Infinity and the Kalam Cosmological Argument

If you haven't seen the video on the Kalam argument, please have a look at the video. It is short, and gives the background needed to understand the rest of this post:


Infinity

In response to the arguments against the infinite, opponents have stated that there is a fully coherent system of transfinite mathematics which proves the system coherent. The problem is that coherence in a formal system is quite different from coherence in reality. The main point of this article is that mathematicians can either hold to a realist view of numbers or can hold to infinity as a legitimate mathematical concept, but not both. For an introduction to infinity, I recommend this video by Bill Shillito. The actual lecture starts at 2:00.

Formal Systems vs. Sciences
We need to draw a distinction between a formal system and an investigation of reality. A formal system is a bit like a game. You have objects, rules for manipulating those objects, and various starting positions which are called "axioms." Think of the game of chess. You have pieces such as the king, rook, and knight. You have an object called the chessboard. You have rules for manipulating those objects. You also have a starting board position, which serves as your axiom.

  Theorems are like board positions in chess other than the starting board position.

You can prove that a board position is a theorem of the chess game by showing all the moves that get you from the starting point to that position.

You can also prove that board positions, like this one, are impossible given the rules of chess and the starting position. For example, you can show that both sides start with 16 pieces, and then go through the rules one by one to show that none can increase the number of pieces that a side has. Therefore, a board position where one side has 40 pieces is provably not a theorem of the system.

The fun thing about formal systems is that you can make the rules as arbitrary as you want. For example, you can stipulate that there is such a thing as "all the black squares" and that there is such a thing as "all the white squares" while denying that there is such a thing as "all the black and white squares."

That's what formal systems are: games based on stipulated rules which may or may not have anything to do with reality. This is distinct from investigations of reality, where you have to give some account as to why you think it works a certain way. You cannot just make things up as you go along, the way you can when creating a formal system.

Georg Cantor, the inventor of the mathematics of infinity, believed that he was investigating facts about reality. His system was based on naive set theory, which defined numbers as kinds of sets, and sets as "any well-defined collection of things." So, you could have a set of book on the table, or a set of marbles in the drawer, or a set of other sets, which is how set theory defines numbers.

Paradoxes
The problem with naive set theory is that it has been proven to contain contradictions. Here are just three of the paradoxes that arise from this set theory:

Burali-Forti Paradox: Set theory defines an ordinal as the order type of a set. The set of all ordinals would also have an ordinal number. But if the set of all ordinals had an ordinal number, then that ordinal would have to be in the set, and hence would not be the order type of all ordinals. The set of all ordinals is part of naive set theory (since it is a well-defined collection of things) and yet is self-contradictory.

Cantor's Paradox: A cardinal number is the number of elements in a set. The power set is the set of all subsets in a set. Since Cantor proved that the power set must always be greater than the set, what about the set of all cardinals? Again, the set of all cardinals is a well-defined collection of things, but for any value you give it, you can always derive a higher cardinality from the power set operation. So the set of all cardinals is also self-contradictory.

Russell's Paradox: An impredicative definition is one that includes itself. 'Phrases in the English language' is an example, since that is itself a phrase in the English language. The set of all sets would be a member of itself. The set of all horses would not, since sets are not horses. What about the set of all sets that are not members of themselves? If it is a member of itself, then it fails the definition, and should not be included. If it is not a member of itself, then it fits the definition and should be included. In other words, if it is included, then it should not be included. If it is not included, then it should be included. Contradiction.

Escaping the Paradoxes
Mathematicians escaped these paradoxes by creating axiomatic set theories, such as Zermelo-Fraenkel or New Foundations. In these set theories, a set is something defined by the axioms. Axiomatic set theory turns set theory into a formal system, like the game of chess. Rules are stipulated as arbitrarily as you want. Axiomatic set theory can rescue set theory from the paradoxes, but it cannot rescue mathematical realism from these paradoxes.

The problem with mathematical realism is that there is no way to avoid these paradoxes. If sets really existed, then there is no justification to say that sets cannot be members of themselves. Clearly, there are categories that are members of themselves, such as the category of things in this post. And that is the difference between formal systems and reality. In formal systems, you can make up whatever rules you want, no matter how arbitrary. Reality is not so cooperative. For example, if sets existed, then we could categorize them in this fashion:

Or this fashion:

Or this fashion:

And in any of these three cases, our categorization brings rise to each of the three paradoxes. The only way around this is to stipulate that you cannot divide things into these categories. That works for formal systems, and that's the fun about playing a game. You can say that within the rules of your game, you can have sets that are members of themselves and sets that are not members of themselves, but make it illegal to create a category of all sets that are not members of themselves. When you are talking about the real world, however, you cannot just make things up like that when you are talking about reality.

For a fun video on mathematical realism and the arguments against it, I recommend this video on mathematical platonism by Kane Baker.