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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:
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.