What's your favorite axiomatization of the real numbers?
The usual way to do it seems to be to view them as a totally ordered field that is Dedekin-Complete, or as Cauchy Sequences of rational numbers.
Personally I like Tarski's axiomatization of the real numbers.
1. Symmetry
If x < y, then not y < x
2. Denseness
If x < z, there exists a y such that x < y and y < z
3. Separation
For all X, Y ? R, if for all x ? X and y ? Y, x < y, then there exists a z such that
for all x ? X and y ? Y, if z ? x and z ? y, then x < z and z < y.
Or in plain language:
"If a set of reals precedes another set of reals, then there exists at least one real number separating the two sets."
4. Associativity
x + (y + z) = (x + z) + y.
5. Closed under addition
For all x, y, there exists a z such that x + z = y.
6. Exclusion
If x + y < z + w, then x < z or y < w.
Axioms for one (primitives: R, <, +, 1):
7. Non-emptyness
1 ? R.
8. Property of 1
1 < 1 + 1.
There we go. With just 8 axioms we defined the real numbers. The standard way of doing it takes 14 different axioms, and they're nested into different groups for fields, order etc. And here, we didn't even have to define multiplication. The fact that two sets of numbers, where all elements in one collection is larger than all in the other, have a number between them also seems more intuitive than the Dedekin-Completeness of saying "all nonempty sets that are bounded have a least upper bound."
What do you think? Is this axiomatization underestimated?
If you have a definition for the natural numbers N, and allow set operations, then I also like that the real numbers can just be seen as the set of all subsets of N.
The usual way to do it seems to be to view them as a totally ordered field that is Dedekin-Complete, or as Cauchy Sequences of rational numbers.
Personally I like Tarski's axiomatization of the real numbers.
1. Symmetry
If x < y, then not y < x
2. Denseness
If x < z, there exists a y such that x < y and y < z
3. Separation
For all X, Y ? R, if for all x ? X and y ? Y, x < y, then there exists a z such that
for all x ? X and y ? Y, if z ? x and z ? y, then x < z and z < y.
Or in plain language:
"If a set of reals precedes another set of reals, then there exists at least one real number separating the two sets."
4. Associativity
x + (y + z) = (x + z) + y.
5. Closed under addition
For all x, y, there exists a z such that x + z = y.
6. Exclusion
If x + y < z + w, then x < z or y < w.
Axioms for one (primitives: R, <, +, 1):
7. Non-emptyness
1 ? R.
8. Property of 1
1 < 1 + 1.
There we go. With just 8 axioms we defined the real numbers. The standard way of doing it takes 14 different axioms, and they're nested into different groups for fields, order etc. And here, we didn't even have to define multiplication. The fact that two sets of numbers, where all elements in one collection is larger than all in the other, have a number between them also seems more intuitive than the Dedekin-Completeness of saying "all nonempty sets that are bounded have a least upper bound."
What do you think? Is this axiomatization underestimated?
If you have a definition for the natural numbers N, and allow set operations, then I also like that the real numbers can just be seen as the set of all subsets of N.
