Surreal numbers

[scruffy]

There's real numbers, hyper-real numbers, and surreal numbers.

The question is how we deal with SCALE.

Calculus has the concept of an "infinitesimal", which means a number so small that it's smaller than the smallest real number.

And, it has the concept of "infinity", a number so big it's off the charts.

Unfortunately, calculus is not very precise. It confuses people by mistakenly equating infinities with singularities, and it calls division by zero an "infinity" which is completely wrong.

To address these woeful misrepresentations, the mathematicians Georg Cantor and John von Neumann developed "surreal" sets. This is what a surreal number tree looks like:

It contains both infinities and infinitesimals.

There is a "halfway" concept called "hyper-real", which adds an infinity and an infinitesimal to the real number line. But the full implementation allows for the description of things that are smaller than an infinitesimal, and bigger than an infinity.

Surreal number - Wikipedia

en.wikipedia.org

Why does this matter?

It matters because of topology, which means physics. For example:

The physicists and mathematicians who developed general relativity around the turn of the 20th century, were forced to grapple with preservation of measure. In the specific case of relativity this means things like dot products, and norms. The Lorentz transformations provide an example.

A relativistic "norm" is a distance in spacetime. The rules say this distance has to be the same for all observers, which means the norm has to be preserved through any kind of geometric transformation, including rotations, reflections, translations, and so on.

But the norm, is what they call a quadratic "form", for example in ordinary Euclidean geometry it would be sqrt(x^2 + y^2). So how can we preserve this in the case where x and y are infinitesimals, which means numbers that aren't even on the real number line?

The answer required a deep dive into set theory. In sets, a fundamental concept is the uniqueness of elements. To establish uniqueness, you label, and you count. You take each element and you assign a number to it, it's called an ordinal. You say, this is the first element, that's the second, and so on - and you keep labeling till you get to the end, till you've exhausted all the elements. Then you ask "how many" elements there are, which is the "size" of the set, that's called the cardinal. If you use the real number line for labels, starting with 0, the cardinal (size) becomes the smallest unused number after you're done labeling. Very easy.

What Cantor and von Neumann showed, is there is a way to formalize infinities in a countable manner, which resolves a paradox in the usual Zermelo-Frankel approach they teach in college. The full formal set is called NBG, instead of ZF/C. It stands for Neumann-Bernays-Godel. It requires "sur"-real numbers, so you can count tiny things and huge things. The benefit of NBG is it's fully axiomatizable, whereas ZF/C is not.

Von Neumann–Bernays–Gödel set theory - Wikipedia

en.wikipedia.org

NBG let's you make intelligent statements about infinities and infinitesimals, and in some cases even let's you describe singularities. It is the foundation of much of modern mathematics, including group theory which links algebra and geometry through algebraic topology. It provides axiomatic "ordering" of sets, without which we couldn't calculate the distance between points, which means important concepts like area and volume become useless, and induction becomes impossible.

 

[shockedcanadian]

There's real numbers, hyper-real numbers, and surreal numbers.

The question is how we deal with SCALE.

Calculus has the concept of an "infinitesimal", which means a number so small that it's smaller than the smallest real number.

And, it has the concept of "infinity", a number so big it's off the charts.

Unfortunately, calculus is not very precise. It confuses people by mistakenly equating infinities with singularities, and it calls division by zero an "infinity" which is completely wrong.

To address these woeful misrepresentations, the mathematicians Georg Cantor and John von Neumann developed "surreal" sets. This is what a surreal number tree looks like:

View attachment 1007480

It contains both infinities and infinitesimals.

There is a "halfway" concept called "hyper-real", which adds an infinity and an infinitesimal to the real number line. But the full implementation allows for the description of things that are smaller than an infinitesimal, and bigger than an infinity.

Surreal number - Wikipedia

en.wikipedia.org

Why does this matter?

It matters because of topology, which means physics. For example:

The physicists and mathematicians who developed general relativity around the turn of the 20th century, were forced to grapple with preservation of measure. In the specific case of relativity this means things like dot products, and norms. The Lorentz transformations provide an example.

A relativistic "norm" is a distance in spacetime. The rules say this distance has to be the same for all observers, which means the norm has to be preserved through any kind of geometric transformation, including rotations, reflections, translations, and so on.

But the norm, is what they call a quadratic "form", for example in ordinary Euclidean geometry it would be sqrt(x^2 + y^2). So how can we preserve this in the case where x and y are infinitesimals, which means numbers that aren't even on the real number line?

The answer required a deep dive into set theory. In sets, a fundamental concept is the uniqueness of elements. To establish uniqueness, you label, and you count. You take each element and you assign a number to it, it's called an ordinal. You say, this is the first element, that's the second, and so on - and you keep labeling till you get to the end, till you've exhausted all the elements. Then you ask "how many" elements there are, which is the "size" of the set, that's called the cardinal. If you use the real number line for labels, starting with 0, the cardinal (size) becomes the smallest unused number after you're done labeling. Very easy.

What Cantor and von Neumann showed, is there is a way to formalize infinities in a countable manner, which resolves a paradox in the usual Zermelo-Frankel approach they teach in college. The full formal set is called NBG, instead of ZF/C. It stands for Neumann-Bernays-Godel. It requires "sur"-real numbers, so you can count tiny things and huge things. The benefit of NBG is it's fully axiomatizable, whereas ZF/C is not.

Von Neumann–Bernays–Gödel set theory - Wikipedia

en.wikipedia.org

NBG let's you make intelligent statements about infinities and infinitesimals, and in some cases even let's you describe singularities. It is the foundation of much of modern mathematics, including group theory which links algebra and geometry through algebraic topology. It provides axiomatic "ordering" of sets, without which we couldn't calculate the distance between points, which means important concepts like area and volume become useless, and induction becomes impossible.

Based on this theory perhaps, as I believe in due time; Man will solve how to transport ourselves faster than the speed of light. The calculations involved to determine this is immense. What will be even more challenging will be to find a means in which our body and/or the capsule can survive such speeds not just actually identifying a manner to travel faster than light.

I anticipate the early adoption would be to send a one way capsule with a video and other details about our location to a foreign, alien planet. Then there is the problem of not colliding with anything unless we are able to calculate the surreal number precisely as to send a small item to an exact planet, depending on how we are able to move this cargo. It may not even be in a straight, continuous line but simply "move item A to from coordinate X to coordinate Y" and somehow the movement would be instant without traveling physically across the universe.

If we are able to master and produce such capsules and produce them as we do today drones, we could send them to endless locations around the universe.

Quantum computers will help find these answers. The knowledge gained from these super computers will either assist our species greatly or the weaknesses from within our species will ensure the knowledge is obtained and applied in a manner to in which we destroy ourselves.

As an aside, I've been under the impression that governments and/or the wealthy are leveraging these super computers to mine bit coin etc (assuming it is cost effective relative to the cooling/energy costs required to run quantum computers). We would never be aware of these operations but it would allow governments to avoid the destruction of currency and debt if they would leverage this instead of further debt.

 

[yidnar]

I wonder if the infinitesimal realm is as limitless as the infinite ??

 

[Sherlock Holmes]

There's real numbers, hyper-real numbers, and surreal numbers.

The question is how we deal with SCALE.

Calculus has the concept of an "infinitesimal", which means a number so small that it's smaller than the smallest real number.

And, it has the concept of "infinity", a number so big it's off the charts.

Unfortunately, calculus is not very precise. It confuses people by mistakenly equating infinities with singularities, and it calls division by zero an "infinity" which is completely wrong.

To address these woeful misrepresentations, the mathematicians Georg Cantor and John von Neumann developed "surreal" sets. This is what a surreal number tree looks like:

View attachment 1007480

It contains both infinities and infinitesimals.

There is a "halfway" concept called "hyper-real", which adds an infinity and an infinitesimal to the real number line. But the full implementation allows for the description of things that are smaller than an infinitesimal, and bigger than an infinity.

Surreal number - Wikipedia

en.wikipedia.org

Why does this matter?

It matters because of topology, which means physics. For example:

The physicists and mathematicians who developed general relativity around the turn of the 20th century, were forced to grapple with preservation of measure. In the specific case of relativity this means things like dot products, and norms. The Lorentz transformations provide an example.

A relativistic "norm" is a distance in spacetime. The rules say this distance has to be the same for all observers, which means the norm has to be preserved through any kind of geometric transformation, including rotations, reflections, translations, and so on.

But the norm, is what they call a quadratic "form", for example in ordinary Euclidean geometry it would be sqrt(x^2 + y^2). So how can we preserve this in the case where x and y are infinitesimals, which means numbers that aren't even on the real number line?

The answer required a deep dive into set theory. In sets, a fundamental concept is the uniqueness of elements. To establish uniqueness, you label, and you count. You take each element and you assign a number to it, it's called an ordinal. You say, this is the first element, that's the second, and so on - and you keep labeling till you get to the end, till you've exhausted all the elements. Then you ask "how many" elements there are, which is the "size" of the set, that's called the cardinal. If you use the real number line for labels, starting with 0, the cardinal (size) becomes the smallest unused number after you're done labeling. Very easy.

What Cantor and von Neumann showed, is there is a way to formalize infinities in a countable manner, which resolves a paradox in the usual Zermelo-Frankel approach they teach in college. The full formal set is called NBG, instead of ZF/C. It stands for Neumann-Bernays-Godel. It requires "sur"-real numbers, so you can count tiny things and huge things. The benefit of NBG is it's fully axiomatizable, whereas ZF/C is not.

Von Neumann–Bernays–Gödel set theory - Wikipedia

en.wikipedia.org

NBG let's you make intelligent statements about infinities and infinitesimals, and in some cases even let's you describe singularities. It is the foundation of much of modern mathematics, including group theory which links algebra and geometry through algebraic topology. It provides axiomatic "ordering" of sets, without which we couldn't calculate the distance between points, which means important concepts like area and volume become useless, and induction becomes impossible.

Why post all this? anyone interested can easily study this, why do you plagiarize these websites and try to fool people that you're somehow relevant?

You remind me of the guy in the training class (there's always one) who eagerly answers when the tutor is asked a question.

Fellow students think such people are dingbats and tutors think such people are dingbats.

 

[Sherlock Holmes]

Here, post something that might stimulate people without stressing how "Oh so wonderful" you are:

 

[scruffy]

I should probably add an explicit note of caution, in relation to the 'labeling' operation.

In the example I used numbers from the real number line, as labels. Which implies an "order" or "ordering", that doesn't really exist.

In the example, the labels are arbitrary, they could for example be letters of the alphabet, in any order whatsoever. The purpose of the labels is to uniquely identify the members of the set, not to order them.

HOWEVER, sometimes we "can" order them, depending on the construction of the set. The easiest way to create an order is to use the <= operator, in which case we have a "partially ordered set" (poset). The relation <= is binary, it is reflexive, transitive, and antisymmetric. The reason it's "partial" is to allow some pairs that are comparable and some that aren't - for example, you could have the set { A, B, C1, C2, D } but maybe you're only interested in the first letter. So <= allows cases like this.

Partially ordered set - Wikipedia

en.wikipedia.org

On the other hand, a "total" order uses < which is a much stronger relation. In addition to being reflexive, transitive, and antisymmetric, a total orders adds the requirement of being "strongly connected". This means it relates "every" pair of elements.

Total order - Wikipedia

en.wikipedia.org

Connected relation - Wikipedia

en.wikipedia.org

In any set there are two special sets, the empty set and the entire set. These are helpful in defining "openness" and "closure", which is how we start being able to do math on sets. Because an ordering still doesn't define the important concept of "near", and it doesn't define the distance between subsets (subsets can be treated as "points" with the right kind of labeling and ordering, which means we can use "operators" to do math on them). For that we need "open" sets.

Open sets are the foundation of topology. In the example, we can build "subsets" out of one or more of the member elements. An open set is one that contains every union of its members, every finite intersection of its members, the empty set, and the set itself. Any set that meets these conditions is called a topological space.

Open sets provide a concept of "near-ness", without defining an actual distance (measure). Open sets provide continuity, connectedness, and compactness, which is sufficient for topology but not necessarily for geometry. If we have an actual measure space, we can do geometry - examples being distance on the real line, angle on a sphere, and hyperbolic sections in Minkowski space. The infinitesimal "d" (as in dx) can be defined with or without a geometry.

A "manifold" is an example of a topological space without a (global) measure. Even though it's locally Euclidean (and therefore points have local coordinate systems we can use to do math with), it may not have a definable metric - if it does, we can measure distances and angles and map them to the point coordinate systems and vice versa - but for example a self crossing curve (like a figure 8) is not a manifold by this definition, because one of the points can not be charted this way. However if all points can be charted this way we have a "differentiable" manifold and the infinitesimal dx can be used for geometry

Open set - Wikipedia

en.wikipedia.org

On the other hand, a closed set contains it's boundary, and is therefore complementary to an open set. Both openness and closure are relative to the topology, so a set that is closed in one topology may be open in another. It depends on the embedding. Some sets are "absolutely closed", these include the compact Hausdorff spaces.

Closed set - Wikipedia

en.wikipedia.org

And finally - if we do have a measure ("distance between subsets"), we can build a "measure space" ("sigma algebra") consisting of subsets closed under complement and countable unions and intersections. Sigma algebras are necessary for integration. (And for the assignment of probabilities). An infinitesimal differential dx is not necessarily integrable, for it to be integrable the concept of "area" has to make sense. A quick way to get a sigma algebra from a set is to partition it, in this case the collection of all unions of subsets will be a sigma algebra.

σ-algebra - Wikipedia

en.wikipedia.org

 

[scruffy]

Further reading and some cool history:

Infinitesimal - Wikipedia

en.wikipedia.org

 

[Nobody911]

There's real numbers, hyper-real numbers, and surreal numbers.

The question is how we deal with SCALE.

Calculus has the concept of an "infinitesimal", which means a number so small that it's smaller than the smallest real number.

And, it has the concept of "infinity", a number so big it's off the charts.

Unfortunately, calculus is not very precise. It confuses people by mistakenly equating infinities with singularities, and it calls division by zero an "infinity" which is completely wrong.

To address these woeful misrepresentations, the mathematicians Georg Cantor and John von Neumann developed "surreal" sets. This is what a surreal number tree looks like:

View attachment 1007480

It contains both infinities and infinitesimals.

There is a "halfway" concept called "hyper-real", which adds an infinity and an infinitesimal to the real number line. But the full implementation allows for the description of things that are smaller than an infinitesimal, and bigger than an infinity.

Surreal number - Wikipedia

en.wikipedia.org

Why does this matter?

It matters because of topology, which means physics. For example:

The physicists and mathematicians who developed general relativity around the turn of the 20th century, were forced to grapple with preservation of measure. In the specific case of relativity this means things like dot products, and norms. The Lorentz transformations provide an example.

A relativistic "norm" is a distance in spacetime. The rules say this distance has to be the same for all observers, which means the norm has to be preserved through any kind of geometric transformation, including rotations, reflections, translations, and so on.

But the norm, is what they call a quadratic "form", for example in ordinary Euclidean geometry it would be sqrt(x^2 + y^2). So how can we preserve this in the case where x and y are infinitesimals, which means numbers that aren't even on the real number line?

The answer required a deep dive into set theory. In sets, a fundamental concept is the uniqueness of elements. To establish uniqueness, you label, and you count. You take each element and you assign a number to it, it's called an ordinal. You say, this is the first element, that's the second, and so on - and you keep labeling till you get to the end, till you've exhausted all the elements. Then you ask "how many" elements there are, which is the "size" of the set, that's called the cardinal. If you use the real number line for labels, starting with 0, the cardinal (size) becomes the smallest unused number after you're done labeling. Very easy.

What Cantor and von Neumann showed, is there is a way to formalize infinities in a countable manner, which resolves a paradox in the usual Zermelo-Frankel approach they teach in college. The full formal set is called NBG, instead of ZF/C. It stands for Neumann-Bernays-Godel. It requires "sur"-real numbers, so you can count tiny things and huge things. The benefit of NBG is it's fully axiomatizable, whereas ZF/C is not.

Von Neumann–Bernays–Gödel set theory - Wikipedia

en.wikipedia.org

NBG let's you make intelligent statements about infinities and infinitesimals, and in some cases even let's you describe singularities. It is the foundation of much of modern mathematics, including group theory which links algebra and geometry through algebraic topology. It provides axiomatic "ordering" of sets, without which we couldn't calculate the distance between points, which means important concepts like area and volume become useless, and induction becomes impossible.

Explanation of Hyper-real and Surreal Numbers

While hyper-real and surreal numbers provide powerful tools in mathematics, particularly in analysis and number theory, they come with their own set of challenges and limitations.

Cons of Hyper-real Numbers

1. Complexity:
- The concept of hyper-real numbers can be quite complex and may be difficult for many students and mathematicians to grasp. This complexity can hinder their widespread acceptance in certain mathematical circles.

2. Non-standard Analysis:
- Hyper-real numbers are an integral part of non-standard analysis, which is not universally accepted or utilized within the mathematical community. Many mathematicians prefer classical analysis, which does not rely on infinitesimals.

3. Lack of Familiarity:
- Many practitioners are not familiar with hyper-real numbers, which can limit their practical application in fields that could benefit from them.

Cons of Surreal Numbers

1. Intuitive Understanding:
- Surreal numbers are constructed through a highly abstract recursive process, making them difficult to intuitively understand, especially compared to real numbers.

2. Complexity in Use:
- While surreal numbers include a broader set of numbers (including infinitesimals and infinite numbers), this complexity can complicate calculations and proofs, potentially leading to errors or misunderstandings.

3. Application Limitations:
- Surreal numbers are not as widely used in practical applications as real numbers or even hyper-real numbers, limiting their impact in certain areas of mathematics and its applications.

Conclusion

While these number systems offer unique advantages, they also present challenges and limitations that can affect their understanding, acceptance, and practical application in mathematics.

Hyper-real numbers can be likened to imagination and creativity in everyday life. Just as hyper-real numbers extend the real number system to include infinitesimally small and infinitely large quantities, imagination and creativity allow us to go beyond the constraints of reality and explore limitless possibilities.

Similarly, surreal numbers can be compared to dreams and aspirations. Surreal numbers are a unique extension of the number system that include both real and non-real elements, much like how dreams combine elements of reality with fantastical elements.

Just as surreal numbers can represent complex and abstract concepts, dreams and aspirations push us to strive for goals that may seem beyond reach but are still within the realm of possibility.

In both cases, hyper-real and surreal numbers serve as a reminder that there is more to life than what we perceive on the surface. They encourage us to think outside the box, push boundaries, and embrace the unknown.

Just as mathematicians explore these abstract number systems to expand their understanding of mathematics, we can embrace imagination, creativity, dreams, and aspirations to enrich our everyday lives.