"Cantor's Theorem and the Problem of Infinity" (2024) Jennifer Lee Lawson, MA

 Cantor’s Theorem and the Problem of Infinity

Jennifer Lee Lawson

Infinity. Try for a moment to grasp it. Numbers with no end. Is it that it exists and we cannot
grasp it, or is it that it simply does not exist? In this paper, I will argue that infinite cardinal
numbers may not exist. I will analyze Cantor’s Theorem to show this. First, I will explain
Cantor’s Theorem. Then, I will argue why I believe, when tried for infinite Sets, it is truly possible
that he is incorrect. I will argue without using many mathematical or logical symbols, for ease of
reading.

Cantor’s Theorem is well known in mathematics, logic and philosophy of mathematics. It is a
theorem in Set Theory. In the end, Cantor’s Theorem aims to show there is no largest cardinal
number. In other words, there is no largest infinity.

The concept of infinity is taught, in the United States, in public schools in our Math classes. On
our finite chalkboards, we use < — > to symbolize numbers that go on forever in both directions.
Infinity, the concept, however, is rarely ever fully explained. Cantor helps us actually visualize it
with the following picture:

The bottom circle in the picture is called an Empty Set, which is a Set with nothing in it. Just

above that, there are more circles, and the letters inside symbolize cardinal numbers. Those are

called Sets. The top circle appears to show infinite cardinal numbers. An apparent paradox

occurs when one tries to put a circle around the entire picture, otherwise known as The Set of

All Sets. The Set of All Sets contains every number. In other words, if The Set of All Sets is

possible, cardinal numbers may not go on forever. There may be a vast amount of them; too

many for a human being to ever count, possibly; but they may, in fact, end at some point.

Let us not use a chalkboard, then, and simply visualize with our minds. Take a look at the

picture above, Cantor’s Theorem. Just by considering it a graphic, we can see that, clearly, it is

possible to draw a circle around all of the Sets. Is it impossible? No. It is not. We could simply

draw. There is not one thing stopping us from doing that. When we draw a circle around it, is

that paradoxical? No. It is not. It’s simply a bunch of circles and letters with a bigger circle

encompassing them. When we do this, it is The Set of All Sets, which has, for a very long time,

been considered impossible or paradoxical.

Can we conceive of The Set of All Sets now? Maybe you can, especially if you come at this

problem with no prior assumption about the concept of infinity and simply use your hand,

supposing you have one, to just draw a circle.

"Radical Logic" (2024) Jennifer Lee Lawson, MA

 

Jennifer Lee Lawson

Radical Logic

In Jonathan Lear’s book Radical Hope, he argues for flexibility in adapting to cultural breakdown. He makes a great case. But what does it require? In this short paper, I argue that it requires adaptable logics.

There are several different logics to choose from. In philosophy, we typically try to find out which one secures truth the best, or which one is correct. But what if the application, in cases like cultural breakdown, require that we use almost all of them? This would have huge ramifications for philosophy, as we live in a time when breakdowns of many kinds are becoming more frequent. Philosophy education and familiarity with logics can provide us with the adaptability we need to survive.

We tend to think of logics as theoretical. But what about their applications? Applied logic is typically only thought of with regards to informal logic. But what about other logics, like dialethism?

The artist frequently uses different logics in their art. Poetry, music, photography. These forms of art capture different logics. This is how logics are applied. We need, then, the ability to make art or have access to the arts in order to make our way through a breakdown, even if it’s something singing.

When 9/11 occurred, which is what Jonathan Lear was referring to for things like modern-day breakdowns, people turned to the arts to mourn, discuss, reflect, and more. As for myself, I wrote a poem which won an award at my undergraduate university.

In a breakdown, we lose many things. But we need our concepts to adapt and move. This doesn’t always happen in normal conversations.

So, what about logics? I’m going to assert that basically every logic except informal logic and dialethism are to be applied at will when a breakdown occurs. This requires exceptional prior education.

People who do not have these things would unlikely survive a breakdown. They would be unmovable, just like Sitting Bull in Radical Hope. They would be too rigid, too basic. Thus, excellent education, like what Plenty Coups in Radical Hope argued for, and, perhaps especially, philosophy and logic education, are foundational to our very survival as a species.

Who knew logic was so important?