Gödel's incompleteness wrt weakend versions of ZFC

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This discussion centers on the implications of removing the axiom of infinity from Zermelo-Fraenkel set theory with the Axiom of Choice (ZFC). It concludes that ZFC minus certain axioms, specifically the axiom of infinity, leads to a theory that lacks the capacity to express elementary arithmetic, aligning closely with Peano Arithmetic. The participants agree that any subtheory of an incomplete theory remains incomplete, affirming Gödel's incompleteness theorems in this context.

PREREQUISITES
  • Understanding of Gödel's incompleteness theorems
  • Familiarity with Zermelo-Fraenkel set theory (ZFC)
  • Knowledge of Peano Arithmetic (PA)
  • Basic concepts of mathematical logic and axiomatic systems
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  • Research the implications of removing axioms in ZFC on mathematical completeness
  • Study the relationship between ZFC and Peano Arithmetic in detail
  • Explore the nuances of Gödel's first and second incompleteness theorems
  • Investigate alternative set theories that do not include the axiom of infinity
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Mathematicians, logicians, and students of mathematical foundations who are interested in the implications of Gödel's theorems and the structure of set theories.

phoenixthoth
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Suppose for the sake of argument that we look at ZFC with the axiom of infinity removed.

http://en.wikipedia.org/wiki/Gödel's_incompleteness_theorems#First_incompleteness_theorem

http://en.wikipedia.org/wiki/Gödel's_incompleteness_theorems#Second_incompleteness_theorem

We would then be in a position where the hypotheses of Gödel's theorems are not satisfied, correct? Basically, I want to remove, for the sake of argument, a minimal amount of axioms of ZFC so that ZFC minus some axiom(s) leads to a theory that does not include arithmetical truths and is not capable of expressing elementary arithmetic.

Is it possible that ZFC- (my shorthand for ZFC with some axiom(s) removed) is consistent and complete?
 
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No, obviously any subtheory of an incomplete theory is also incomplete.

Also, ZFC without the axiom of infinity is basically Peano Arithmetic (or rather, replacing the Axiom of infinity by its negation yields a theory which is bi-interpretable with PA).
 

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