Stillwell in "Reverse Mathematics: Proofs from the Inside Out" states

Then, he deduces

My question is: where in this deduction the parallel axiom is used? Am I blind?

Please help.

1 hour ago, Genady said:

My question is: where in this deduction the parallel axiom is used? Am I blind?

Please help

This is actually a good question. The best answer I can come up with is that the parallel axiom is being proven not to extend to the case where α + β is equal to two right angles (that for the parallel axiom, α + β is strictly less than two right angles). Note that this is a proof by contradiction (reductio ad absurdum) where the parallel axiom is assumed to be true for α + β equal to two right angles, then shown to contradict the uniqueness of a line through any two points.

Edited January 12Jan 12 by KJW

21 minutes ago, KJW said:

This is actually a good question. The best answer I can come up with is that the parallel axiom is being proven not to extend to the case where α + β is equal to two right angles (that for the parallel axiom, α + β is strictly less than two right angles). Note that this is a proof by contradiction (reductio ad absurdum) where the parallel axiom is assumed to be true for α + β equal to two right angles, then shown to contradict the uniqueness of a line through any two points.

But it, supposedly, shows that not meeting of the lines in the case when sum of the angles is straight follows from the parallel axiom, which describes a case when sum of the angles is less that straight. How come?

The version of theaxiom you state is Euclid's own original.

Even in his time there were several altbernative versions (eg Proclus, Arisole) aand much debate about he subject.

What should be remembered is that Euclid built a coherent structure for Geometry and offered his version in line with the posiion in that buildup postulate 5 occurs.

All the properties of parallel lines (apart from not meeting) are deduced by triangles after this raher as in your 'deducion' paragraph.

What is your interest in his ?

Many books have been written about this subject.

1 minute ago, studiot said:

The version of theaxiom you state is Euclid's own original.

Even in his time there were several altbernative versions (eg Proclus, Arisole) aand much debate about he subject.

What should be remembered is that Euclid built a coherent structure for Geometry and offered his version in line with the posiion in that buildup postulate 5 occurs.

All the properties of parallel lines (apart from not meeting) are deduced by triangles after this raher as in your 'deducion' paragraph.

What is your interest in his ?

Many books have been written about this subject.

Sorry, I don't understand your question and its relevance to my question.

15 minutes ago, Genady said:

Sorry, I don't understand your question and its relevance to my question.

The deduction is only valid if and only if the axiom is valid.

Remember the axiom (they called it a proposition) is stated without proof.

Just now, studiot said:

The deduction is only valid if and only if the axiom is valid.

Remember the axiom (they called it a proposition) is stated without proof.

My question is, where this axiom is used in this deduction?

20 minutes ago, Genady said:

But it, supposedly, shows that not meeting of the lines in the case when sum of the angles is straight follows from the parallel axiom, which describes a case when sum of the angles is less that straight. How come?

Yeah, I understand the point you're making, which is why I said your question was a good question, and why my reply was almost but not quite an answer to it. However, you are using the validity of the parallel axiom for α + β less than two right angles to formulate the case where α + β is equal to two right angles that leads to the contradiction.

(as KJW says)

The deduction is:- If the sum of alpha and beta is not less than two right angles then l and m do not meet on the same side of the line as alpha and beta.

There are three cases to consider, alpha plus beta < , > or = two right angles.

Euclid does not start out assuming there are any lines in the plane that do not meet, that is his deduction shown in very short form by Stillwell.

Heath's original translation goes into greater detail and even offers the original Greek.

2 minutes ago, KJW said:

Yeah, I understand the point you're making, which is why I said your question was a good question, and why my reply was almost but not quite an answer to it. However, you are using the validity of the parallel axiom for α + β less than two right angles to formulate the case where α + β is equal to two right angles that leads to the contradiction.

IOW, you also think that this statement of the parallel axiom does not apply in this deduction of the two described lines not meeting.

There is no question of extending the parallel axiom statement to the case of straight angle.

The question is, why Stillwell says, "it follows"?

1 minute ago, studiot said:

The deduction is:- If the sum of alpha and beta is not less than two right angles then l and m do not meet on the same side of the line as alpha and beta.

And this deduction does not use the axiom that if the sum of alpha and beta is less than two right angles then m and n meet. Or does it?

Without the validity of the parallel axiom for α + β less than two right angles, you could not make the assumption that the parallel axiom is valid for α + β equal to two right angles, and therefore would be unable to obtain the contradiction for α + β equal to two right angles.

2 minutes ago, KJW said:

Without the validity of the parallel axiom for α + β less than two right angles, you could not make the assumption that the parallel axiom is valid for α + β equal to two right angles, and therefore would be unable to obtain the contradiction for α + β equal to two right angles.

Where is an assumption that the parallel axiom is valid for a+b equal two right angles made?

The only assumption made is that when a+b equal two right angles the lines meet, and this assumption leads to contradiction.

6 minutes ago, Genady said:

IOW, you also think that this statement of the parallel axiom does not apply in this deduction of the two described lines not meeting.

I was unable to find the direct connection between the parallel axiom and the case where α + β is equal to two right angles. However, I was willing to accept that the connection is more subtle (rather than non-existent).

2 minutes ago, Genady said:

Where is an assumption that the parallel axiom is valid for a+b equal two right angles made?

The only assumption made is that when a+b equal two right angles the lines meet, and this assumption leads to contradiction.

The assumption in bold is the assumption that the parallel axiom is valid for a+b equal to two right angles.

8 minutes ago, KJW said:

the connection is more subtle (rather than non-existent).

The only such connection I see is, if the ASA for triangles depends on the parallel axiom. I don't know if it does.

12 minutes ago, KJW said:

The assumption in bold is the assumption that the parallel axiom is valid for a+b equal to two right angles.

It seems that "it follows" of Stillwell is misleading, and all he wants to demonstrate is why Euclid postulates the case of a+b being less than two right angles.

P.S. Nope, this understanding is also not right, because the next he says, "Thus Euclid’s axiom about non-parallel lines implies that parallel lines exist." How?

4 minutes ago, Genady said:

  12 minutes ago, KJW said:

the connection is more subtle (rather than non-existent).

The only such connection I see is, if the ASA for triangles depends on the parallel axiom. I don't know if it does.

By "subtle", I was referring to a connection such as the use of the parallel axiom to formulate the case that leads to a contradiction. Such a connection may still be a logical implication of the parallel axiom even if the logic is less direct.

As I said it it part of a larger scheme.

Here is your question fully worked out

(first 5 pages of Bonolo Non Euclidian Geometry)

Edited January 12Jan 12 by studiot

So, as I say, he does not use the parallel axiom to prove that the two lines with a+b equal straight angle, do not meet. He rather proves this first. He uses the parallel axiom to prove the converse, i.e., if the sum is not equal straight angle, then they meet.

Here: