# Finitism

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After 2 votes and with 2 points ahead, the winner is...

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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.

**1. Infinity as it pertains to mathematics.**

**2. Infinity as it pertains to reality.**

*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.**

__The Concept of Infinity__

__Time and Infinity__- The null set {} is an element of N.
- If S is an element of N, then S U {S} is an element of N.
- If properties 1 and 2 do not imply that S is an element of N, then S is not an element of N.

*introductory*set theory textbook, chapter 7 talks about infinity, infinite sets and infinite numbers in quite a bit of detail.

**Vote Con!**

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.