- #31
gufiguer
- 15
- 2
Here is a philosophical approach (a formal approach is a little different)
For sets you can read Jech, Halmos and Enderton. You can focus on ordinals.
All textbooks on analysis that I know don't pay much attention on using the axiom of choice and the recursion theorem on ordinals. I feel sad about this.
The Peano axioms is an example of a inductive system. You can found the Peano axioms in Landau, but the inductive systems in Mendelson is better.
Here is interesting you define some algebraic structures like groups, rings, fields, modules and vector spaces and then work on some properties, without using more than set theory, the ordinals and what you can define and prove just with the inductive system you have. The set of ordinals N is a good approach for the natural numbers of our intuition.
Constructing a model for the integers is just standard and can be assisted by a book on elementary theory of numbers or elementary algebra. Then you have a set Z that works like the integers of our intuition (the pure idea of negative numbers) and you can see some properties of this set and that it is a ring (an integral domain) and you can construct his field of fractions (every integral domain has one). This field of fractions is the set Q and is a good model for the rational numbers of our intuition (the fractions) and you can work some properties and define it's absolute value function.
Then you can proceed analogous for a "completion for a metric space", but this expression is not rigorous because metric spaces are defined with the reals and you don't have the reals yet. But the construction is the same and can be performed to get a set R that corresponds well to the real numbers of our intuition (those numbers used to measure lenghts). An alternative is to construct a set R' via Dedekind cuts and verify that R and R' are isomorphic fields. Define properly the order and then verify that R are the unique ordered complete field up to order-preserving isomorphisms.
The construction of the real numbers can be made with pure geometry, using the Hilbert axioms for Euclidean geometry. You can use set theory and it is good for notation, but it is not needed.
Then construct the complex numbers C (not seen by our intuition) with the cartesian product of R with itself and embedding all N, Z, Q and R in C. Then work on properties of complex numbers. A lot of them.
Then define a topology for a set X and just work on topology and set theory for a period. Then define metric spaces and normed vector spaces and see a lot of properties, propositions, theorems etc.
Then, IMO, you can name it "Analysis", not before. :)
For sets you can read Jech, Halmos and Enderton. You can focus on ordinals.
All textbooks on analysis that I know don't pay much attention on using the axiom of choice and the recursion theorem on ordinals. I feel sad about this.
The Peano axioms is an example of a inductive system. You can found the Peano axioms in Landau, but the inductive systems in Mendelson is better.
Here is interesting you define some algebraic structures like groups, rings, fields, modules and vector spaces and then work on some properties, without using more than set theory, the ordinals and what you can define and prove just with the inductive system you have. The set of ordinals N is a good approach for the natural numbers of our intuition.
Constructing a model for the integers is just standard and can be assisted by a book on elementary theory of numbers or elementary algebra. Then you have a set Z that works like the integers of our intuition (the pure idea of negative numbers) and you can see some properties of this set and that it is a ring (an integral domain) and you can construct his field of fractions (every integral domain has one). This field of fractions is the set Q and is a good model for the rational numbers of our intuition (the fractions) and you can work some properties and define it's absolute value function.
Then you can proceed analogous for a "completion for a metric space", but this expression is not rigorous because metric spaces are defined with the reals and you don't have the reals yet. But the construction is the same and can be performed to get a set R that corresponds well to the real numbers of our intuition (those numbers used to measure lenghts). An alternative is to construct a set R' via Dedekind cuts and verify that R and R' are isomorphic fields. Define properly the order and then verify that R are the unique ordered complete field up to order-preserving isomorphisms.
The construction of the real numbers can be made with pure geometry, using the Hilbert axioms for Euclidean geometry. You can use set theory and it is good for notation, but it is not needed.
Then construct the complex numbers C (not seen by our intuition) with the cartesian product of R with itself and embedding all N, Z, Q and R in C. Then work on properties of complex numbers. A lot of them.
Then define a topology for a set X and just work on topology and set theory for a period. Then define metric spaces and normed vector spaces and see a lot of properties, propositions, theorems etc.
Then, IMO, you can name it "Analysis", not before. :)
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