Cont: Deeper than primes - Continuation 2

[doronshadmi]

This does not follow.

This does not follow only if the notion of actual infinity is used in The Pythagorean Theorem.

Since potential infinity is used in http://www.internationalskeptics.com/forums/showpost.php?p=12126277&postcount=2799 , we are working within the domain.

What you call "normal arithmetic" is no more than a particular mathematical framework, which can't be considered as the one and only one way to do arithmetic.

You actually ignore any possible alternative about the notion of infinity, since you accept only the Cantorean notion of infinity (known as actual infinity).

 

[jsfisher]

This does not follow only if the notion of actual infinity is used in The Pythagorean Theorem.

It does not follow no matter what version of infinity you attempt to apply to the Pythagorean Theorem.

 

[doronshadmi]

It does not follow no matter what version of infinity you attempt to apply to the Pythagorean Theorem.

Well, this is exactly the predicted reply of anyone that rejects the notion of potential infinity (an endless increased value).

By fixed-only-value notion one can't agree that symbols like a² or c² do not have a fixed value as b² has, in the considered case.

 

[jsfisher]

Well, this is exactly the predicted reply of anyone that rejects the notion of potential infinity (an endless increased value).

Be as dismissive as you like, but you still cannot prove things by misusing the Pythagorean Theorem as you have.

 

[doronshadmi]

Be as dismissive as you like, but you still cannot prove things by misusing the Pythagorean Theorem as you have.

The Pythagorean Theorem does not hold in case of using actual infinity, but by using potential infinity it perfectly holds has clearly seen in http://www.internationalskeptics.com/forums/showpost.php?p=12126277&postcount=2799.

You are still do your "best" in order to ignore the fact that by fixed-only-value notion one can't agree that symbols like a² or c² do not have a fixed value as b² has, in the considered case.

And in my considered case c² (which is infinite) > a² (which is infinite) > b² (which is finite) and > 0, and a²+b²=c² (The Pythagorean Theorem holds).

 

[jsfisher]

The Pythagorean Theorem does not hold in case of using actual infinity, but by using potential infinity it perfectly holds has clearly seen in http://www.internationalskeptics.com/forums/showpost.php?p=12126277&postcount=2799.

Repeating your mistake does not make it correct. The Pythagorean Theorem applies to right triangles in an Euclidean plane. You've erroneously applied it to something that isn't a right triangle in an Euclidean plane. You made other errors, too, but the first requires remediation before you can advance.

 

[doronshadmi]

The Pythagorean Theorem applies to right triangles in an Euclidean plane.

Exactly, but unlike your fixed-only framework, which according to it a², b² and c² must have fixed values, my framework has relations among fixed and non-fixed values, such that non-fixed value c² > non-fixed value a² by fixed value b², in an endless (non-fixed) Euclidean plane.

In my framework non-fixed values are potential infinities, where fixed values are finite values, such that finite and infinite values have impact on each other (unlike in your fixed-only framework, where both finite and infinite values are fixed, and as a result finite values are inaccessible to infinite values (have not impact on infinite values), exactly because infinity in your fixed-only framework is defined in terms of actual infinity (it is defined in terms of boxes of completeness, which literally eliminates the notion of infinity as an open endless non-enthropic framework)).

Repeating your enthropic fixed-only framework does not make it correct in terms of non-fixed and fixed framework (which is a non-enthropic framework).

You made other errors, too, but the first requires remediation before you can advance.

Please show them in details.

 

[jsfisher]

Exactly, but unlike your fixed-only framework, which according to it a², b² and c² must have fixed values, my framework has relations among fixed and non-fixed values, such that non-fixed value c² > non-fixed value a² by fixed value b², in an endless (non-fixed) Euclidean plane.

You can make up whatever framework you like, but you don't get to misapply established theorems you think convenient.

If you want your version of the Pythagorean Theorem in your framework and have it apply to things that aren't triangles in the Euclidean plane, then you'll be needing to provide definitions for your framework and proof for your theorems.

History would indicate you will do neither.

 

[doronshadmi]

If you want your version of the Pythagorean Theorem in your framework and have it apply to things that aren't triangles in the Euclidean plane

My version of the Pythagorean Theorem is defined in terms of potential infinity in triangles that are in the Euclidean plane.

Since you define infinity only by fixed values, you simply blind to triangles that are defined by potential infinity, which are non-fixed values, as already defined in my previous posts.

In case that you are still missing it, potential infinity is an endless increased value, which is naturally non-fixed.

Warning: It can't be understood in terms of actual infinity.

History would indicate you will do neither.

History indicates that if one forcing fixed infinite values on any mathematical framework, one can't get anything beyond this forcing.

You made other errors, too, but the first requires remediation before you can advance.

Details jsfisher, please support your argument in details.

If once again your reply will be in hands waving style, I'll simply ignore it.

 

[jsfisher]

My version of the Pythagorean Theorem is defined in terms of potential infinity in triangles that are in the Euclidean plane.

Great! but you haven't provided any actual definitions, yet. We can't move on to the proof of your version of the Pythagorean Theorem until you do.

 

[doronshadmi]

Great! but you haven't provided any actual definitions, yet. We can't move on to the proof of your version of the Pythagorean Theorem until you do.

Potential infinity is an endless increased value is an actual definition among the Pythagorean Theorem a²+b²=c², such that a² and c² are potential infinities, where c² > a² exactly by finite value b², in the considered case.

Total smoothness is a property of a non-composed element, and therefore it can't be defined in terms of collections.

In other words angle ab has a fixed value of 90⁰, where ac and bc angles are totally smooth complements of each other, such that ab+ac+bc=180⁰.

These total smooth complements can't be defined in terms of collections (where Cantorean set theories are some particular case of collection).

So, the Pythagorean Theorem a²+b²=c² holds under total smoothness, and it is definitely not closed under the notion of actual infinity, as used by the Cantorean set theories.

So, Cantorean set theories are not the foundation of any possible interesting mathematical framework, in spite of the claims of the majority of, so called, contemporary mathematicians.

 

[Hevneren]

Total smoothness is a property of a non-composed element, and therefore it can't be defined in terms of collections.

Is that all there is to "total smoothness"? It's a property of a non-composed element? Doesn't sound like a definition to me. You could probably substitute "non-composed" for it (whatever "non-composed" means).

In other words angle ab has a fixed value of 90 degreees, where ac and bc angles are totally smooth complements of each other, such that ab+ac+bc=180 degrees.

Yes, we all learnt that in fourth grade. The angles of a triangle add up to 180 degrees.

These total smooth complements can't be defined in terms of collections (where Cantorean set theories are some particular case of collection).

You're repeating yourself. A "totally smooth" element can't, by your own definition, be defined in terms of collections.

So, the Pythagorean Theorem a2+b2=c2 holds under total smoothness, and it is definitely not closed under the notion of actual infinity, as used by the Cantorean set theories.

How does that follow?

 

[Hevneren]

Potential infinity is an endless increased value is an actual definition among the Pythagorean Theorem a²+b²=c², such that a² and c² are potential infinities, where c² > a² exactly by finite value b², in the considered case.

Total smoothness is a property of a non-composed element, and therefore it can't be defined in terms of collections.

In other words angle ab has a fixed value of 90⁰, where ac and bc angles are totally smooth complements of each other, such that ab+ac+bc=180⁰.

These total smooth complements can't be defined in terms of collections (where Cantorean set theories are some particular case of collection).

So, the Pythagorean Theorem a²+b²=c² holds under total smoothness, and it is definitely not closed under the notion of actual infinity, as used by the Cantorean set theories.

So, Cantorean set theories are not the foundation of any possible interesting mathematical framework, in spite of the claims of the majority of, so called, contemporary mathematicians.

Short version: Angles aren't collections, therefore the Pythagorean Theorem holds for infinite values.

 

[jsfisher]

Potential infinity is an endless increased value is an actual definition among the Pythagorean Theorem....

Infinity, potential or otherwise, appears nowhere in the Pythagorean Theorem. If you want to use the theorem, you will first need to prove it is valid for these "endless increased value" triangles you have yet to define.

 

[doronshadmi]

Infinity, potential or otherwise, appears nowhere in the Pythagorean Theorem.

By your fixed-only notion, you have no clue what is potential infinity.

 

[jsfisher]

By your fixed-only notion, you have no clue what is potential infinity.

Bare assertions and insults do not serve to prove your version of the Pythagorean Theorem.

 

[doronshadmi]

Bare assertions and insults do not serve to prove your version of the Pythagorean Theorem.

Why insults? Don't you a proud member of the group of, so called, contemporary mathematicians, which define anything by using fixed-only values, including infinity (for example: ℵ₀, ℵ₁ etc.)?

 

[doronshadmi]

Doesn't sound like a definition to me.

It is not a definition.

Yes, we all learnt that in fourth grade. The angles of a triangle add up to 180 degrees.

Did you also learned in fourth grade how a²+b²=c² in case that a² and c² are potential infinities?

You're repeating yourself. A "totally smooth" element can't, by your own definition, be defined in terms of collections.

In this case you are missing my argument that Cantorean set theories can't be considered as the foundation of mathematics, exactly because they can't deal with totally smooth elements, nor with endlessly increased collections, as given, for example, in http://www.internationalskeptics.com/forums/showpost.php?p=12043625&postcount=2798.

They can't deal with infinite collections exactly because infinite collections are endless increased things that can't be measured by fixed values like ℵ₀, ℵ₁, etc.

How does that follow?

Infinity, whether it is increased smoothly or not, can't be measured by fixed values.

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Total smooth increasing can't be defines in terms of collections, and endless increased collections can't be defined by fixed values (as wrongly done by using Cantor's notions of, so called, actual infinity in therms of collections).

Total smooth increasing (shown in http://www.internationalskeptics.com/forums/showpost.php?p=12126277&postcount=2799) and endless increased collections (shown in http://www.internationalskeptics.com/forums/showpost.php?p=12043625&postcount=2798) are naturally defined by potential infinity, where potential infinity is defined as an endless increased value, whether it is smooth, or not.

 

[jsfisher]

Why insults?

That is my question to you. Rather than offer an argument of substance, you pony up insults. It doesn't help build your case.

Don't you a proud member of the group of, so called, contemporary mathematicians, which define anything by using fixed-only values, including infinity (for example: ℵ₀, ℵ₁ etc.)?

What you presume I be or not be is of no relevance. The only thing needed is from you, and it would be the proof for your version of the Pythagorean Theorem that covers your yet undefined special triangles.

 

[doronshadmi]

What you presume I be or not be is of no relevance.

I do not presume anything. I clearly claim that your fixed-only value notion, prevents from you to grasp the notion of a² and c² as totally smooth and endless increasing (potential infinities) values of the Pythagorean Theorem a²+b²=c², in addition to the finite notion of it, as currently learned.

The notion of potential infinity enables one to understand how a given thing can be considered as a whole and yet incomplete.

For example, in terms of totally smooth endless increasing values a²+b²=c² is a whole and yet incomplete.

Also in terms of discrete endless increasing values http://www.internationalskeptics.com/forums/showpost.php?p=12043625&postcount=2798 is a whole and yet incomplete.

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As for actual infinity, it is both whole AND complete AND can't be defined in terms of collections (Cantor wrongly defined it in terms of collections).