Is Infinity Essential in Set Theory's ZF Axioms?

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SUMMARY

The discussion centers on the relevance of infinity within the Zermelo-Fraenkel (ZF) axioms of set theory. Participants argue that while ZF axioms are foundational for understanding infinite sets, they may be perceived as redundant for those focused solely on finite sets. The conversation highlights the role of information quanta in modeling mathematical concepts and questions the necessity of ZF axioms in explaining infinity. Ultimately, the consensus suggests that ZF axioms are essential for rigorous exploration of infinite sets.

PREREQUISITES
  • Understanding of Zermelo-Fraenkel (ZF) axioms
  • Familiarity with set theory concepts, particularly infinity
  • Knowledge of information theory and information quanta
  • Basic grasp of mathematical modeling techniques
NEXT STEPS
  • Research the implications of the Axiom of Infinity in ZF set theory
  • Explore the historical development of set theory from Cantor to Cohen
  • Study the relationship between information theory and mathematical modeling
  • Investigate alternative set theories that focus on finite sets
USEFUL FOR

Mathematicians, theoretical computer scientists, and anyone interested in the foundations of set theory and the concept of infinity.

mustang19
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Obviously information exists. From the concept of information quanta we can create physical models of mathematical concepts. Are ZF axioms redundant?
 
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mustang19 said:
Obviously information exists. From the concept of information quanta we can create physical models of mathematical concepts. Are ZF axioms redundant?
In the end you can always define them as a certain magnetic configuration of a computer, preferably a TM, achieved by saving them in some editor. Good luck with handling them when you have to explain anything.
 
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mustang19 said:
Obviously information exists. From the concept of information quanta we can create physical models of mathematical concepts. Are ZF axioms redundant?
ZF stands for Zermelo-Fraenkel, right? How would you, for instance, from concept of information quanta arrive to the conclusion that infinity exists (axiom of infinity)?

By the way, the whole set theory (from Cantor to Zermelo and Fraenkel to Cohen) is devised with the main intention to rigorously understand infinity. For those who are only interested in finite sets there is no much use of abstract set theory and ZF axioms.
 
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