First steps to infinity and beyond

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Axioms

In math, there are several philosophical liberties that could be taken. The axioms declare the absolute rules of math; what can and can't be done, what exists and what doesn't, and what statements are true and false, are all determined by them. There are many different ways to axiomatize mathematics. These different methods are called Set Theories, and there are quite a few. For example: Zermelo-Fraenkel, Morse-Kelley, Peano Arithmetic, Neumann-Bernays-Gödel, Kripke-Platek, and more.

If there is a "universe" in which axioms are true, then that universe is called a model of those axioms. This is an important part of model theory, the study of the formalization of this idea.

Finitism

Peano Arithmetic is the standard set of axioms which define the natural numbers. Every model of Peano Arithemtic only contains "natural-number like"-things, so the universe of all real numbers or all complex numbers aren't models of Peano Arithmetic. Peano Arithmetic does not declare the existence of an infinite set or number; in fact, the standard model of choice, $\mathbb{N}$, contains only finite numbers and sets.

Finitism is the idea that there are no infinite sets or infinite numbers in mathematics. Although this idea may seem somewhat reasonable for most people, if axioms declare there are infinite sets, mathematics becomes incredibly beautiful. More things are provable, things about proving things are provable, model theory is formalized in some way, the strangeness and oddities of infinity become formalized in pure mathematics. The intuition of infinity is often preserved if we let it exist.

Letting Infinity Exist

When one lets an infinite set exist and powerset exist in the axioms (that is, for any set, there is a set which contains exactly all of its subsets), then this beauty of infinity is shown. It is against intuition to not allow powerset to exist, so this is only assumed natural and absolute.

With this, an infinite set and its powerset never have a one-on-one correspondence to each other (this correspondence is also known as a bijection). That is, there is no way to give each element of the infinite set exactly one unique element of its powerset. This intuitively makes the powerset "bigger" than the original set. The proof of this is known as Cantor's theorem.

When there is a bijection between two sets, we say the sets have the same cardinality. We then assign numbers to sets based upon their cardinality; a measure of size.

Cardinals and Ordinals

The ordinals are defined in a way that extends the natural numbers.

  • The smallest ordinal is $0=\{\}$.
  • The next ordinal is $1=\{0\}=\{\{\}\}$.
  • The next ordinal is $2=\{0,1\}=\{\{\},\{\{\}\}\}$.

...

Each ordinal is the set of all smaller ordinals. Of course, that begs the question: what is the set of all natural numbers? With this pattern, it should be an ordinal. It is in fact, and it is called $\omega=\{0,1,2,3...\}$. The next ordinal is called $\omega+1$, and then $\omega+2$, and so on. Eventually, one gets to $\omega\cdot 2$, which is simply the the set $\{0,1,2,3...\omega,\omega+1,\omega+2...\}$. Note that $2\cdot\omega$ is not $\omega\cdot 2$, and $1+\omega$ is not $\omega+1$. If you would like a more detailed explanation of ordinal arithmetic, it would be suggested that you should search a reliable source.

The cardinality of a set $X$ (denoted $|X|$) is the smallest ordinal which has a bijection onto $X$. An ordinal is a cardinal if it is the cardinality of some set.

Every finite ordinal is a cardinal. However, no set has cardinality $\omega+1$. In fact, $\omega+1$ has a bijection onto $\omega$. $\omega$ can match to $0$, and then $0$ can match to $1$, $1$ can match to $2$, and so on. Every element of $\omega$ is, with this mapping, given exactly one unique element of $\omega+1$. So, $|\omega+1|=|\omega|=\omega$.

The smallest cardinal is denoted $\aleph_0$. The next smallest cardinal is denoted $\aleph_1$. The next smallest cardinal is denoted $\aleph_2$. This pattern continues. The smallest cardinal larger than all $\aleph_n$ for finite $n$ is $\aleph_\omega$, and etcetera. (The alternative notation $\omega_\alpha$ is also used.)

For ordinals, $\alpha,\beta,\gamma,$ and $\delta$ are often used as variables. For cardinals, $\kappa,\lambda,$ and $\mu$ are often used, though $\mu$ is used less often.

The Intuition of Large Cardinal Axioms

Large cardinal axioms may have some intuition behind them. For example:

  • The correct cardinals $\kappa$ are those who have a specific set of cardinality $\kappa$ which acts almost exactly like the collection of all sets.
  • The worldly cardinals $\kappa$ are those who have a specific set of cardinality $\kappa$ which acts like its own little universe or "world".
  • The inaccessible cardinals $\kappa$ are those who cannot be reached from smaller cardinals by addition of cardinals or taking the cardinality of the powerset of ordinals.
  • The Mahlo cardinals $\kappa$ are those who are inaccessible and have many, many, inaccessibles below them.
  • The weakly compact cardinals $\kappa$ are those for which certain sets of information of cardinality $\kappa$ can be "compacted" to smaller sets of information and still retain the basic properties of the original set.
  • The indescribable cardinals $\kappa$ are those which are so large that one begins to run out of interesting properties for them because for most properties of $\kappa$ there is a smaller ordinal $\alpha$ which has that property.
  • The measurable cardinals $\kappa$ are those who have many very large subsets.
  • The strongly compact cardinals $\kappa$ are those for which certain sets of information of cardinality at least $\kappa$ can be "compacted" to smaller sets of information and still retain the basic properties of the original set.
  • Though it is not a cardinal property, Vopěnka's principle implies that in every collection which has no cardinality (i.e. it is too large for any cardinal), there are two elements which are incredibly similar to each other.

It turns out that, although these axioms look intuitive, they cannot be proven to exist with standard mathematics. In fact, standard mathematics proves that they cannot be proven to exist. This is why they are axioms.