguity. Lastly, this axiom meets the standard of existence, and that truth is tied to infinite complexity. By linking (K) (truth of known knowledge), ¬N (existence over nothingness), and (D) (infinite potential discoveries), the axiom suggests that truth and existence are bound to an infinitely complex process of knowing and discovering. (D) implies unending layers of truth. Its epistemic framing directly ties truth to an infinite, evolving complexity.

Lastly, the fourth axiom: Extended propositional axiom (K ∧ ¬N) ∧ D ∧ (D → (K′ ∧ D′)) fully meets the multidimensional aspects of infinity. These components span epistemic (knowledge), existential (not nothing), and potential (future discoveries) dimensions, with the implication adding a recursive, layered structure. This makes the axiom inherently multidimensional, capturing present, future, and existential aspects of infinity. The addition of (D → (K’ ∧ D’)) enhances multidimensionality by explicitly modeling an ongoing, iterative process of discovery, reflecting multiple stages of infinity. This axiom partially meets the validity model and is being formally managed. The axiom is expressed in propositional logic, a formal system with well-defined rules. However, limitations can arise as the terms (K), (N), (D), (K’), and (D’) are abstract and lack precise definitions (e.g., what constitutes “known” or “yet to be discovered”). This vagueness reduces formal rigor compared to set-theoretic axioms, where terms like sets and membership (∈) are strictly defined. Lastly, the axiom fully meets the standard of existence and truth tied to infinite complexity. Truth tied to known knowledge and existence is grounded in a non-empty reality. The implication explicitly models infinite complexity by suggesting a perpetual cycle of discovery, where truth and existence are intertwined with an ever-growing, layered system.

The four axioms: standard set-theoretic, modified set-theoretic, propositional, and extended propositional, all fully meet infinite complexity, tying existence and truth to unbounded systems. The propositional and modified set-theoretic axioms fully meet multidimensionality with epistemic and existential layers; the standard axiom partially meets it due to its linear focus. Only the standard axiom fully meets formal management within ZF set theory; others partially meet it due to vague terms or non-standard notation. Positives include rich conceptual depth and dynamism (especially the extended propositional). Limitations involve ambiguity in terms and non-standard operations.

There are still additional questions to observe. Such as the prescription I subscribe to regarding truth claims: Popperian Falsification. A potential challenge is presented as these axioms may not be suitable for Popperian Falsification, given a host of abstractions in the axioms. The set-theory axioms would be far away from falsification, while the propositional logic axioms, notably K ∧ ¬N: “Everything known and not nothing is true”, could be testable; however, falsifying “everything yet to be discovered is true” would require showing that some future discovery is false. Since (D) refers to an infinite, unspecified set of future knowledge, it’s practically impossible to test comprehensively.

Going forward, these axioms offer a conceptual framework for thinking about infinity not just mathematically, but philosophically and epistemologically. While the standard set-theoretic axiom and expanded propositional axiom remain the most rigorous, the alternative axioms invite exploration into how we define knowledge, existence, and complexity in an infinite context. Their multidimensional scope suggests new ways of modeling infinite systems that go beyond linear succession. However, the tension between abstraction and falsifiability—especially under a Popperian lens – highlights a core limitation: we may propose profound structures, but testing them remains elusive. This doesn’t nullify their value; instead, it challenges us to refine definitions, develop clearer symbolic logic, and determine where conceptual richness outweighs empirical verifiability. These axioms serve as both a philosophical proposition and a call for precision – urging us to embrace complexity while staying accountable to formal rigor. Future work might focus on bridging formal systems with epistemic models, crafting axioms that are both expansive and testable.

---

[1] Hofstadter, D. (1979). Gödel, Escher, Bach: an Eternal Golden Braid. Basic Books: New York

[2] Hilbert, D. (1926). On the Infinite. Congress of the Westphalian Mathematical Society: Munster, GER. Delivered June 4, 1925, p. 183-201.

[3] Euler, L. (1988). Introduction to the Analysis of Infinite: Book I. Springer-Verlag: New York. (Original work published, 1748).

[4] Merlo, G. (2012). Complexity, Existence and Infinite Analysis. The Leibniz Review, 22. 9-36.

[5] Leibnizian Analysis: A method of mathematical reasoning that emphasizes generalized components rather than classical mathematical proofs.