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

1. ## 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.

2. 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?

3. Here is a model of the axiom set. The x's are in red.

4. 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.

5. 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.

6. This chapter in my geometry book is before the finite geometry chapter, but it makes a little more sense now. Thanks all.

7. 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.

8. 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.