Incompleteness of Euclidean Geometry: Proving the Parallel Postulate

In summary, The parallel postulate, which states that if a line segment intersects two lines with angles less than 90 degrees, the lines must intersect, cannot be proven or disproven from the remaining axioms in Euclidean geometry. This is due to the existence of other equivalent axioms and geometries, such as hyperbolic geometry, where the parallel postulate does not hold true. Stronger postulates could allow for the proof of the parallel postulate, but it is impossible to do so using only Euclidean geometry's axioms.
  • #1
junglebeast
515
2
"For example, Euclidean geometry without the parallel postulate is incomplete; it is not possible to prove or disprove the parallel postulate from the remaining axioms."

http://en.wikipedia.org/wiki/Gödel's_incompleteness_theorems

The parallel postulate says that, if a line segment intersects 2 lines that both have angles less than 90 degrees, then those two lines must intersect.

http://en.wikipedia.org/wiki/Parallel_postulate

Why is it be impossible to prove this postulate? This seems intuitively obvious and seems like it would be very easy to prove on the basis of simply calculating the intersection point.
 
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  • #2
junglebeast said:
"For example, Euclidean geometry without the parallel postulate is incomplete; it is not possible to prove or disprove the parallel postulate from the remaining axioms."

http://en.wikipedia.org/wiki/Gödel's_incompleteness_theorems

The parallel postulate says that, if a line segment intersects 2 lines that both have angles less than 90 degrees, then those two lines must intersect.

http://en.wikipedia.org/wiki/Parallel_postulate

Why is it be impossible to prove this postulate? This seems intuitively obvious and seems like it would be very easy to prove on the basis of simply calculating the intersection point.

What axioms are you going to use to prove it? What assumptions do you make in your calculation?

There are quite a number of axioms which are equivalent to the parallel postulate, in the sense that they can be proved from the postulate, and which allow you to prove the postulate.

There are also geometries which satisfy all the other axioms of Euclid, and in which the parallel postulate is not true. For example... a hyperboloid geometry. This shows it must be impossible to prove the postulate from the other axioms.

Cheers -- sylas
 
  • #3
junglebeast said:
Why is it be impossible to prove this postulate?

It's not.

It's impossible to prove it from Euclid's postulates. Stronger postulates could allow it to be proven. Examples:
* Trivially: Euclid's axioms + the parallel postulate
* Less trivial: Euclid's axioms + "a rectangle exists"

One nice way to think of it is that hyperbolic geometry is a model of Euclid's axioms where the parallel postulate fails.
 

Related to Incompleteness of Euclidean Geometry: Proving the Parallel Postulate

1. What is the parallel postulate in Euclidean geometry?

The parallel postulate in Euclidean geometry states that given a line and a point not on that line, there is exactly one line through the point that is parallel to the given line.

2. What is the significance of the parallel postulate?

The parallel postulate is significant because it is one of the basic axioms or assumptions upon which Euclidean geometry is built. It allows for the construction of parallel lines, which are essential in many geometric proofs and constructions.

3. How can the parallel postulate be proven?

The parallel postulate cannot be proven within the framework of Euclidean geometry. This is known as the "incompleteness of Euclidean geometry" and was famously proven by mathematician David Hilbert in the late 19th century.

4. What implications does the "incompleteness of Euclidean geometry" have?

The "incompleteness of Euclidean geometry" means that the parallel postulate cannot be proven using other axioms or assumptions within the framework of Euclidean geometry. This led to the development of non-Euclidean geometries, which do not rely on the parallel postulate and can model curved spaces.

5. How does the "incompleteness of Euclidean geometry" impact our understanding of geometry?

The "incompleteness of Euclidean geometry" demonstrates that our understanding of geometry is not limited to Euclidean geometry and that other geometries, such as non-Euclidean geometries, also have valid and useful applications. It also highlights the importance of axioms and assumptions in determining the properties and rules of a mathematical system.

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