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In ZF, the axiom of infinity says that the set of natural numbers exists. I was wondering if there was a (finitist?) weakening of ZF that included the axiom "the class of natural numbers exists".
ZF Axiom of Infinity: Natural Numbers Exist
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SUMMARY
The Zermelo-Fraenkel (ZF) set theory includes the axiom of infinity, which asserts the existence of the set of natural numbers. The discussion explores the possibility of a finitary version of ZF that would include the statement "the class of natural numbers exists." However, it concludes that this notion is redundant, as the existence of natural numbers is inherently derived from the axiom of infinity within ZF.
PREREQUISITES- Understanding of Zermelo-Fraenkel set theory
- Familiarity with axiomatic systems in mathematics
- Knowledge of the concept of classes in set theory
- Basic grasp of the axiom of infinity
- Research the implications of the axiom of infinity in ZF set theory
- Explore finitary approaches to set theory and their limitations
- Study the definition and properties of classes in set theory
- Investigate alternative axiomatic systems that address natural numbers
Mathematicians, logicians, and students of set theory interested in the foundations of mathematics and the properties of natural numbers within axiomatic frameworks.
- #2
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Wait, this is a dumb question, isn't it. This actually follows from a suitable definition of classes and ZF - the axiom of infinity.
Sorry.
Sorry.
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