What are the Fundamental Logical Axioms?

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SUMMARY

The discussion centers on the fundamental logical axioms underlying axiomatic systems and mathematics. It establishes that there is no definitive list of axioms applicable to all systems, as demonstrated by Russell and Whitehead's unsuccessful attempts. The Zermelo-Fraenkel axioms of Set Theory, along with the Axiom of Choice (ZFC) and Peano's axioms, form the basis of modern mathematics. Additionally, while there are axiom systems for logic that are both correct and complete, such as those referenced in Lou Van den Dries' lecture notes, their completeness remains uncertain.

PREREQUISITES
  • Understanding of Zermelo-Fraenkel axioms and Axiom of Choice (ZFC)
  • Familiarity with Peano's axioms of arithmetic
  • Basic knowledge of formal logic systems
  • Ability to read DVI-files for accessing academic notes
NEXT STEPS
  • Research the Zermelo-Fraenkel axioms and their implications in set theory
  • Study Kurt Gödel's incompleteness theorems and their impact on axiomatic systems
  • Explore Lou Van den Dries' lecture notes on logic for detailed axiom systems
  • Investigate the differences between complete and consistent axiom systems in logic
USEFUL FOR

Mathematicians, logicians, philosophy students, and anyone interested in the foundations of mathematical logic and axiomatic systems.

Atran
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Hi, I've been recently reading about logic. Is there a list of the exact logical axioms underlying all axiomatic systems, postulates and mathematics?

Thanks...
 
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Atran said:
Hi, I've been recently reading about logic. Is there a list of the exact logical axioms underlying all axiomatic systems, postulates and mathematics?

Thanks...

The short answer is no. Russell and Whitehead tried to create (or discover) a formal logic for all of mathematics and essentially failed. The formalization of modern mathematics is based to a large extent on the Zermelo-Fraenkel axioms of Set Theory together with the Axiom of Choice (ZFC) and Peano's axioms of arithmetic.

http://mathworld.wolfram.com/Zermelo-FraenkelAxioms.html
 
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SW VandeCarr is certainly correct that there is no axiom system which is both correct and complete. This was proven by Kurt Godel in the 1930's. The mathematicians proceeded by just making up axiom system which looked correct and the started using them. They didn't care whether they are complete or not.

If you talk about logic, then the situation is quite different. There are axiom systems for logic which are both correct and complete. A good reference for this are the lecture notes of Lou Van den Dries: http://www.math.uiuc.edu/~vddries/ click on "Logic Notes" (you will need to be able to open DVI-files for this).
An axiom system for logic is described on page 38. But I don't think that it is known to be correct and complete...
 

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