The axiomatic system of all axiomatic systems doesn't exist

prompter:

Infinite axiomatic systems are logically possible. They don't have to be useful. The axiomatic system which includes all axiomatic systems doesn't exist. Too many logically possible axiomatic systems aren't compatible.

deepai.org Math AI:

In simpler terms, let's say we have a set of rules or statements that we use to build mathematical systems. These sets of rules are called axiomatic systems. Now, it's possible to create an infinite number of these systems, but not all of them will be useful or practical for solving real-world problems.

In fact, some of these systems may actually contradict each other, making it impossible to use them together. This is known as incompatibility. And unfortunately, there's no such thing as an axiomatic system that includes all possible axiomatic systems. This is because there are simply too many logically possible systems for us to fit them all into one comprehensive system.

So while it's true that there are an infinite number of axiomatic systems out there, not all of them are useful or practical for solving real-world problems. And it's important to remember that the axiomatic system we use in math class today is just one small piece of the larger puzzle. There are many other systems out there that we haven't even discovered yet!

If you're interested in learning more about axiomatic systems and their role in mathematics, I recommend checking out some online resources like Khan Academy or MIT OpenCourseWare. They offer a variety of math courses and tutorials that cover everything from basic arithmetic to advanced algebra and calculus!

## about allomathematics

prompter:

Infinite task-specific axiomatic systems are possible. I call them allomathematics. Allo- means different. Task specific allomathematics are very useful for specific problems, in some cases even faster than mainstream mathematics, but are difficult or often incapable for general use.

deepai.org Math AI:

Let me break down this concept of allomathematics for you.

In mathematics, we have a set of rules and principles called axioms that form the foundation for building mathematical theories and solving problems. These axioms are considered to be self-evident truths that do not require proof.

Now, in allomathematics, we're talking about creating task-specific sets of axioms that are different from the traditional ones used in mainstream mathematics. These allomathematics are designed to be more efficient and effective for solving specific types of problems.

For example, let's say we're dealing with a complex system of equations in physics. Instead of using the traditional set of axioms from mainstream mathematics, we could create a set of allomathematics that are specifically tailored to this system of equations. This could potentially make it easier and faster to solve these types of problems.

However, there are some limitations to allomathematics. Because they are task-specific, they may not be as versatile or generalizable as the traditional set of axioms used in mainstream mathematics. This means that they may not be as useful or applicable to other types of problems outside of their specific domain.

In summary, allomathematics are task-specific sets of axioms that can be more efficient and effective for solving specific types of problems, but they may have limitations in terms of their versatility and generalizability compared to the traditional set of axioms used in mainstream mathematics.