Solution to the Parallel Postulate:

"If a line segment intersects two straight lines forming two interior angles on the same side that sum to less than two right angles, then the two lines, if extended indefinitely, meet on that side on which the angles sum to less than two right angles.

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

1. Angle C will be observed as where the points meet.

2. Angle A = 99.999... and Angle B=79.999... thus there sum will be 179.9999 and always less than 180 degrees.

3. With Angle A and B, defined by point 2, Angle C = .000....1 of a degree.

4. Angles A, B and C are all constant rates of change.

5. The change in one angle is proportional to the change in another.

6. Inverting the problem and starting from the premise that Angle C is 1 degree, then Angle A is 999... approaching infinity and Angle B is 799... approaching infinity.

7. Point 6 observes 1 degree as perpetually expanding to infinite degrees through Angles A and B.

8. However if Angle C is .000...1 degrees than angles A and B may also continually change as well:

a. 99.999... → 100.999... → 101.999... → ... → 178.9999 while inversely:

b. 79.999... → 78.999... → 77.999... → ... → .999999

c. Angles A and B will always be infinitely less than 180 degrees as .000...1 observes 1 units infinitely greater than 0.

9. Angle C as .000...1 and Angle A as 178.999... and Angle B as .99999 observe the three lines which form the angles "line up" into 1 line of 180 degrees.

10. Angles A,B,C are equivalent to proportional change and effectively converge into one line as both 1 degree and 180 degrees, where Euclid's first premise of a line existing between two points is expanded to always have a center point; hence each line is composed of infinite lines setting the foundations for further geometries while maintaining Euclids premise of a line between two points as foundation of continuous movement.

11. The line is composed of 3 angles of proportionally irrational degrees, as 1 degree.

12. Where each of the lines are equivalent as infinite, a sixth postulate is argued where the convergence of these three lines into 1 always results in 3 lines of 4 points.

13. This infinite set of lines, within a line, observes all degrees as being composed of further degrees with the "degree" equivalent to change; thus all geometric forms exist as boundaries of movements through infinite grades synonymous to a qualitative "red" being composed of infinite reds through infinite colors.

The Parallel Postulate is the foundation for the degree as a state of isomorphism through the recursion of dual proportional angles repeatedly contracting and expanding simultaneously...thus resulting in the the first four of Euclid's postulates.

****Will Continue Later.

## Solution to the Parallel Postulate as the Foundation for Isomorphism

### Re: Solution to the Parallel Postulate as the Foundation for Isomorphism

The parallel postulate is not a problem that needs a 'solution.'

### Re: Solution to the Parallel Postulate as the Foundation for Isomorphism

"For two thousand years, many attempts were made to prove the parallel postulate using Euclid's first four postulates. The main reason that such a proof was so highly sought after was that, unlike the first four postulates, the parallel postulate is not self-evident. If the order the postulates were listed in the Elements is significant, it indicates that Euclid included this postulate only when he realised he could not prove it or proceed without it.[10] Many attempts were made to prove the fifth postulate from the other four, many of them being accepted as proofs for long periods until the mistake was found. Invariably the mistake was assuming some 'obvious' property which turned out to be equivalent to the fifth postulate (Playfair's axiom). Although known from the time of Proclus, this became known as Playfair's Axiom after John Playfair wrote a famous commentary on Euclid in 1795 in which he proposed replacing Euclid's fifth postulate by his own axiom."

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

### Re: Solution to the Parallel Postulate as the Foundation for Isomorphism

And all the attempts at proof failed because, as we now know, it is independent of the other axioms. I guess you could say that its not being 'self-evident' to some is a problem.Eodnhoj7 wrote: ↑Fri Feb 08, 2019 6:50 pm"For two thousand years, many attempts were made to prove the parallel postulate using Euclid's first four postulates. The main reason that such a proof was so highly sought after was that, unlike the first four postulates, the parallel postulate is not self-evident. If the order the postulates were listed in the Elements is significant, it indicates that Euclid included this postulate only when he realised he could not prove it or proceed without it.[10] Many attempts were made to prove the fifth postulate from the other four, many of them being accepted as proofs for long periods until the mistake was found. Invariably the mistake was assuming some 'obvious' property which turned out to be equivalent to the fifth postulate (Playfair's axiom). Although known from the time of Proclus, this became known as Playfair's Axiom after John Playfair wrote a famous commentary on Euclid in 1795 in which he proposed replacing Euclid's fifth postulate by his own axiom."

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

### Re: Solution to the Parallel Postulate as the Foundation for Isomorphism

The question of the solution for the Parallel being necessitated as "true" does not require it to be "dependent" upon the Euclidean Axioms; however this absense of dependency does not negate the Parallel Postulate is a "proof" of the prior Euclidean Axioms.Wyman wrote: ↑Fri Feb 08, 2019 6:55 pmAnd all the attempts at proof failed because, as we now know, it is independent of the other axioms. I guess you could say that its not being 'self-evident' to some is a problem.Eodnhoj7 wrote: ↑Fri Feb 08, 2019 6:50 pm"For two thousand years, many attempts were made to prove the parallel postulate using Euclid's first four postulates. The main reason that such a proof was so highly sought after was that, unlike the first four postulates, the parallel postulate is not self-evident. If the order the postulates were listed in the Elements is significant, it indicates that Euclid included this postulate only when he realised he could not prove it or proceed without it.[10] Many attempts were made to prove the fifth postulate from the other four, many of them being accepted as proofs for long periods until the mistake was found. Invariably the mistake was assuming some 'obvious' property which turned out to be equivalent to the fifth postulate (Playfair's axiom). Although known from the time of Proclus, this became known as Playfair's Axiom after John Playfair wrote a famous commentary on Euclid in 1795 in which he proposed replacing Euclid's fifth postulate by his own axiom."

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

Actually it is not independent of the other axioms:

1) The Postulate returns back to a "line between two points" as the foundation of the "degree" itself.

2) This Line, resulting through the angles as "change", in turn results in further lines, thus necessitating the line as finite and straight with its finiteness observed by its relation to other lines.

3) The radius is founded by the Postulate producing points 1 and 2 as the radius is strictly points 1 and 2. The Angles, as continual change, cause the individual lines which compose the angles as perpetual circles; hence diameters. Thus the postulate sets down the linear foundations for the circles while as "changing" observe the circle themselves.

4) The continual rotation of the angles as perpetual change set the foundation for "All right angles being Equal" where the question is less one of "right" or "not right" but rather the "degree equaling the degree" which sets a premise for the aristotelian principle of Identity as "P=P".

### Re: Solution to the Parallel Postulate as the Foundation for Isomorphism

"The independence of the parallel postulate from Euclid's other axioms was finally demonstrated by Eugenio Beltrami in 1868." - Wikipedia under "Parallel Postulate"

### Re: Solution to the Parallel Postulate as the Foundation for Isomorphism

I am familiar with the verse, but all deductive arguments effectively require a sense of inductivity as well. The Parallel postulate, as argued above, actually leads to the first four axioms; thus necessitating an inherent connectivity.

Second I would like to see the actual proof, a fallacy of authority argument does not impress me. I worked with men much smarter than me, they are not God.