Theories in maths which do not use the axiom of choice.

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SUMMARY

Many mathematical theories operate without the Axiom of Choice (AC), particularly in discrete mathematics, real analysis, and algebra. Discrete mathematics, which focuses on finite sets, inherently does not require AC. Real analysis also functions effectively without invoking AC, and most algebraic operations can be performed without it, especially when dealing with finite dimensional vector spaces. The discussion highlights that a significant portion of mathematics can be conducted without reliance on the Axiom of Choice.

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  • Understanding of discrete mathematics and finite sets
  • Familiarity with real analysis concepts
  • Basic knowledge of algebra and vector spaces
  • Awareness of set theory and axiomatic systems
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Mathematicians, logicians, educators, and students interested in foundational mathematics and the implications of the Axiom of Choice in various mathematical theories.

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i read that there are some logicians who do not use the axiom of choice in their axioms systems. i wonder what is the math that isn't using the axiom of choice, or what theories do not use it?
 
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A lot of theories do not use the axiom of choice. Pretty much any discrete mathematics (i.e. with finite sets only) won't use it. Most of real analysis has no need to invoke the axiom of choice. Most algebra gets by without needing the axiom of choice to be invoked. You don't really ever need an algebraic closure of anything, for instance: start with a ground field, decide what polys you need to have roots, this will normally only be a finite number, and add in the roots accordingly. The vast majority of algebra only ever deals in finite dimensional vector spaces.

Incidentally, it is not just logicians who chose to use or ignore the axiom of choice specifically. Completely unjustified statistic alert: you can probably get by doing 99% of maths without ever needing to invoke the axiom of choice.
 
i should have been clearer, i meant theories of sets which do not use AC.
 

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