I will begin by making three points. To be clear, the first point is to demonstrate that infinity is an important and meaningful component of modern mathematics, the second point is to demonstrate that infinity in modern mathematics is important to real life, and the third point is to demonstrate that infinity does exist in a meaningful way.
1. Infinity as it pertains to mathematics.
Most of modern mathematics is based on the system of axioms known as
ZFC. As far as we know, these axioms are consistent. They have provided us with a very rich theory of mathematics. It should be noted that mathematicians do not strictly conform to these axioms, as areas such as proof theory are independent of ZFC, and commonly study
other potential systems of axioms. It should also be noted that the axioms of ZFC are not considered to be objectively true, but rather, to be more like definitions. In a formal mathematical context, sets are meaningless without ZFC, which lays out the properties we wish for them to satisfy. The axiom of infinity is problematic for finitists, as it states that at least one infinite set exists. I have already pointed out that the axioms of ZFC function not as a standard of objective truth, but as a definition for sets in the formal mathematical context. In this sense, the axiom of infinity should be accepted if and only if it makes sense and is useful. On this point, without the axiom of infinity mathematics would be greatly handicapped. We would no longer be able to refer to "the set of all positive integers" or "the set of all real numbers." We would have to constantly tip-toe around these sets while doing math. A lot of math relies on referring to the existence of some number or element of a (often infinite) set, and removing the axiom of infinity would require us to avoid such statements. Even the definition of a real number using
Dedekind cuts requires infinite sets. There is no reason to get rid of almost all of our most useful mathematical concepts, just because of a "lack of existence" of infinity.
2. Infinity as it pertains to reality.
One could argue that mathematics dependent upon infinity is nonsense, and thus we should get rid of it. The point I would like to make now is that math as we know it is quiet useful to our reality. Even if there aren't any everyday examples of infinite objects in our real lives, most of the technology we have today is dependent upon areas such as calculus, which is dependent upon the concept of infinity. (If you disagree with this, take note of my first point regarding the importance of infinity to modern mathematics.) If we gave up all math with anything to do with infinity, we would be forced to give up the majority of our modern technological advancements. This would be an absurd sacrifice.
3. The existence of infinity and infinite mathematical objects.
One could argue that something only exists in a meaningful way if it exists in the physical world. Under this view, infinity does exist as far as we know, as the universe is thought to be infinite. If you believe in God, God's power, love, and wisdom are generally considered to be infinite. Regardless of any existence within physical reality, I assert that the most important way that infinity exists is not within physical reality. Existence can be a hazy concept. I would argue that conceptual existence is a very real kind of existence. Infinity exists in the same way that knowledge, love, and math exist.
Typo. In the definition of injective: I meant to write "The idea is that f is one-to-one in that it never sends two different values to the same value."
First, it's pedantic-yet-important correction time: Not all numbers, all real numbers. There is no such thing as all numbers. If you are a bit confused by the distinction, it is important to realize that "real numbers" is a misnomer. They aren't the only ones out there. Also, it isn't that the set of real numbers isn't itself "greater" than the set of natural numbers, it is that there are more real numbers than natural numbers. This is often phrased as "some infinities are larger than others," which I assume is what doesn't make sense to you.
I'm going to assume that you know what a function is, and that you are familiar with the notation f: A --> B. If you aren't, I suggest you read this: https://www.mathsisfun.com/sets/function.html. In the notation f: A --> B, A is the domain, and B is the codomain.
We say f is injective whenever f(x) = f(y) implies x = y. The idea is that f is one-to-one in that it never sends two different values to the same
We say f is surjective whenever for any y in B there is x in A such that f(x) = y. The idea is that no element of B is left out.
We say f is bijective when it is both injective and surjective. The idea is that f provides a one-to-one correspondence between the two sets. For every element of A there is an element of B, and visa versa.
For two sets A and B, it makes sense to say that A and B are the same size if and only if there is a bijection f: A --> B. We represent the "size" of the set A by |A|, and call it the cardinality of A. Numbers such as |A| are called cardinal numbers. Thus for cardinal numbers |A| and |B|, |A| = |B| whenever there is a bijection f: A --> B. We say that |A| < |B| (also written |B| > |A|) whenever there is an injection f: A --> B but no bijection g: A --> B. It is in this way that "greater" is still meaningful.
Cantor proved that |R| > |N|, where N is the set of natural numbers, and R is the set of real numbers. This is what is meant when it is said that there are more real numbers than rational numbers.
I agree that the set of all numbers, all even numbers, all odd numbers, etc. is infinite.
What I don't understand is saying that the set of all numbers is greater than the set of all even (or odd) numbers.
To my way of thinking, the idea of "greater" becomes meaningless when comparing infinite sets.
What is lim(x->0) 1/xx ?
Exactly.
Might as well reject the entire concept of limits, integrals, derivatives, series, the entire field of calculus and everything else based on that.
If you can't comprehend infinity, like the average 5th grader in the US, there is a possibility that you will flunk every other math exam because you don't know what dx means.
I am not towards either side, but merely to "finitism" as descriped. Such a thought would be in itself absurd.
Friendly reminder: Your time to make your argument is running low!
I was talking to my friends about this just the other day! excited to see how this goes. Also, I didnt know there was a word for that. I suppose I am a finitist.