People wait all week for friday,
all year for summer,
all life for happiness.

And again, I gave a talk for the best mathematics club of the multiverse, for "younger" students, meaning that this text is not scientific. This is the script for the talk.

We first want to have a look at ants. One behavior of ants is to build "chains" in which the ants follow each other. The single ant is relatively simple, and can only have simple instructions, and a canonical instruction is to just follow the ant in front. In nature (as far as I remember), ants use trails of pheromones, while the "follow the frontman" behavior is more common to other insects like the Oak processionary, but the biological details are not that important for the talk, and ants are a more well-known insect and therefore didactically more suitable (and there is a unicode symbol for ants).

Now, our leading intuition for this talk will be that we have a swarm of infinitely many ants which will just randomly be dropped and organizes itself according to increasingly complex rules. Our goal is to have a chain of ants that starts somewhere and does not end:

One line of ants following a leader

We can get this result if we tell the ants to follow some other ant. However, we might as well get a configuration which is infinite on both sides, and has no leader:

One infinite line of ants on both sides, no leader

The solution to this problem is rather simple: We can introduce one ant, which knows that it has no predecessor. Let us call this ant the zero-ant (which is totally not a suggestive name). Now, if we are lucky, we get the desired result. However, the following configuration is still possible:

The ants assemble in tree form, the zero ant being the leader

As you can easily see, every ant follows some other ant, but sometimes ants are followed by more than one other ant. We have to teach the ants that every ant has at most one successor. That is, every ant knows that it has one successor, and every ant except the zero-ant has a predecessor. Again, we could get the desired result. But we could also get the following result:

One line that follows the zero ant, but some circles of ants besides that line

Besides the main line of ants which is as we desire, we have cycles. The problem is that the ants have no way of knowing whether at some point in the future the zero-ant comes. We now leave the realm of theoreticall biological realizability, and approach the realm of mathematics. To break cycles, we have to introduce a more complicated principle, which talks about sets of ants. Assuming we have some set B of ants, we call an ant m\in B a non-successor, if it is not successor of any other element in B. In general, this means that m is what we would intuitively call a "minimum". We now say that every finite non-empty set of ants knows an ant which is not the successor of any other ant in this set. A cycle of ants would contradict this. Therefore, we cannot have cycles anymore. But we can still have the following situation:

One line that follows the zero ant, but some other lines which are infinitely long on both sides

The line that is infinite on both sides has ants that "think" they are farther away from the zero-ant than every ant that is connected to the zero-ant. We call the ants which are connected to the zero-ant the standard ants, the others we call non-standard ants.

To really get what we want, we would need the same principle for every non-empty set of ants. Then we could just apply it to our set of non-standard ants, and get a contradiction. At least then all hives look "the same".

At this point, we need to get more mathematical. A hive of ants is given by a pair (A,0,\sigma) of the set A of its members, the zero-ant 0\in A, and a successor function \sigma:A\rightarrow A. Since \sigma is a function, this already implies that there can be at most one successor for every ant, and that there actually is such a successor. We call a hive (A,0,\sigma) a Peano hive, if it satisfies the above axioms, that is, more formally:

  • \forall_x. x = 0 \leftrightarrow \forall_y \sigma y \neq x – an element is 0 if and only if it has no predecessor.
  • \forall_{x,y\in A}.\sigma x=\sigma y\rightarrow x=y – every ant to have at most one predecessor.
  • \forall_{B\subseteq A}\exists_{x\in B}\forall_{y\in B} x \neq
  \sigma yevery non-empty subset of A has a non-successor

Theorem: Let two Peano hives (A,0_A,\sigma) and (B,0_B,\tau) be given. Then there is a function b : A
  \rightarrow B with b(0_A)=0_B and forall a\in A we have b(\sigma a) = \tau(b(a)), and b is bijective.

Proof: We prove that b exists, and is surjective and injective.

To prove its existence, we would have to prove the recursion theorem; to not overcomplicate things, we will be sloppy at this point: Assume b was not well-defined, then the set of ants in A for which b is not well-defined has a non-successor m. By definition, m\neq 0_A. But if m =
\sigma m', then b(m) = b(\sigma m') = \tau b(m') would be well-defined. Contradiction.

Assume b wasn't surjective. Then the set B' := B \backslash
\operatorname{Im} b of ants in B which are not in the image of b is non-empty, and therefore contains a non-successor m. This non-successor cannot be 0_B by definition. Therefore, there must be some m' \in B such that m = \tau m'. But since m is a non-successor, m'\not\in B', so there is some n\in A such that b(n)
= m'. But then b(\sigma n) = \tau b(n) = \tau m' =
m. Contradiction.

Assume b wasn't injective. Then the set A' := \{ x \in A
\mid \exists_{y \in A} b(x) = b(y) \} of ants that are not mapped uniquely is non-empty, and therefore contains a non-successor n. Let \underline{n} be another ant mapped to b(n). Clearly, \underline{n}\neq 0_A, and therefore, b(n) has a predecessor m', and therefore, n \neq
0_A, since b(0_A)=0_B. Therefore, both n and \underline{n} has a predecessor, and these predecessors contradict the non-successor property of n. ∎QED

From our non-successor principle, a (more commonly used) principle follows, namely the principle of induction: If B\subseteq A, 0\in B and \forall_{x\in B}\sigma x\in B, then B=A.

Proof: Let B be such a set, and assume A\backslash B \neq
 \emptyset. Then let m \in A\backslash B be a non-successor. Since 0\in B, m \neq 0. But then m has a predecessor q, and q\in B. But then \sigma q = m \in B. Contradiction. ∎QED

It also works the other way around:

Proof: Assume there was a set B \subseteq A that contains no non-successor. Trivially, A \backslash B is non-empty, since it contains at least 0_A, since otherwise, 0_A would be a non-successor in B. Assume x \in A \backslash B. If \sigma x was in B, then, since predecessors are unique by axiom, \sigma x would be a non-successor in B. Therefore, \sigma x \in A \backslash B. But then by induction, A = A \backslash B, and therefore, B =
 \emptyset. ∎QED

As we just proved, these axioms entirely specify the structure of the hives. These hives behave like natural numbers, and in fact, the axioms we just formalized are the Peano axioms.

The problem when doing this is that you need a universal quantifier over all subsets of the ants, that is, you are in second order logic.

It is not even clear what "all subsets" mean. There are uncountably many such subsets

Proof: Assume there was a surjection f:\mathbb{N} \rightarrow
 \mathfrak{P}(\mathbb{N}). Consider the set R:=\{n \mid
 n\not\in f(n)\}. Assume f(r)=R. If r\in R, then by definition, r\not\in f(r)=R. Contradiction. If r\not\in
 R, then r\not\in R=f(r), which is also a contradiction. Such an r cannot exist, and therefore, f cannot be surjective. ∎QED

However, there are only countably many words. For every system to describe subsets, there is at least one plausible subset that we cannot describe. To introduce natural numbers, therefore, one usually wants to refrain from using sets at all, and only quantify over "ants" directly. Notice, we quantify over all possible "ants", and want them to behave like we want natural numbers to be. At this point, natural numbers do not exist yet. We have the constant symbol 0 and the function symbol S, and the axioms

  •  \forall_n. n = 0 \leftrightarrow \forall_m S m \neq n
  •  \forall_n \forall_m. S n = S m \rightarrow n = m

To be able to express a bit more, we add symbol + for addition and a symbol \cdot for multiplication, and define some of its properties:

  • \forall_n 0 + n = n
  • \forall_n \forall_m S n + m = S (n + m)
  • \forall_n 0 \cdot n = 0
  • \forall_n \forall_m S n \cdot m = m + (n \cdot m)

A proposition is a string with a free variable. Such propositions can be regarded as subsets of our ants. For example, we could define the proposition \operatorname{Primes}(x) :\Leftrightarrow
\forall_n\forall_m. n\cdot m = x \rightarrow (n = x \vee m = x), which denotes the set of prime numbers (actually "prime ants" so far, since we cannot be sure to really describe natural numbers here).

Now, while first order logic does not allow quantification over sets of ants, it allows to have infinitely many axioms. For every proposition \varphi(x), we add the non-successor axiom

  •  (\exists_x \varphi(x))\rightarrow \exists_n. \varphi(n)\wedge
  \forall_m. \varphi(m) \rightarrow S m \neq n

However, similar to our system with finite sets above, there are configurations of a hive with non-standard ants, they are called non-standard numbers.