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.