# Abstract Axiomatic System Problem - Proving a number of undefined term

• Sep 6th 2010, 06:30 AM
sedemihcra
Abstract Axiomatic System Problem - Proving a number of undefined term
Consider the following axiom set, in which x's, y's, and "on" are the undefined terms:

Axiom 1. There exist exactly five x's.
Axiom 2. Any two distinct x's have exactly one y on both of them.
Axiom 3. Each y is on exactly two x's.

How many y's are there in the system? Prove your result.

Solution (so far):

I think, there the number of y's is undefined because Axiom 2 and Axiom 3 contradict each other.
• Sep 6th 2010, 06:38 AM
undefined
Quote:

Originally Posted by sedemihcra
Consider the following axiom set, in which x's, y's, and "on" are the undefined terms:

Axiom 1. There exist exactly five x's.
Axiom 2. Any two distinct x's have exactly one y on both of them.
Axiom 3. Each y is on exactly two x's.

How many y's are there in the system? Prove your result.

Solution (so far):

I think, there the number of y's is undefined because Axiom 2 and Axiom 3 contradict each other.

I see no contradiction. How many ways are there to pick two distinct x's?
• Sep 6th 2010, 06:44 AM
Plato
Here is a model of the axiom set. The x's are in red.
• Sep 6th 2010, 06:52 AM
sedemihcra
Quote:

Originally Posted by undefined
I see no contradiction. How many ways are there to pick two distinct x's?

There would be 10 ways to pick two distinct x's.
• Sep 6th 2010, 06:58 AM
undefined
Quote:

Originally Posted by sedemihcra
There would be 10 ways to pick two distinct x's.

And there are 10 edges in the graph provided by Plato.
• Sep 6th 2010, 07:03 AM
sedemihcra
This chapter in my geometry book is before the finite geometry chapter, but it makes a little more sense now. Thanks all.
• Sep 6th 2010, 07:07 AM
Ackbeet
I should point out that precisely because Plato was able to produce a model that satisfies the axioms (you can easily see that the model works by seeing the edges as y's), the axioms are consistent.
• Sep 6th 2010, 08:57 AM
undefined
Alternatively we can find a bijection between {1,...,10} and the 2-subsets of {1,...,5} which is what prompted my first response.

In other words, label the x's and call the set X = {1,2,3,4,5}

List the 2-subsets lexicographically as usual

y1 = {1,2}
y2 = {1,3}
...
y10 = {4,5}

See that this satisfies the axioms and that it is unique up to labeling.

Plato's model looks nicer though.