00:00

hi everyone welcome back i’m dave uh in today’s episode we’re going to talk

about models of instance geometry what do they look like

and before we get before we begin i wanted to mention that that this episode

is part of the series instance geometry tutorials with step-by-step proofs link

is below in the description so let’s get started

we’re going to begin by reviewing uh briefly what the incident axioms are so

we’re going to let point line and incident be undefined terms

we have incident axiom one where we say that two distinct points

determine a unique line and axiom two every line has at least two points on it

and a three says that three there exists

three distinct points there exists three points that are non-collinear

and so what do we mean by collinear right so three or more points are

collinear if there exists a line through all of them

and we say non-collinear if this is not true

00:01

three or more lines are concurrent if they’re just a point incident with all

of them and then non-concurrent is the negation

and then we have the uh parallel uh definition here so

our definition of parallel means they’re distinct lines and no point is incident

with them so in the previous episode we talked more more about parallelism

uh generally speaking though uh in the first couple of episodes of the series

we went episode by episode and we proved each one of these statements here

in both column format and proof and paragraph proof format so i recommend

checking out all those episodes then um i decided you know hey we’re

pretty good at writing proofs now let’s just prove all these results here and so

we did that in the previous episode as well we went through each of these

proofs here step by step and went through all of them

all right and so another axiom in our previous episode we talked about

00:02

parallelism and we talked about these three statements here we call this

statement right here the elliptic parallel property

and this statement right here the euclidean parallel property and the

statement right here the hyperbolic parallel property and they all have to

do with if you start with an arbitrary line and a point not on the line and

they ask the question how many parallel lines go through that point

but are parallel to l and so this says no lines and this says

a unique line exactly one and this says more than one it always happens that

there’s more than one so um i just wanted to mention that

these are not new axioms for incidents geometry rather they’re just additional

statements that may or may not be satisfied

by a particular model for the incident’s

geometry and so that’s what i’m going to

talk about today are models i’m going to begin our discussion and then we’ll do

some more episodes over additional models for our instance geometry so let’s be

00:03

very clear what i mean by incidence geometry so any collection of points and

lines that satisfy the incident axioms right we have the incident relation

and all three of those terms are defined

by a1 a2 and a3 so we’re not adding this as an axiom but we’re considering it as

a question and whether or not we can or whether or not we even should right so

anyways if we have a1 a2 and a3 this is what i’m going to call incidence

geometry for now now keep in mind also that we’re just working in the plane if

you want to work in higher dimensions well then we’ll need some more axioms

for that so for right now this is what i’m calling incidence geometry

and then we’ll look at higher dimensions later

all right so right now we’ll just have a1 a2 and a3 and these are interesting

statements and we’re going to see if these statements hold and by looking at

various models all right so here’s our first model so

model is an example and i’m going to call this model three-point geometry

00:04

and what is going to be a point and what is going to be a line and what does it

mean to say incident so i need to explain all of that so we’re going to say that

in three point geometry we’re going to mean a point to be a symbol a b or c

and a line to consist of exactly two of these points um and so we have this um

this table right here which kind of helps us understand the geometry right

here so we have three points a b and c and we have exactly three lines so this

is line one which is made up of points a and b and nothing else and line two a

and c and nothing else and line three b and c and nothing else and so what we

can see is that each point is incident with two lines

so for example a is incident with these two lines and b is incident with these

two lines and c is incident with these two lines so we can see that number one

is true we can also see there’s no parallel lines for example these two are

00:05

not parallel because c is on both of them

these two are not parallel because a is on both of them and l1 and l3 are not

parallel because b is on them so there’s

no parallel lines so every time you take a line and a point not on it

the elliptic parallel property holds because there’s no lines that are

parallel through it so what this shows is that the elliptic

parallel property can hold in an instance geometry well we’re going to

see some examples of some other models for example we do four-point geometry

and five-point geometry we’ll see that some of these parallel properties

some of them holds in other models another one might hold

so we’ll look at that here in a couple couple of episodes so this is the first

example of our first model here and so what we want to check though is

in order for us to say a model we can’t just you know put it all down and say it

is we have to actually check every axiom right so for every point p and every

point q not equal to p in other words if

00:06

you pick any two points there’s only one line going through it so there’s

six possibilities to check a and b and here’s the unique line that goes to

a and b a and c and so here’s the unique line and then

b and c and then here’s the unique line and then what if we switch the ordering

right b and a well it does doesn’t matter b and a is the same thing if we

have a and b either way it’s the same line

so all possibilities considered we can see that this axiom holds

and this is just a brute force check every possible combination and we see

that axiom a1 holds when we say here for every

right we went through and checked every p and every q all the possibilities and

we can see that a1 holds uh a two holes does doesn’t that true

for every line so we’ve got to check all

the lines there’s only three lines so we can check all three does each line have

00:07

at least two points on it and the answer

is yes you can just tell that by looking so a2 holds what about a3

uh there exists three distinct points right so three-point geometry is going

to be the minimum because of a3 right there exists three distinct points

and no line goes through all three of them right so there’s no line that goes

through all three of them l1 doesn’t go through all three of them l2 doesn’t go

through all three of them and l3 doesn’t go through all three of them so axiom

one two and three a one eight two and three hold so we can say this is a model

this is an example and here’s two properties that we talked about that

stand out in this model this isn’t two properties that’s gonna hold in every

model as we look at different models we can make additional statements but in

order for us to first call it a model we have to check that all three of these

actually hold all right so now let’s take a step back

and look at the axiomatic method a little bit more detail

00:08

in the first episode of the series where i talked about what instance geometry

was i touched on some of these terms but now i want to go through this a little

bit better or not better but necessarily that you’ve seen what theorems are

you’ve seen what a model is you’ve seen you know a lot of this stuff here now

for several episodes you’re you’re becoming more familiar with it so let me

just go through here and you know rehash or or you know make sure

that you’re okay with all this stuff right here so we have undefined terms

must contain technical terms that are deliberately chosen as undefined

and subject to the interpretation of the

reader and this is what models are going to give us they’re going to give us an

interpretation and so you know because in the

three-point geometry nothing in the axiom says that you can only have three

points that’s what three-point geometry says you only have three points right so

that’s an interpretation the three-point geometry is

00:09

all right um so we also have defined terms such as collinear and concurrent

lines and so we also have defined terms and then we have the set of axioms

system uh system axioms of the system are set as statements

dealing with the defined and undefined terms and remain unproven

so the word postulate is another word for axiom there

all right and then all other statements are called theorems

now the more mature or more advanced your system becomes then you’ll start

distinguishing between limits and theorems and propositions and facts and

things like that i talked about a little bit about that in a previous episode

all right properties of an ectomatic system there are some definite

properties that we want to talk about so um it’s important for the system to

be consistent so inconsistent means you can derive a contradiction

and you don’t want to be able to drive a

00:10

contradiction if your system derives the contradiction then it’s basically

useless it’s not that it’s not undefined maybe your system makes sense

but it’s just not going to be used for anything useful because your system will

say anything and everything so an action system is called consistent

provided a theorem any theorem which can be proven

can never logically contradict any of the axioms or previously proven theorems

so that means it’s consistent another property is is called

independent so an axiom is independent if it cannot be proven from the other

axioms so if we want to actually add one of the parallel postulates

the the elliptic euclidean or hyperbolic if we want to add another axiom to our

system um we would be we would be able to show

that it’s independent by looking at different models um and so

that’s that’s another reason why models can be useful you can use models to help

00:11

you understand if a set of axioms is independent or not from

one statement is independent from the rest of the statements

so generally speaking you’d like to have a

set of axioms that are independent for each other

but it’s it’s not necessarily a necessary requirement

uh sometimes you may have uh two alternative motives for giving your axioms

maybe they’re just easier to get started with whatever

maybe some seem more intuitive so this one is not as an important

property as consistency consistency is a must-have property

independent set of axioms is is useful extremely useful especially for people

who are building and working with axiomatic systems

an axiom set must also be complete so this this word is used a lot of

different ways in mathematics and so the the way i’m

00:12

going to mean it here is you must have enough axiom to prove every theorem

that can possibly be stated using the terms and axioms of the system

that can be proven either true or false in other words if you can write out a

theorem and it it has a true or false value you

need to have enough axioms to actually get to whether or not it is true or false

all right and so um let’s ask the question is incident at geometry

consistent right so instant geometry remember we we said that was axiom a1 a2

and a3 so the immediate question is is it consistent

so let’s look back at our model we can use models to help us answer this

question so three-point geometry we have points a b and c

and we have these two properties that we talked about and here was the geometry

given to us in this table right here so the question is is it consistent though

and here’s how we’re going to answer that question so to answer the question

00:13

we’re going to say maybe it’s inconsistent in fact let’s

just say it’s inconsistent the incidence geometry is inconsistent

in other words if you look at a1 a2 and a3 and you just look at those axioms you

can derive a contradiction from them well

then that would mean that you’d be able to derive any statement at all from the

system and so we’ll be able to have the statement right here if points a and b

are distinct then a and b are the same points so if your system is inconsistent

you’d be able to derive anything from it and you would be able to derive this

statement right here from it if a and b are different then a and b

are the same point right that’s obviously a contradiction

and if we look at this geometry right here it doesn’t actually hold does it

so if this was a theorem in our geometry

then it would have to manifest itself in this model right here and it doesn’t i

00:14

have three distinct points and none of them are the same so

this uh assumption right here that it’s inconsistent doesn’t work out

because if such a proof existed then it would hold in any model of it and it

doesn’t hold in this model so even just coming up with one model of a geometry

is valuable and could be valuable information in particular a proof of

such statement would have to hold in three-point geometry and it doesn’t

so there’s exactly three distinct points there’s no way we’d be able to prove

this statement right here so we can conclude that incidence

geometry is consistent theory so um again you know if it’s

inconsistent then anything can be proven and it’s a useless system

so this model right here gives us an example

of how to show that something uh how or how to get evidence that something is

consistent right here all right so this is um exactly what i

just said right there so all right so now let’s uh talk about

00:15

more models yeah let’s right here click right here and we’ll talk about 4.5

point and we’ll do some more models fianna’s geometry

click there and uh i’ll see you in that episode