More Rigorous Text than Rudin?

[bhagwad]

I was recommended Rudin's "Principles of Mathematical Analysis" as a text that assumes you know nothing and takes it from there. And make no mistake, I think it's an amazing book. I've learned techniques and overcome some hurdles all on my own that make me feel quite good about myself. So this is not a criticism of Rudin's work.

I just feel that it's not rigorous enough for my taste. While starting out, I tried to imagine that Rudin wasn't talking about "numbers" when using terms like "-x" and "0" etc, but this became impossible later on.

For example, while constructing real numbers from the rationals using Dedekind cuts, he suddenly talks about the "Archimedean" property of the set of rationals Q which he hasn't mentioned before. In fact, he seems to take the set of rationals for granted entirely without defining what Integers are, what natural numbers are, and indeed what "numbers" are in the first place.

He also doesn't define a "set" or how they're constructed. Now I know a lot of this since I've worked with analysis before, but for my own satisfaction I want to start with a completely blank slate. Basically assume that I have infinite intelligence (a rash proposition!) but know absolutely nothing.

Can anyone help me out with a text that starts from scratch...absolutely from nothing and then builds up to the construction of the real numbers? Like I said, I love Rudin and plan to continue studying the book but I also feel "incomplete" without having a rigorous understanding of some of the fundamentals that Rudin seems to take for granted.

Any suggestions would be greatly appreciated!

 

[UltrafastPED]

Try "Principia Mathematica" by Whitehead and Russel!
http://en.wikipedia.org/wiki/Principia_Mathematica

 

[bhagwad]

Try "Principia Mathematica" by Whitehead and Russel!
http://en.wikipedia.org/wiki/Principia_Mathematica

Correct me if I'm wrong, but isn't that considered somewhat dated by today's standards?

 

[UltrafastPED]

So? You asked for something that starts at the beginning ...

 

[jgens]

Can anyone help me out with a text that starts from scratch...absolutely from nothing and then builds up to the construction of the real numbers? Like I said, I love Rudin and plan to continue studying the book but I also feel "incomplete" without having a rigorous understanding of some of the fundamentals that Rudin seems to take for granted.

Honestly I think this request is seriously misguided. To save you some cash, however, there is a construction of all these things outlined below:

1. Take 0 = { }, 1 = {0}, 2 = {0,1}, ... , n = {0,...,n-1}, ... and this gives us our set of natural numbers. Order these guys by set membership and it is easily checked that this set is well-ordered. Defining addition is easy and we say that n+m is the unique natural number order isomorphic to the disjoint union n ⊔ m when endowed with the obvious order. Multiplication can now be handled recursively as repeated addition. By the way everything has been defined we get commutativity and associativity and distributivity automatically so we have constructed N.
2. To get the integers start with the cartesian product NxN and define an equivalence class by taking (n₁,m₁) ~ (n₂,m₂) if and only if n₁+m₂ = n₂+m₁. It is easy to show that every equivalence class has a representative of the form (n,0) or (0,n). So take Z to be these equivalence classes and let the positive integers be those with representative (n,0) and the negative integers be those with representative (0,n). Addition and multiplication and the ordering are easy enough to define now since these properties are inherited from the natural numbers.
3. To get the rationals start with the cartesian product Zx(Z-{0}) and define an equivalence relation where (n₁,m₁) ~ (n₂,m₂) if and only if n₁m₂ = n₂m₁. We let Q the equivalence classes of this relation and again the addition and multiplication and ordering are inherited from Z in a natural way.

If you are honestly interested in these constructions, then can fill in the missing details above. They are all pretty trivial and unenlightening. If you want to understand these number systems, then the best way to do that is not by examining their construction, but by examining the consequences of the properties those number systems have.

 

[johnqwertyful]

Well, going from N to Z to Q is really an algebra thing. Analysis takes over when you go from Q to R (Cauchy completeness/Dedekind cuts). The construction of N, Z, Q are not that difficult. They are fairly straightforward.

Natural numbers are well, natural. There's not much to talk about. You can talk about defining a successor function. Google knows that. Going from N to Z is just adding in additive inverses (an algebraic thing). Going from Z to Q is just adding in multiplicative inverses (another algebraic thing).

A book on analysis wouldn't cover algebra, as it is not an algebra text. What you're looking for is probably some more algebra. I recommend Dummit and Foote.

 

[johnqwertyful]

Also "start from scratch". I've thought about books like that. But to truly start from scratch, it's pure logic and philosophy. It would take countless volumes of books to even begin to talk about sets or any math at all. I don't even know if there is a "beginning". Because you can get into logic/philosophy and just keep going back.

Basically every book has to start from SOMEWHERE. Every book has to assume some knowledge, some axioms, some something. Not to mention there's controversy over certain philosophical questions surrounding math. Different approaches, etc.

 

[UltrafastPED]

Also "start from scratch". I've thought about books like that. But to truly start from scratch, it's pure logic and philosophy. It would take countless volumes of books to even begin to talk about sets or any math at all. I don't even know if there is a "beginning". Because you can get into logic/philosophy and just keep going back.

That's why I suggested "Principia Mathematica"!

 

[R136a1]

That's why I suggested "Principia Mathematica"!

It's a horribly outdated book. I wouldn't recommend anybody to read it. There are much better books out there which don't make things as complicated, and which do things the modern way.

I honestly don't know why people even remember the principia. Its relevance and influence on modern math is close to zero.

Anyway, if you want to start at the beginning, but if you still want a rather comfortable book, get Hrbacek and Jech's book on set theory: https://www.amazon.com/dp/0824779150/?tag=pfamazon01-20

It's not a book on analysis though. For analysis, I doubt there are books which do books as deeply as you want, but check "foundations of real and abstract analysis" by Bridges, or "Real Analysis" by McShane and Botts.

 

[bhagwad]

It's a horribly outdated book. I wouldn't recommend anybody to read it. There are much better books out there which don't make things as complicated, and which do things the modern way.

I honestly don't know why people even remember the principia. Its relevance and influence on modern math is close to zero.

Anyway, if you want to start at the beginning, but if you still want a rather comfortable book, get Hrbacek and Jech's book on set theory: https://www.amazon.com/dp/0824779150/?tag=pfamazon01-20

It's not a book on analysis though. For analysis, I doubt there are books which do books as deeply as you want, but check "foundations of real and abstract analysis" by Bridges, or "Real Analysis" by McShane and Botts.

Thank you. Yours is the first actual suggestion :). I appreciate it.

 

[bhagwad]

Honestly I think this request is seriously misguided. To save you some cash, however, there is a construction of all these things outlined below:

1. Take 0 = { }, 1 = {0}, 2 = {0,1}, ... , n = {0,...,n-1}, ... and this gives us our set of natural numbers. Order these guys by set membership and it is easily checked that this set is well-ordered. Defining addition is easy and we say that n+m is the unique natural number order isomorphic to the disjoint union n ⊔ m when endowed with the obvious order. Multiplication can now be handled recursively as repeated addition. By the way everything has been defined we get commutativity and associativity and distributivity automatically so we have constructed N.
2. To get the integers start with the cartesian product NxN and define an equivalence class by taking (n₁,m₁) ~ (n₂,m₂) if and only if n₁+m₂ = n₂+m₁. It is easy to show that every equivalence class has a representative of the form (n,0) or (0,n). So take Z to be these equivalence classes and let the positive integers be those with representative (n,0) and the negative integers be those with representative (0,n). Addition and multiplication and the ordering are easy enough to define now since these properties are inherited from the natural numbers.
3. To get the rationals start with the cartesian product Zx(Z-{0}) and define an equivalence relation where (n₁,m₁) ~ (n₂,m₂) if and only if n₁m₂ = n₂m₁. We let Q the equivalence classes of this relation and again the addition and multiplication and ordering are inherited from Z in a natural way.

If you are honestly interested in these constructions, then can fill in the missing details above. They are all pretty trivial and unenlightening. If you want to understand these number systems, then the best way to do that is not by examining their construction, but by examining the consequences of the properties those number systems have.

There's some good stuff here, but there's a lot I don't understand. What is "well ordered", what is "isomorphic", what is an "equivalence class"? So far, Rudin hasn't addressed these concepts in his book and I don't know if he will do so later on. Besides, since he already assumes we know about natural numbers it wouldn't be of much use anyway.

 

[Jorriss]

There's some good stuff here, but there's a lot I don't understand. What is "well ordered", what is "isomorphic", what is an "equivalence class"? So far, Rudin hasn't addressed these concepts in his book and I don't know if he will do so later on. Besides, since he already assumes we know about natural numbers it wouldn't be of much use anyway.

If you don't know those then the issue is not a lack of rigor by Rudin but that you do not have sufficient background for the book. R136 gave a good suggestion with Jech.

You may as also be interested in an algebra book. Virtually any undergraduate level algebra book would cover the essentials of set theory that one uses in analysis. Further, an isomorphism is an algebraic concept (in this context).

 

[bhagwad]

If you don't know those then the issue is not a lack of rigor by Rudin but that you do not have sufficient background for the book. R136 gave a good suggestion with Jech.

You may as also be interested in an algebra book. Virtually any undergraduate level algebra book would cover the essentials of set theory that one uses in analysis. Further, an isomorphism is an algebraic concept (in this context).

I'm not sure about the background part. In the preface, he says "This book is intended to serve as a text for the course in analysis that is usually taken by advanced undergraduates or by first year students who study mathematics". It would seem to me to be the starting point since I completed my undergraduation with a course in Math.

Besides, it's not as if I don't understand what "Archimedean" is for example. It's just that I haven't seen those concepts proved formally and that was what I was looking for. Are you sure that everyone who reads Rudin knows about Peano's axioms for example? In short does everyone who studies Rudin know everything about the number theory of natural numbers?

 

[WannabeNewton]

A typical freshman year honors calculus class would have covered all of that (save for detailed properties of the natural/real numbers-something like the Archimedean property would have definitely been covered as it is a very elementary result) and real analysis a la Rudin would have been the next step. It's a waste of time to go through things like the construction of the naturals in a real analysis book when such things are done in texts on more foundational aspects of math (i.e. set theory). It would be much more pragmatic to take such constructions for granted and focus more on something like the construction of the reals, via for example the Cantor construction, because this is actually pertinent to real analysis.

 

[AlephZero]

Are you sure that everyone who reads Rudin knows about Peano's axioms for example? In short does everyone who studies Rudin know everything about the number theory of natural numbers?

That doesn't really matter. The only practical way to learn math is to start somewhere in the middle, not at "the beginning". You started learning the useful properties of integers and rational numbers right back in elementary school, before you could even define what a number was (except for vague ideas like "it's something you use to count things").

If you want to digress from real analysis into investigating the completeness and consistency of various axiom systems for mathematics, that's fine, just so long as you remember that isn't what real analysis is about.

IMO the main problem with Russell and Whitehead is not so much that it "starts at the beginning", but that R&W were trying to figure out where the beginning actually was. With the benefit of almost 100 years of hindsight, it's possible to write a more coherent (and shorter) version. Reading the introductions to the sections is an interesting eyewitness account of history being made, though.

There are better ways to learn calculus than to first learn Latin (or German) and then read the first editions of Newton and Liebniz. The same applies to R&W.

 

[WannabeNewton]

^Indeed, well put! It's like going through a differential topology text and expecting the author to develop the entire theory of integrals starting from measure spaces and working down to the Riemann integral before delving into integration of differential forms.

 

[bhagwad]

That doesn't really matter. The only practical way to learn math is to start somewhere in the middle, not at "the beginning". You started learning the useful properties of integers and rational numbers right back in elementary school, before you could even define what a number was (except for vague ideas like "it's something you use to count things").

If you want to digress from real analysis into investigating the completeness and consistency of various axiom systems for mathematics, that's fine, just so long as you remember that isn't what real analysis is about.

IMO the main problem with Russell and Whitehead is not so much that it "starts at the beginning", but that R&W were trying to figure out where the beginning actually was. With the benefit of almost 100 years of hindsight, it's possible to write a more coherent (and shorter) version. Reading the introductions to the sections is an interesting eyewitness account of history being made, though.

There are better ways to learn calculus than to first learn Latin (or German) and then read the first editions of Newton and Liebniz. The same applies to R&W.

Thank you - that was a very insightful comment. Perhaps what I'm really looking for after all is not real analysis after all, but the study of the foundations of mathematics. I just picked up Rudin because it was recommended to me as a text that assumes you know nothing, so I was speaking from that perspective.

 

[jgens]

Are you sure that everyone who reads Rudin knows about Peano's axioms for example?

I certainly knew nothing about PA when I first read Rudin, but that hardly impacted my ability to understand what was going on in the book. The book provides a set of properties that characterize the real numbers uniquely up to isomorphism and this is all you need.

In short does everyone who studies Rudin know everything about the number theory of natural numbers?

No one knows close to everything about the theory of natural numbers. There are still questions about N that are active topics of research. The one thing I can guarantee you though, is that understanding properties of N is not going to come from understanding one particular construction of it.

 

[atyy]

There is no beginning. An analogy from Cori and Pelletier's "Mathenmatical Logic" I like is that it's a spiral. You always begin on the Nth level and there are levels above and levels below.

 

[mathnerd15]

is the Hubbard Analysis a good text? Rudin proves everything in the slickest way possible, which maybe matters to some people

 

[bhagwad]

If you don't know those then the issue is not a lack of rigor by Rudin but that you do not have sufficient background for the book. R136 gave a good suggestion with Jech.

You may as also be interested in an algebra book. Virtually any undergraduate level algebra book would cover the essentials of set theory that one uses in analysis. Further, an isomorphism is an algebraic concept (in this context).

I may also add that in Rudin's text on P.21 in Step 9 of the derivation of real numbers from the rationals, he defines isomorphism by saying: "This fact may be expressed by saying that the ordered field Q is isomorphic to the ordered field Q* whose elements are the rational cuts.

Given this, it's quite clear that Rudin does not assume that the student is familiar with isomorphism etc and so it's clearly wrong to claim that those who don't know it do not have sufficient background for Rudin.

 

[jgens]

I did not mean that kind of isomorphism in my post. The relevant type of isomorphism when talking about order-type is order-preserving bijection.

 

[Jorriss]

Given this, it's quite clear that Rudin does not assume that the student is familiar with isomorphism etc and so it's clearly wrong to claim that those who don't know it do not have sufficient background for Rudin.

I don't understand how that shows what you are saying. If he didn't define an isomorphism earlier, then the fact he is using it there shows he likely does assume the reader is familiar with the concept.

Given this, it's quite clear that Rudin does not assume that the student is familiar with isomorphism etc and so it's clearly wrong to claim that those who don't know it do not have sufficient background for Rudin.

You're right though, if one doesn't know what an isomorphism is that can be easily remedied by simply looking it up and moving on - but my point was that missing prerequisites are not a lack of rigor on behalf of the author. Though, you did mention not knowing what an equivalence class is and that does make me question whether your background is sufficient.

 

[bhagwad]

I don't understand how that shows what you are saying. If he didn't define an isomorphism earlier, then the fact he is using it there shows he likely does assume the reader is familiar with the concept.


You're right though, if one doesn't know what an isomorphism is that can be easily remedied by simply looking it up and moving on - but my point was that missing prerequisites are not a lack of rigor on behalf of the author. Though, you did mention not knowing what an equivalence class is and that does make me question whether your background is sufficient.

So far I haven't seen anything I couldn't quickly look up - no mention of equivalence classes so far and I've just finished with the construction of real numbers using Dedekind cuts. My only worry was that even though I can look stuff up, my knowledge may not be absolutely rigorous and I might be fooled into thinking I know something when I really don't.

 

[bhagwad]

For those interested, I've found what seems to be a pretty awesome book which Rudin himself recommends at the end of the construction of real numbers. It's called Foundations of Analysis by Edmund Landau - The English translation of Grundlagen der Analysis.

Seems to be just what I'm looking for. Thanks for all the answers guys. I appreciate it!

 

[jgens]

My only worry was that even though I can look stuff up, my knowledge may not be absolutely rigorous and I might be fooled into thinking I know something when I really don't.

I can guarantee you that this problem will not be fixed by studying constructions of the various number systems. These sorts of constructions are the kind of things you see once and then never think of again. The list of properties uniquely characterizing these number systems is way more important than what those numbers actually are.

 

[Jorriss]

My only worry was that even though I can look stuff up, my knowledge may not be absolutely rigorous and I might be fooled into thinking I know something when I really don't.

Based on what you've said I still think a few chapters of an abstract algebra book or of Jech's set theory book (the one R136 recommended) would be good options.

BTW this is a bit off topic but in my idea math is about ideas more than anything and Rudin is, imo, a very poor textbook for a first serious exposure to real analysis because he motivates so little (Also this won't mean much now but if you stick with Rudin, his last two chapters are terrible). I think you might enjoy Pugh's analysis text more.

 

[bhagwad]

Based on what you've said I still think a few chapters of an abstract algebra book or of Jech's set theory book (the one R136 recommended) would be good options.

BTW this is a bit off topic but in my idea math is about ideas more than anything and Rudin is, imo, a very poor textbook for a first serious exposure to real analysis because he motivates so little (Also this won't mean much now but if you stick with Rudin, his last two chapters are terrible). I think you might enjoy Pugh's analysis text more.

What a coincidence - I was just checking out Pugh's book :)

 

[mathwonk]

Classical standard references for your question are: J.L. Kelley, General Topology, Appendix, for axiomatic set theory and construction of the integers, and E. Landau, Foundations of Analysis, for construction of the real numbers from the integers.


As a matter of language, to me Rudin is "rigorous", in that his development is logically sound and he proves completely what he is trying to prove; he just starts from a more advanced set of assumptions than you wish to do. Everyone should start wherever he finds it appealing to start.

Not everyone has an appetite for the kind of foundations you are asking about, but if you do, then I think you may enjoy the treatments I suggested. Landau is a bit dry for my taste and writes in a brutal "satz-beweis" style ("theorem-proof"), similar to Rudin of course..

 

[bhagwad]

Classical standard references for your question are: J.L. Kelley, General Topology, Appendix, for axiomatic set theory and construction of the integers, and E. Landau, Foundations of Analysis, for construction of the real numbers from the integers.


As a matter of language, to me Rudin is "rigorous", in that his development is logically sound and he proves completely what he is trying to prove; he just starts from a more advanced set of assumptions than you wish to do. Everyone should start wherever he finds it appealing to start.

Not everyone has an appetite for the kind of foundations you are asking about, but if you do, then I think you may enjoy the treatments I suggested. Landau is a bit dry for my taste and writes in a brutal "satz-beweis" style ("theorem-proof"), similar to Rudin of course..

Thank you very much! I've been reading up Landau myself now.

 

[gufiguer]

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. :)