Zeno’s Paradox of Achilles and the Tortoise

Can we divide a distance or substance infinitely until we uncover an atomic unit? In nature, it seems possible; we sometimes discover the smallest piece of an object. But does this apply universally? What if the process of division never ceases, perpetually yielding more subdivisions?

This encapsulates Zeno's paradox.

We can infinitely bisect space, leading to the conclusion that a finite distance can emerge as the infinite sum of finite quantities.

Thus, it's remarkable that:

We can sum finite amounts infinitely and still arrive at a finite total.

Could it be that finite numbers possess an infinite aspect hidden within?

Hilbert’s Paradox of the Grand Hotel

Infinity can be larger than itself (and also smaller), suggesting there may be various types of infinite quantities.

Imagine an infinite hotel occupied entirely by guests. If we instruct each guest to move to the room with double their current number, we create an abundance of vacancies!

Mathematically, this translates to inf=2*inf, where half of the new spaces are now available, and the other half remains occupied.

Thus, it seems astonishing that:

Infinity can reside within another infinite (or perhaps itself).

Conclusion of Philosophical Questions

Is it even accurate to assert that infinity exists? Some argue against its existence. For instance, if infinite mass were real, it would exert infinite gravitational force, which would contradict our existence.

“Infinity does not exist; it is merely a practical concept, one we cannot observe in reality—it remains a mathematical abstraction.”

Aristotle was among the first to address this, distinguishing between a platonic infinity that does not genuinely exist (actual infinity) and a potential infinity that can grow indefinitely but is never fully realized.

We can observe that infinity is intricately tied to repetition, recursion, cycles, and perhaps rotations.

When we can repeat an action, we encounter infinity. This suggests invariance under specific procedures, meaning we can perform a mathematical operation and revert to the initial conditions, allowing for endless repetition.

The Mathematical Journey Through Infinity

We can articulate the philosophical questions posed above through numbers and mathematics. This permits us to analyze them in a formal and objective manner.

Series

The initial method for studying infinity mathematically is through infinite summation, known as a Series:

Grandi’s Series (an alternating series)

Harmonic Series

The Paradox of the Sum of the Reciprocals of the Squares of the Integers

As you observe, you replace the initial number n=1, the figure at the base of the ? (sigma) in the expression, and iteratively add +1 to _n_ until reaching infinity (as indicated above).

Numerous intriguing series exist; here are a few of my favorites:

• Geometric Series
• Taylor Series (and Maclaurin)
• Fourier Series
• Dirichlet Series
• Alternating Series

One of the most renowned series is associated with a 1 million dollar prize:

Why is this significant? The Series serves as a generalization of all those mentioned above, meaning that understanding this "ultimate" series will enable comprehension of nearly all others. While more complex series can be defined, we have struggled with this one for years, leading to a $1M reward for its resolution.

Note: The Dirichlet Series generalizes the Zeta function (its definition coincides with the Series above).

Future Related Stories ~ Cross-over

I must conclude here—this could extend infinitely. I plan to discuss various topics in future writings, but they deserve separate attention.

Sets

In the late 19th century, there was a movement to ground mathematics on robust foundations and axioms, seeking the cornerstone of mathematics—the philosopher’s stone of math. The objective was to formalize all mathematics under a few rigorous axioms.

One significant approach was through Sets, with Cantor being a pivotal figure in its development. He distinguished between countable infinity and non-countable infinity.

In essence, countable infinity allows for a bijective connection (one-to-one) with natural numbers (from one to infinity). If such a connection cannot be established, the infinity is deemed non-countable.

For instance, stars may be infinite, but we can associate each star with a number by selecting them with our fingers. However, what if there are other universes? Which number would we assign after counting all the stars in our universe?

Cantor also made a crucial distinction between cardinal numbers (global size) and ordinal numbers (internal index). Cardinal numbers represent the size of a set (cardinality), while ordinal numbers indicate that order is essential since sets lack an internal order among elements.

From both cardinal and ordinal numbers, we can evolve into transfinite numbers, which represent initial steps toward this concept.

> Here lies a potential discussion on formalism vs intuitionism. This pertains to the foundations of mathematics and understanding core logic.

> Transfinite numbers

> p-adic numbers

On Gödel (This section may be nonsensical without context)

I suspect there’s a misunderstanding surrounding Gödel’s incompleteness theorem. (I admit I haven’t fully grasped it, but I have some intuitive insights.)

It relates to Infinity, as you may have heard. The error isn’t in the proof itself but in its interpretation. People often state, “Some statements can be true but unprovable.” While that’s accurate, it doesn’t capture the entire reality.

My interpretation leans more towards “Only axioms are unprovable; if you can’t prove something, you’re likely missing the correct axioms.” In simpler terms, “there is no absolute truth.” Any proof relies on its axioms, and those axioms can be freely chosen.

I’m uncertain if I accurately conveyed my thoughts on what’s amiss; I’ll leave another spoiler:

Gödel’s incompleteness theorem only applies within recursion (note that recursion produces infinity, as it can be repeated indefinitely). This ties into the foundational concepts of mathematics involving sets (or perhaps another framework). Another significant spoiler: Mathematically, recursion equates to a singularity. (Indeed, we face an infinite issue there.)

In another intriguing twist, particles can also be viewed as singularities in a mathematical sense.

(This is a philosophical hypothesis; you may disregard it. I might one day elaborate on how I arrived at this conclusion, provided I don’t forget—it's been somewhat of a mathematical exploration.)

Statements can be equivalent to a set via a morphism connection. Within this framework, discussing a set within itself becomes logically inconsistent, as Gödel’s proof hinges on recursion (the set containing itself). The only set that can reference itself is the unity; those theorems merely assert that we cannot prove axioms (the unity’s equivalence to a number). Claiming that there are true but unprovable statements can be misleading, as only axioms fall into that category.

I might have misrepresented the definition of an axiom here, so allow me to clarify: an axiom serves as the fundamental building block of logic; it’s akin to a gene in DNA. An axiom is to a particle what a statement is to matter (with matter comprising numerous particles).

In a similar vein, we can "prove" or more accurately, "discover" the laws of nature and matter, just as we can prove statements. However, we cannot prove why particles exist (this is a matter for metaphysics), nor can we prove the validity of an axiom (which pertains to metamathematics).

The MAIN ISSUE in grasping infinity stems from this misunderstanding. This confusion generates complications in numerous thought experiments, including:

The Turing machine, undecidability, the concept of computability (raising questions about whether consciousness is computable), the halting problem, and Russell's paradox—all of which are inconsistent (and often paradoxical) due to misinterpretations of infinity.

Calculus

With Cantor and set theory, we began to comprehend the nature of infinities, but in calculus, we actively engage with them.

Historically, we have primarily dealt with finite numbers, but upon introducing series—a form of discrete domain—we find ourselves adding infinitely finite amounts. However, what if we could sum infinite infinitesimals?

This is where calculus comes into play; we often associate infinity with immensely large numbers, neglecting its aspect as a minuscule quantity, which we refer to as infinitesimals.

Infinitesimals are nearly equal to zero, yet not quite zero.

The summation symbol for series, representing summation, evolves into a smooth notation that also signifies summation over infinitesimal values, denoted by _dx_.

Since Newton’s time, calculus (the study of continuous change, including infinitesimal changes) has emerged as a pivotal topic in mathematics, physics, and engineering. Why is this the case?

We tend to view infinity as outside our reality, leading to the belief that it may not genuinely exist within it (as Aristotle suggested, only potential infinity exists). However, as we observe, utilizing infinity or comprehending it in calculus enables us to create better models for understanding reality. Why does infinity align so harmoniously with reality? So, what if...

Spoiler: Perhaps our reality revolves around infinity? Finite numbers might be the false constructs.

Singularity

One of calculus’s most effective tools for examining infinity is the singularity of a function. Most people aren’t taught about this concept, as contemporary calculus primarily focuses on real analysis (using real numbers with functions), while singularity properly fits within complex analysis (involving complex numbers with functions).

In calculus, we achieved favorable results by summing infinitesimal values, but singularities posed challenges, as they are points where a function is undefined. Thankfully, we discovered methods to resolve this challenge—at least for poles (a specific type of singularity).

What if we extend our analysis further and connect singularities with transfinite and p-adic numbers?

Oh! I must not forget to relate this back to what I previously discussed; there’s a thread to follow:

Singularity (mathematics)

Singularity is a captivating subject, and I’ll compile related discussions.

“Null space: The black hole of Math” serves as an introduction; the behavior of null space closely resembles that of singularities (in terms of deleting information), though I’m not equating it to singularities within black holes (but the analogy stands). Perhaps they share a commonality at their core?

“What does it mean to have a determinant equal to zero?” introduces this infinity or singularity issue, where the concept is not merely vast, but undefined. This implies that there may be no answer or potentially multiple answers simultaneously.

“det(A)=0 is no longer a problem” explores ways to navigate this issue in linear algebra. We can also relate calculus singularities to det(A)=0, but that will be a topic for future discussion.

That’s all for today—thank you for reading!