Mon, 12 Dec 2016 14:44:17 GMT

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:

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:

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:

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:

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 of ants, we call an ant a non-successor, if it is not successor of any other element in . In general, this means that 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:

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 of the set of its
members, the zero-ant , and a successor function
. Since 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 *Peano hive*, if it satisfies the above axioms,
that is, more formally:

- – an element is if and only if it has no predecessor.
- – every ant to have at most one predecessor.
- –
*every*non-empty subset of has a non-successor

**Theorem:** Let two Peano hives and
be given. Then there is a function with and forall we
have , and is bijective.

*Proof:* We prove that 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 was not well-defined, then the set of ants in for which is not well-defined has a non-successor . By definition, . But if , then would be well-defined. Contradiction.

Assume wasn't surjective. Then the set of ants in which are not in the image of is non-empty, and therefore contains a non-successor . This non-successor cannot be by definition. Therefore, there must be some such that . But since is a non-successor, , so there is some such that . But then . Contradiction.

Assume wasn't injective. Then the set of ants that are not mapped uniquely is non-empty, and therefore contains a non-successor . Let be another ant mapped to . Clearly, , and therefore, has a predecessor , and therefore, , since . Therefore, both and has a predecessor, and these predecessors contradict the non-successor property of . ∎QED

From our non-successor principle, a (more commonly used) principle
follows, namely the principle of *induction*: If ,
and , then
.

*Proof:* Let be such a set, and assume . Then let be a
non-successor. Since , . But then
has a predecessor , and . But then
. Contradiction.
∎QED

It also works the other way around:

*Proof:* Assume there was a set that contains no
non-successor. Trivially, is non-empty, since
it contains at least , since otherwise, would be
a non-successor in . Assume . If
was in , then, since predecessors are unique
by axiom, would be a non-successor in
. Therefore, . But then by
induction, , and therefore, .
∎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 . Consider the set . Assume . If , then by
definition, . Contradiction. If , then , which is also a
contradiction. Such an cannot exist, and therefore,
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 and the function symbol , and the axioms

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

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 ,
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 , we add the non-successor axiom

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.