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Finitism is generally defined as a belief that infinity is an invalid concept, or a rejection of infinite mathematical objects. There are various strengthened versions and variants of this belief, (classical finitism, strict finitism, extreme finitism, etc.) and I will debate just about any of them, as I believe that infinity is an entirely valid, sound, and useful concept. If your beliefs are stronger, weaker, or more specific than simply rejecting infinite mathematical objects, further specification would be preferable. Without any further specification on what exactly your beliefs are on this topic, I will assume that we are debating the validity of the concept of infinity, or at least whether infinite mathematical objects exist or are a reasonable concept.

Round 1
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.
The Concept of Infinity
We can use semantics or certain words to try and describe infinity, but as there is no quantifiable way to measure it to explain what it actually even means, then not only is it beyond human comprehension but we have no way of examining if it really exists. 

Time and Infinity
The only thing that could possibly be infinite is time, as we don’t know when it started. Our current standards of mathematics are good at explaining things as finite or limited, but as it currently stands, our research cannot find anything infinite.

Round 2
I disagree with both of your claims. I will start with the second claim, as my refutation of it is relatively short. Time is not the only thing in the universe that could be infinite. I mentioned that space is thought to be infinite in round one. I will now elaborate on this. Just saying that space is thought to be infinite may have lead you to believe that that is because we have not found any end to it. This is not the case. Evidence suggests that the universe is "flat." This is of course not to suggest that the universe is somehow two dimensional. What this means is that evidence suggests that the universe has zero curvature. This would force the universe to be infinite. Discussion of the meaning of positive, negative, and zero curvature, as well as the existence of evidence suggesting that the universe has zero curvature can be found here: Even if it turns out that space is finite, this goes back to conceptual existence, which as I said before is a very real kind of existence. You did also attempt to refute infinity's conceptual existence in your first claim which I will now refute.

As I discussed in round 1, modern mathematics is primarily based around set theory. In mathematics, the concept of infinity is equated to that of infinite sets. An infinite set is a set which is not finite, and so it would seem that I have described infinity. Nonetheless, this is a debate on infinity's existence, and so I will now describe an infinite set to drive this point home. If you are not familiar with set notation, you may want to visit this website: Consider the set {}, the null set. This is a well defined set: It is defined as having no elements. For a set S, consider S U {S}. If S is a set, then S U {S} is also a set. Now consider a set N defined by the following:

  1. The null set {} is an element of N.
  2. If S is an element of N, then S U {S} is an element of N.
  3. If properties 1 and 2 do not imply that S is an element of N, then S is not an element of N.
This set is a set of sets, and each set in it is finite. By iterating S U {S}, we get an set with one more element than the previous, forcing N to be infinite.

Round 3
I extend all arguments made previously, although I will take this opportunity to point something out that I had neglected to before. The claim that modern mathematics does not have a good grasp on infinity is simply false: In this introductory set theory textbook, chapter 7 talks about infinity, infinite sets and infinite numbers in quite a bit of detail.
Round 4
Round 5
Extend. Pro has forfeited all but one round so far. Vote Con!