Infinity Definition Essay Samples

In general, infinity is the quality or state of endlessness or having no limits in terms of time, space, or other quantity. In mathematics, infinity is the conceptual expression of such a "numberless" number. It is often symbolized by the lemniscate (also known as the lemniscate of Bernoulli ), which looks something like the numeral 8 written sideways ( ). This symbol for infinity was first used in the 1600s by the mathematician John Wallis.

Infinity can be defined as the limit of 1/ x as x approaches zero. Sometimes people say that 1/0 is equal to infinity, but technically, division by zero is not defined. Another notion is that infinity is a quantity x such that x + 1 = x . The idea is that the quantity is so large (either positive or negative) that increasing its value by 1 does not change it.

A set (see set theory ) can be defined as infinite if there exists a one-to-one correspondence between that set and a proper subset of itself. According to this definition, the set of integer s is infinite because its elements can be paired off one-to-one with all the even integers:

...-3-2-1123...
...-6-4-2246...

The converse of the above statement is not always true. Some infinite sets have infinite proper subsets such that they cannot be paired off one-to-one. An example is the set of real number s and its proper subset, the set of integers.

In the 1800s, Georg Cantor defined infinity in terms of the cardinalities of infinite sets. The cardinality of a set is the number of elements in the set. In this sense, the cardinality of the set of integers is smaller than the cardinality of the set of real numbers, even though both sets are infinite. The set of integers is denumerable (its elements can all be accounted for by means of a listing scheme), while the set of real numbers is not denumerable.

In a more down-to-earth sense, the words "approaches infinity" are used in place of the words "increases without limit." Thus, it is said that the limit of 1/ x , as x approaches infinity, is equal to zero. In this context, infinity does not represent a defined quantity, but is merely a convenient expression.

Also see Mathematical Symbols .

1:1:1 (Part 2)
Section 1: Fundamentals of Existence
Chapter 1: The Creator
Point 1

Every Jew must believe and know that there is a first existence, primal and eternal, which brought all things into existence and continues to sustain them. This is G-d.

Let us continue our exploration of the Ramchal's opening line of The Way of G-d. The Ramchal uses three different adjectives that all revolve around the concept of infinite. So let's define: What does "infinite" actually mean?

Literally, infinite means: not finite. When we describe something as being infinite, we're saying that it has no limitations, no "edges."

Infinite also implies that it has no specific properties, because if it were hot, red or funny, that specific characteristic would serve as a limitation. If something is red, it is not blue, nor green, nor white.

Even though in daily discourse we express ideas like, "He's infinitely rich" or "I've been waiting for this phone call forever," these are just popular idioms. Can a person's net assets be infinite? Impossible. We might mean that a person is so wealthy that it's difficult to imagine the vastness of his wealth, but the bottom line is a certain sum of money. If this person's investment portfolio would double over the next year, he'll be more wealthy, which means he could not have been infinitely wealthy. Something big can always potentially get bigger. Something infinite cannot get bigger. If it could, it must not have been infinitely big to begin with.

We face the same issue with the phrase "forever." Forever describes a time that will never come. There's no such moment as forever.1

There are three adjectives being used to describe this infinite G-d:


(1) First Existence

By calling G-d "first" ("matzui rishon" in Hebrew) the Ramchal is essentially saying that something must exist that caused the existence of everything else. No matter how far back we go in the history of the universe, we're faced with the question, "What came before that?"

The evolutionist feels secure in the theory that we descended from earlier hominids, which evolved from earlier life forms, which came from a primordial soup, which came from inorganic matter that spontaneously formed into reproducible life, which came from energy. But the question still remains: What came before that? As long as the response is some other finite cause, we can probe further back until we arrive at the first cause.2 (Where did G-d come from? That will be answered in a future essay.)


(2) Primal

Calling the infinite existence "first" is not as significant as calling it "primal" (Kadmon in Hebrew). Primal is the backdrop in front of which everything else can come into existence. It points to the notion that G-d cannot not be. This will be further explored in a future essay.


(3) Eternal

Calling the infinite existence "eternal" (Nitzchi in Hebrew) refers to G-d's future existence. Because G-d was never brought into being, there is nothing that can bring G-d out of being. A table has the possibility of going out of existence by the same or similar agency that brought it into being. Similarly, as the moon came into being due to a combination of forces of nature, those forces could also lead to the destruction of the moon.


Active Sustainer

"...and continues to sustain them"

Here the Ramchal is going beyond the idea of G-d as Creator, to describe G-d as Sustainer. The distinction has huge implications. When relating to G-d as Creator, we are seeing G-d in a historical role of having, once upon a time, created the world. This offers no implication that G-d still has active involvement in His creation.

It's not altogether different than believing that someone created the chair you're now sitting on. Does the creator of the chair have any ongoing involvement with it? If he passed away tomorrow, will something happen to the chair? We would say that the chair needed the maker, in a manner of speaking, in order to come into existence. But once the chair exists, it no longer needs the involvement of its maker.

If we related to G-d only as Creator, we would have the same view. We would view G-d as having, once upon a time, created a world and set it into motion. He "programmed" it with laws of nature and now it runs on its own, like computer software.

The Ramchal is using the opening line of The Way of G-d to present a radically different paradigm. G-d not only created, but is now still sustaining (present tense) the world He created. It's almost like holding a cup in your hand: You not only created its suspension in the air, you are now, actively, still sustaining the cup in the air. It's an ongoing connection.

The implications are huge. For example, how do we understand natural disasters? According to the "G-d as Creator-only" theology, G-d may have been there at the beginning of time to set up the initial conditions of the universe, but after that He "stepped back" (or lost interest, or died) and no longer has any involvement in His creation.

The Ramchal is saying two important things: 1) G-d is now actively sustaining the entire creation, and 2) one must believe and know that this is so. Like in class #2, the Ramchal is proposing that I can have knowledge -- not a leap of faith -- that G-d not only exists, but that He is sustaining every aspect of reality at every moment.

This idea presents many difficulties. While a person may be able to accept the existence of a being that created a world many eons ago, it's harder to accept a being that's continuously involved in a moment-to-moment basis. Especially when we look at the evil and injustice in the world, it may be difficult -- logically and emotionally -- to believe in a deity who allows for tsunamis to drown hundreds of thousands of people. It's all the more difficult to believe in a being that actively and directly controls the waters that drown those men, women and children.

These are the kinds of questions that the Ramchal anticipates as being on our minds, and in due course he will present the Torah's answers to them. But to build a strong foundation toward clarity on these issues, we still have more work to do in appreciating the nature of an infinite being. The Ramchal will use the rest of Chapter 1 to elaborate and clarify. Stay tuned.

Even though we said above that finite things cannot be infinite, the one finite area that seems to allow for the concept of infinite is in mathematics, especially the theoretical end of mathematics.

For example, we talk about a function on a graph that "approaches infinity." Or the fact that two parallel lines will "infinitely" never intersect. Or that the decimal expansion of the number pi is infinite (3.1415927...). Mathematics also says that the set of real numbers R={0,1,2,3,...}, is also an infinite set that goes on "forever." There's no "last" or "biggest" number.

We have to keep in mind that the reason the concept of infinity works here is precisely because it is nothing more than that -- a concept. When we talk about the set of real numbers, we're not describing an actual thing that exists in our physical world. We're describing a theoretical notion. There is nothing in the real world of finite objects that is actually infinite. It's logically impossible.

In fact, the whole notion of theoretical infinite has its own internal paradoxes. How many numbers are there? An infinite number, right? Then how many even numbers exist? Half as many? But still an infinite number? Hmmm.

These are the kinds of paradoxes and self-contradictions that one runs into when trying to take a concept like "infinite" and use it within the realm of the finite, like real numbers and even numbers. There are many concepts that work well within their own realm of application, but it would be wrong to extend these theoretical notions into the domain of the real physical world in which we live. In our discussion, we want to probe the existence of the actual, real "infinite source to finite existence."

• Is there anything in this world that actually has an infinite quality to it?

• Why is it so fundamental to the Jewish approach to G-d, to see G-d as Sustainer, rather than just as Creator?

• Why is it not possible to believe in a G-d that has some limitations or deficiencies?


Watch an 8-minute video: "The Question of God's Existence" with Rabbi Moshe Zeldman


Next class in this course:
Unimaginable Perfection

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