Treatment of axioms of formal axiomatic theory

In summary, the conversation discusses the proper treatment of results obtained from outside a formal axiomatized theory, specifically in the case of using induction in Robinson Arithmetic. It is determined that these results are not added to the axioms of Q when applied in a formal proof, and are used in the same manner as other theorems of Q. Further details and references are requested for clarification.
  • #1
agapito
49
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What is the proper treatment of results about a formal axiomatized theory which are obtained from outside the theory itself? For example, there are 9 results dealing with the "≤" relation for Robinson Arithmetic, some of which are established by using induction, which is not "native" to Q.

Are these Q-sentences added to the axioms of Q when applied in some formal proof? Otherwise, where or how do they appear?

Thanks for any help.
 
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  • #2
agapito said:
Are these Q-sentences added to the axioms of Q when applied in some formal proof?
No.

agapito said:
Otherwise, where or how do they appear?
I may have an idea about what you are asking, but I am not totally sure. It would be nice if you provided more details. Please give references to the proofs of theorems of Q that are proved by induction. And most importantly, please explain what you mean by "where or how do they appear?". They are used in exactly the same way as other theorems of Q that are proved without induction.
 

FAQ: Treatment of axioms of formal axiomatic theory

What is the purpose of treating axioms in formal axiomatic theory?

The treatment of axioms in formal axiomatic theory is essential for establishing the foundation and structure of a mathematical or logical system. Axioms serve as the starting point for deriving theorems and other logical statements, and therefore, must be carefully defined and organized to ensure the validity and consistency of the system.

How are axioms selected for a formal axiomatic theory?

The selection of axioms for a formal axiomatic theory is a highly debated and complex process. Generally, axioms are chosen based on their simplicity, self-evident truth, and ability to cover the desired scope of the theory. They should also be independent, meaning that they cannot be derived from other axioms in the system.

Can axioms be proven or disproven?

No, axioms cannot be proven or disproven within the context of their own system. They are accepted as true without requiring proof, and any attempt to prove or disprove them would lead to circular reasoning. However, they can be evaluated based on their consistency with other axioms and their ability to accurately describe the desired concept.

What happens if an axiom is found to be inconsistent?

If an axiom is found to be inconsistent with other axioms in the system or leads to contradictions, it must be revised or removed. This can significantly impact the entire theory and may require a reassessment of other axioms and theorems. However, it is a crucial step in ensuring the validity and reliability of the theory.

How do axioms contribute to the development of a formal axiomatic theory?

Axioms are the building blocks of a formal axiomatic theory. They provide the foundation for deriving theorems and other logical statements, and their organization and structure determine the structure and scope of the theory. Without axioms, a formal theory would lack the necessary structure and logical consistency to be considered a valid mathematical or logical system.

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