Of course, that is then called non-standard analysis — ssu
See, that's the problem here. I think math is filled with a lot of things that a) work b) are totally obvious at some level and c) to make a rigorous proof why they are is problematic. For example, just how many different fields of math can you find something similar to the Axiom of Choice? Just look how much it has created discussion in mathematical circles.Not a "simple" "intuitive" with "beautiful reasoning" in my opinion. — jgill
Who benefits from the History of Math or the Philosophy of Math? Not many I would say.A colleague of mine tried teaching the subject at the U of Colorado some years ago, and neither he nor his students benefited. — jgill
A colleague of mine tried teaching the subject at the U of Colorado some years ago, and neither he nor his students benefited. — jgill
Who benefits from the History of Math or the Philosophy of Math? Not many I would say.
Usually students aren't interested in the fascinating history of a debate in mathematics. — ssu
For example, just how many different fields of math can you find something similar to the Axiom of Choice? — ssu
Then there simply is no time for philosophy. You have to go through all the work done by mathematicians and get to the sophisticated ways mathematicians use them. It's simply a matter of time.It's not a math history course. It's a sophisticated real analysis course, including calculus, based upon a rigorous concept of infinitesimals. — jgill
There's a book by Herman Rubin and Jean E. Rubin called "Equivalents of the Axiom of Choice", which states about 150 statements in mathematics that are equivalent to the axiom of choice.I'm not up to speed in contemporary abstract math, particularly foundations, but I would guess few, if any. — jgill
Set theory
Well-ordering theorem: Every set can be well-ordered. Consequently, every cardinal has an initial ordinal.
Tarski's theorem about choice: For every infinite set A, there is a bijective map between the sets A and A×A.
Trichotomy: If two sets are given, then either they have the same cardinality, or one has a smaller cardinality than the other.
Given two non-empty sets, one has a surjection to the other.
The Cartesian product of any family of nonempty sets is nonempty.
König's theorem: Colloquially, the sum of a sequence of cardinals is strictly less than the product of a sequence of larger cardinals. (The reason for the term "colloquially" is that the sum or product of a "sequence" of cardinals cannot be defined without some aspect of the axiom of choice.)
Every surjective function has a right inverse.
Order theory
Zorn's lemma: Every non-empty partially ordered set in which every chain (i.e., totally ordered subset) has an upper bound contains at least one maximal element.
Hausdorff maximal principle: In any partially ordered set, every totally ordered subset is contained in a maximal totally ordered subset. The restricted principle "Every partially ordered set has a maximal totally ordered subset" is also equivalent to AC over ZF.
Tukey's lemma: Every non-empty collection of finite character has a maximal element with respect to inclusion.
Antichain principle: Every partially ordered set has a maximal antichain.
Abstract algebra
Every vector space has a basis.
Krull's theorem: Every unital ring other than the trivial ring contains a maximal ideal.
For every non-empty set S there is a binary operation defined on S that gives it a group structure. (A cancellative binary operation is enough, see group structure and the axiom of choice.)
Every set is a projective object in the category Set of sets.
Functional analysis
The closed unit ball of the dual of a normed vector space over the reals has an extreme point.
Point-set topology
Tychonoff's theorem: Every product of compact topological spaces is compact.
In the product topology, the closure of a product of subsets is equal to the product of the closures.
Mathematical logic
If S is a set of sentences of first-order logic and B is a consistent subset of S, then B is included in a set that is maximal among consistent subsets of S. The special case where S is the set of all first-order sentences in a given signature is weaker, equivalent to the Boolean prime ideal theorem; see the section "Weaker forms" below.
Graph theory
Every connected graph has a spanning tree.
If you call that few if any, well... — ssu
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