Other than ZFC, what other axiomatic systems are useful?

In summary, there are other axiomatic systems besides ZFC that have been shown to be useful in the real world, such as the Peano Axioms. However, Godel's theorem reminds us that no system of logic can prove all statements to be true.
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Other than ZFC, what other axiomatic systems have been shown to be useful to the real world?

I'm a layman that is reading through the Wikipedia articles and it seems like the axioms are becoming more Philosophy than Math. We can use a different axiomatic system other than ZFC, but are there any that are actually useful?
 
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FAQ: Other than ZFC, what other axiomatic systems are useful?

What is ZFC and why is it important in mathematics?

ZFC stands for Zermelo-Fraenkel set theory with the axiom of choice. It is an axiomatic system that serves as the foundation for most of modern mathematics. It provides a rigorous and formal framework for defining sets, functions, and other mathematical concepts.

Are there any other commonly used axiomatic systems besides ZFC?

Yes, there are several other axiomatic systems that are commonly used in mathematics, such as ZF (Zermelo-Fraenkel set theory without the axiom of choice), NBG (von Neumann-Bernays-Gödel set theory), and MK (Morse-Kelley set theory). These systems are often used as alternatives to ZFC and have their own set of axioms and rules.

What are the advantages of using alternative axiomatic systems?

One advantage is that alternative axiomatic systems can provide a different perspective or approach to studying mathematical concepts. For example, NBG set theory allows for the existence of proper classes, which are too large to be considered sets in ZFC. Additionally, different systems may be better suited for certain areas of mathematics, such as category theory.

Can different axiomatic systems lead to different mathematical results?

Yes, in some cases, different axiomatic systems can lead to different mathematical results. This is because axioms determine the rules and limitations of a mathematical system, so different axioms can lead to different conclusions. However, in most cases, the results obtained from different axiomatic systems are equivalent.

How do mathematicians decide which axiomatic system to use?

There is no one answer to this question as it depends on the specific problem or area of mathematics being studied. Some mathematicians may prefer to use ZFC as it is the most widely accepted and studied system, while others may choose to use a different system that better suits their needs or research interests. Ultimately, the axiomatic system chosen should be consistent and allow for the development of a coherent and meaningful mathematical theory.

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