Thread: Proving axiom of incidence

Originally Posted by topsquark

I think I know what this is about, but some statements are poorly made. I hope I can explain clearly what I am seeing here.

I.1 Every line contains at least two distinct points.
I take this as a given: in fact a line is made from a continuum of distinct points, so a line contains an infinite number of them.

Now look at I.3. I don't agree with the statement as it is written. I think what it is trying to say is "Given two distinct points there exists at most one line containing these points." This is the only statement I can write that doesn't either repeat or contradict I.1.

However, I don't understand why I.2 is there at all. I can't find any reason for it. If I am correct in my guess about I.3 we can simply rewrite I.3 (or I.2) to explicitly state "Given two distinct points there exists exactly one line containing these points." This seems to me to be a much more explicit axiom, it replaces two of the given axioms by one, and we would want the fewest axioms to work with in building such a logical system. Otherwise all I.2 is stating is that a line exists that contains two points; points which may not even be distinct (distinctness is not mentioned in I.2). This is already taken as a given in I.1. I am open to another interpretation but, as written, this one just looks bad to me.

-Dan

What if you re-arrange axioms, so first look at I.2 then I.1 and then I.3.

I.2 is there to insure existence of line as an element, so if we begin with that axiom we insure that there is a line if there are two points whether they are distinct or not, even if its the same point then through one point there is also at least one line. If this axiom wouldn't be then line wouldn't even exist at all!

I.1 states that every line has two distinct points so that axiom determines line in space.

I.3 insures that every line that has two distinct points is unique line.

I generally agree with Plato and topsquark here, but I'll give this a shot.

Originally Posted by OReilly

I.3 insures that every line that has two distinct points is unique line.

I.3 does not say that. You rearranged the sentence to make it say that. The original I.3 is

Originally Posted by OReilly

I.3: There exists at most one line that contains two distinct points.

Take an example. Suppose line L contains distinct points A and B and line L' contains different distinct points A' and B'. Then L and L' are both lines that contain two distinct points. So by I.3, L = L'. But that is not the meaning you want. You want the points A and B to be the same on the two lines, and then conclude they are actually the same line.

You in effect rearranged I.3 like this:

If a "line that contains two distinct points", then "there exists at most one."

Or as you put it "every line that has two distinct points is unique."

But that rearranging changes the meaning of the sentence and is not valid mathematical logic.

So to get I.3 to say what you want, you have to rewrite it, which similar to topsquark could be done like this:

I.3': Two lines that contain the same two distinct points are equal.

But as originally written, I.3 does not say this.

I.1: Every line contains at least two distinct points.
I.2: There exists at least one line that contains two points.
I.3': Two lines that contain the same two distinct points are equal.

Even with that correction, the basic question of existence is problematic.
There is no way to establish the proposition: “Two points determine a line”.

Originally Posted by JakeD

I generally agree with Plato and topsquark here, but I'll give this a shot.

I.3 does not say that. You rearranged the sentence to make it say that. The original I.3 is

Take an example. Suppose line L contains distinct points A and B and line L' contains different distinct points A' and B'. Then L and L' are both lines that contain two distinct points. So by I.3, L = L'. But that is not the meaning you want. You want the points A and B to be the same on the two lines, and then conclude they are actually the same line.

You in effect rearranged I.3 like this:

If a "line that contains two distinct points", then "there exists at most one."

Or as you put it "every line that has two distinct points is unique."

But that rearranging changes the meaning of the sentence and is not valid mathematical logic.

So to get I.3 to say what you want, you have to rewrite it, which similar to topsquark could be done like this:

I.3': Two lines that contain the same two distinct points are equal.

But as originally written, I.3 does not say this.

I.3 doesn't mention DIFFERENT two distinct points, maybe there should stand "same" for clarity.

Originally Posted by Plato

I.1: Every line contains at least two distinct points.
I.2: There exists at least one line that contains two points.
I.3': Two lines that contain the same two distinct points are equal.

Even with that correction, the basic question of existence is problematic.
There is no way to establish the proposition: “Two points determine a line”.

Why can't we conclude that from I.1 and I.3?

If there are two distinct points and every line contains at least two distinct points then by I.1 line can contain that two points and by I.3 thats the only one line.

“If there are two distinct points and every line contains at least two distinct points”
What makes you think: “it is the same line”? That is not the way logic works!
You have a lot of problems in understanding the way mathematics works:
I see that you have deleted you response to http://www.mathhelpforum.com/math-he...-equation.html.

Originally Posted by Plato

“If there are two distinct points and every line contains at least two distinct points”
What makes you think: “it is the same line”? That is not the way logic works!
You have a lot of problems in understanding the way mathematics works:
I see that you have deleted you response to http://www.mathhelpforum.com/math-he...-equation.html.

By I.3: "There exists at most one line that contains two distinct points"

Please, even though you disagree with each other, could you at least argue like professionals? The last two posts show a disturbing (childish) trend...

-Dan

Originally Posted by topsquark

Please, even though you disagree with each other, could you at least argue like professionals? The last two posts show a disturbing (childish) trend...

-Dan

I see that those two posts are erased.
I never first insulted anyone, Plato insulted me several times before and I didn't respond but know I couldn't restrain myself of not replaying.