 For the last several years, I've been on the hunt. I've been searching for an explanation for the popularity of irrational beliefs. People casually accept logical contradictions into their worldview. They're convinced that paradoxes exist. I've been trying to understand why. Their arguments frequently end up appealing to mistaken interpretations of quantum physics or the liar's paradox. But there's a deeper and more foundational error that I've become convinced is the root of so much confusion and it comes from the most unexpected place, mathematics. Specifically, the mathematical conception and treatment of infinity. Much to my surprise, a fairly basic logical error is rooted at the very foundations of modern mathematics and the error is now ubiquitous. It's become an unquestioned premise in mathematical reasoning and as with most unquestioned premises, it's become dogma and the vast majority of professional mathematicians accept the error as truth. The mistake is simple. It's a self-contradictory concept. The existence of actually infinite things. As I will demonstrate, this concept is incoherent. No different than the concept of a square circle or a married bachelor. Upon simple examination, it will become clear why there cannot be actually infinite distances, densities, forces, numbers, lines, circles, sets, or anything else. Now, first of all, we have to define our terms. Infinity or infinite means without end or never completed or without boundaries. An infinite distance can never be covered by definition of what we mean by infinite. There's no end to an infinite series. If the series ends, it is finite by definition. Now, consider the question, how many positive integers are there? Most people intuitively answer, there are infinitely many positive integers, meaning there isn't some upper limit on the size of numbers. You can't think of a number that one cannot be added to. This conveys the general concept of infinite. Now, by the term actual, I mean fully realized or completed or totally encapsulated. So here we find the elementary error in the conception of an actual infinite. I realize my refutation would appear impressive and profound if it were complex, if it were some difficult abstract chain of reasoning disproving a century of mathematical thinking. Well, that would surely impress people. But unfortunately, the refutation is not complex. It's outrageously simple. So simple, it is anti-climatic. What is never completed is never completed. I'll rephrase this in several ways. Infinity is by its definition never fully encapsulated. And the word actual means by its definition totally encapsulated. What is infinite is never fully realized by its definition. And what is actual is fully realized by its definition. What is infinite has no boundaries. What is actual has boundaries. Therefore, an actual infinite is a simple contradiction in terms, no different than a square circle. If this isn't intuitively obvious, I will give a few examples and then explain why in a purely logical sense, infinite things cannot exist. Consider a clear example. Try to imagine a circle with an infinite radius. The radius isn't really big. It's actually infinite. Is this possible? I am certain that you cannot imagine such a thing, and I will demonstrate why. It's the same reason you can't imagine a square circle. The concept is incoherent and one question illustrates. What is the curvature of a circle with an infinite radius? Take any line segment of your infinite circle. Does that segment differ in any way from a straight line? In other words, does it have any curvature whatsoever? The answer must be no if the radius is actually infinite. After all, if the circle has any curvature at all, the circle could be eventually completed and would be therefore finite. Thus, we arrive at an explicit contradiction. A circle with an infinite radius is a straight line. Now, believe it or not, some mathematicians will say, that's not a contradiction. This just shows the incredible nature of infinities. Paradoxes exist and you've just proved it. But in reality, they just demonstrate their irrationalism. No circle is a straight line. Every circle, by definition, has a curvature and is therefore finite. This points at a deeper truth. All circles are finite circles by simple necessity of being a circle in the first place. I'm sure somebody will object by saying, oh, well, the circle isn't actually a straight line. It simply converges with a straight line. And this is only half true. And it's the subject of my earlier piece on calculus. Convergence is a waffle word. Only one of two possibilities is true. Either the circle fully converges, in other words, it becomes identical with a straight line, or it gets ever so close, but not quite. If it never becomes identical, then it must have a curvature and must therefore be finite. There is no third option. Now, though calculus can be rescued easily from logical contradictions, set theory cannot. The set theoreticians are absolutely explicit. According to them, some infinities can be fully completed. They have some kind of actual size. The paradoxes of infinity are not just exclusive to lines and circles. It's not just that an infinite circle is a contradiction. It's that an infinite x is a contradiction, regardless of what x is. And there's an underlying logical reason why. Actually, infinite things cannot exist. It's a simple law of logic. Things are the way that they are. As I've explained in previous articles, everything is certainly exactly what it is. And those things are not what they are not. This is a logical principle, and it's called the law of identity. In an abstract form, it says that A is A. This is literally true for everything in existence. Now, keep this in mind and re-examine the concept of infinity. Infinite means never-ending, incomplete, or always bigger than. Always bigger than is another way of saying not merely A, but more than A. In other words, the very term infinite is an explicit denial of identity. Therefore, an infinite thing is a thing which is itself and more than itself at the same time, an outright contradiction. Think about it. If a thing is not more than itself, then it is complete, and therefore finite. So no things are infinite by virtue of being things in the first place. Every thing, if it's an actual thing, has boundaries and is therefore finite. If you can reference it, then it is always finite by logical necessity. Thus, it becomes clear why infinite circles do not exist, the concept is logically contradictory. The same is true for infinite sets or infinite magnitudes, infinite distances, or infinite anything else. If a set is a set, then it is itself and no more. If a magnitude is a magnitude, then it is itself and no more. If a circle is a circle, then it is only itself and nothing more. Any actual distance by virtue of being an actual distance is a finite distance. It isn't greater than itself. So is there any way then to rescue the concept of infinity? I would say certainly. We simply have to abandon the contradictory concept of completed infinities. Instead, infinite must be a word that makes a claim about inherent limitations. So think again about circles. What is the largest conceivable circle? Well, it's simple. There isn't one. There is no circle which you can conceive of, which I cannot double in size. There is no inherent limitation to the size of circles that we can imagine. So we might shorten this in a sensible way by saying there are an infinite number of sizes to circles. Now, that doesn't mean we can reference an actually infinite circle. It doesn't mean that an actually infinite amount of circles exists. It just means that we can't reference the biggest possible circle because such a thing does not exist. Think again about numbers. What is the largest positive integer? Again, it's simple. There isn't one. Now, it's not that infinity is the biggest integer. Obviously, such an idea is contradictory. It's not that there are an actually infinite number of positive integers and therefore there's no largest one. It's simply there is no such thing as the largest possible integer. There is no inherent limitation to the size of integer you can conceive of. If we want to state this concept in a kind of colloquial shorthand, we might say something like, oh, positive integers are infinite in size. But do we explicit? I'm not saying there is such a size X which represents the biggest possible integer and X is infinity. I'm saying there is no actual size X which represents the largest possible integer. So infinity must strictly be understood as shorthand. It is never an adjective for a concrete noun. And since writing this piece and doing more research, I'm now convinced that we probably should abandon the term infinite altogether. This kind of linguistic ambiguity, I think, causes more harm than good. So consider one more example to illustrate the difference between an incoherent definition of infinity and a coherent one. Imagine a rubber band, not just any rubber band, an extraordinary one. In front of it, there's a sign which says you can stretch this rubber band infinitely. Now we can interpret this sentence two ways. One is coherent and one is incoherent. The coherent interpretation is to say there is no inherent limitation to the stretching of this rubber band. It will stretch as far as you stretch it. The incoherent interpretation is to say the rubber band can be stretched until it reaches an actually infinite size that at some point you'll have completely arrived at an actually infinitely stretched rubber band. This irrational interpretation is how mathematicians conceive of infinite sets. Instead of thinking there is no inherent limitation to the size of set I can create, they think there is such a thing as an actually infinitely sized set. If what I've said is true, then the implications are not slight. They are extreme, so extreme that the last century of mathematics would be demonstrably built on illogical foundations. And indeed, many people have concluded that infinity ultimately shows reason itself to be self-defeating. They argue that logic can be mathematically proven to be either contradictory or fundamentally limited. These exact same arguments are mirrored by people who think that the liar's paradox is a true contradiction or that quantum physics shows reality is paradoxical. And they insist we must accept paradoxes into our worldview. But as I have demonstrated and I will continue to in the future, they are wrong. Logical rules by their very nature are not empirical hypotheses. They will never be disproven because they cannot be disproven. They are presupposed by every thought and every hypothesis. Regardless of how passionately the mathematician wants to complete his infinities, he simply cannot. A simple examination of his concepts would demonstrate the truth. If you like the sound of these ideas, if they resonate with you, then make sure to subscribe. And if you want to help create more content like this, then check out Patreon.com slash Steve Patterson. And you can help support the creation of a more rational worldview. To read this article or to learn about my books, check out Steve-Patterson.com.