So we want to continue looking at things on the subject of fast order and finesse.

I remember what it was called...

So we defined this for two classes of structures, a smaller one and a larger one. So C is definable in D or C times D is definable in F.

A fast order formula phi exists with C, which is from the class of structures in D.

So that's equal to...

So what we want to start with are examples of things that are definable in the non-F.O.

We already saw an example of a non-F.O. definable property, namely the availability of graphs that are not definable in order over a simple compact argument.

This is what Paul has shown. And this compact argument is not defined in the finite model. So we want to show arguments that only work with things that are still in the finite model.

So our first example, the original model of a non-finite model F.O. definable property, or a pair of classes where we only talk about finite things from the front to the sides, is the following.

So, once, place.

These are just finite linear orders. We discussed this before, that an order is generally not necessarily linear, that is, two given elements are not always comparable, and if that is the case, then an order is linear.

And we limit ourselves to the finite ones. Our signature for this is a two-digit relation that we call smaller than equal and that we write in fixed.

So, what do these things look like? Well, you can paint them easily. These things here are made up of a number of points that are smaller than equal here.

And because this is a linear order, they all stand behind each other. We recently mentioned this in the paper discussion.

So I can imagine that the first sons are so many natural numbers in the usual order.

Even is now a class of parts of the order, and that is the one whose basic area has a straight cardinality.

So, that is, from the first four or, in my opinion, the first ten natural numbers.

We want to show...

We want to show that this class of the finite linear order is not defined in the class of finite linear order.

Which may be a bit surprising at first, because it sounds like a harmless property. You should be able to sort the things that correspond to each other,

things that require that they are equal in size or something like that. But no.

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So the figures are exponentially larger in some way or another.

That means they are large enough.

Then they are not distinguishable by the order formula

and therefore not distinguishable by the order of rank M.

We will prove that in a moment.

First of all, what do we have? Why are we done when we have it?

Then we'll just take two large enough figures.

So some figures that I'm not interested in, but which are larger than the order of rank M.

Oh no.

Not yet.

I'm not going to take them yet. I'll need the M first.

We have to send an assumption in advance. We assume that we have a formula phi that defines us as even.

And as M we take the quantor rank of phi.

Now we take two linear equations.

So that A is large enough and B is one size larger.

We can do that evidently.

We take A as the natural numbers from 0 to 2 to the power of M.