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> ZFC is actually the standard set of axioms mathematicians work with every day I'd dispute that mathematicians work with ZFC every day. Most pure but not foun
by rocqua 10y ago
> ZFC is actually the standard set of axioms mathematicians work with every day
I'd dispute that mathematicians work with ZFC every day.
Most pure but not foundational mathematics basically works with intuitionist set theory. By keeping the size and nesting of sets small enough, we never need to worry about ZFC.
Zorn's lemma comes up occasionally, but that is all.
- logicallee 10y agoWikipedia says "Today ZFC is the standard form of axiomatic set theory and as such is the most common foundation of mathematics." What my parenthetical comment that you quoted meant to say (for anyone that was not familiar with it) is thaz ZFC is not some joke, or obscure set of axioms or something irrelevant. Without this remark I thought my comment read that way. I hope you will agree that ZFC is simply "standard math" - even if its axioms are not referenced explicitly by many working mathematicians. If this statement requires further refinement let me know. Obviously I'm not a mathematician!
- rocqua 10y agoCertainly, ZFC is not a pathological constructed example. When a mathematician looks for a foundational set of axioms, ZFC is THE standard choice. It is, and has been, for the last ~90 years at least. My point was that foundational mathematics is rarely touched upon by a lot of normal pure mathematics (say number theory, field extension, graph theory). Interestingly, I believe a lot of people actually dislike the axiom of choice. They find it to be way to 'complex' when compared to the other axioms. It is a lot like euler's 5-th axiom (2 straight lines with the same direction are equidistant everywhere). Interestingly in that case, removing the axiom led to spherical and hyperbolic geometry.