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Naive Set Theory
From the Reviews: "...He (the author) uses the language and notation of ordinary informal mathematics to state the basic set-theoretic facts which a beginning student of advanced mathematics needs to know...Because of the informal method of presentation, the book is eminently suited for use as a textbook or for self-study. The reader should derive from this volume a maximum of understanding of the theorems of set theory and of their basic importance in the study of mathematics." - "Philosophy and Phenomenological Research".
111 pages, Kindle Edition
First published January 1, 1960
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February 20, 2024
The field of set theory originated with the pioneering discoveries of Georg Cantor during the second half of the nineteenth century. Prior to his work, mathematicians were not accustomed to think very much in terms of collections of mathematical objects (particularly abstract ones); the most desirable result of an investigation was a formula in explicit analytical form. After Cantor and his contemporary Dedekind, who introduced infinite sets in an essential way into his arithmetization of the real number continuum, set theory gained in popularity and came, indeed, to be the dominant mode of expression of mathematical ideas during the course of the twentieth century (although it has its competitors today). Everyone knows about the crisis in the foundations of formal logic and mathematics that began in 1901 when Russell hit upon his eponymous paradox. Set theory came to be seen as riddled with paradoxes and potential inconsistencies. There is so much freedom in the possible ways by which to define sets (such as sets whose elements are again other sets) that wild phenomena can happen and one can never be entirely sure that everything one does is free of contradiction. Enormous labors were expended upon techniques by which to tame this riotous chaos, for instance, with a theory of strictly organized types (this reviewer does not pretend to have read the prodigious product of all this ferment, Russell and Whitehead’s three-volume Principia Mathematica published during the 1910’s).
In most ordinary mathematical practice, however, all this is irrelevant. For the purpose of everyday work, one settled upon a concise list of set-theoretic axioms proposed by Zermelo and Fraenkel, which include an axiom of comprehension which allows one to introduce infinite sets in a controlled enough manner. The Zermelo-Fraenkel axioms, possibly supplemented with the axiom choice or the continuum hypothesis (these are controversial among the experts!), are quite sufficient for the great bulk of contemporary mathematics, including analysis, geometry, topology, algebra and so forth. Only a small band of specialists in the foundations of logic much care about alternative schemes of axioms upon which to base set theory, or for foundational approaches that go outside the realm of traditional set theory such as homotopy type theory and higher category theory. Somehow, no one else using ZF has ever encountered any jarring contradiction in the real applications of mathematics, despite the paradoxes that are suspected to be lurking in the theory. Thus, the everyday practicing mathematician can be content to pursue his daily work of proving theorems using set theory more or less along the lines Cantor would have, without being too much concerned about the thorny foundational issues. Set theory as it was practiced in Cantor’s day now goes under the name of ‘naïve set theory’. The great Hilbert quipped in 1926, ‘Aus dem Paradies, das Cantor uns geschaffen, soll uns niemand vertreiben können’.
This much by way of preface to this review of Paul Halmos’ little textbook entitled Naïve Set Theory in the Springer undergraduate mathematics series. Clearly, here is where one has to start and first to gain experience, before graduating to the forbidding complexities of modern research into the foundations of formal logic and set theory. Halmos has tailored his textbook to cover what he judges every aspiring mathematician ought to know about basic set theory before specializing.
The treatment starts out very elementarily, with an introduction to the notation and basic concepts in terms of which professional mathematicians now view their subject: unions, intersections, powers and complements of sets; ordered pairs, functions as mappings of sets and relations as a certain subset; von Neumann’s approach to defining the natural numbers etc. The presentation of these ideas can be deceptive, because the reader might have no inkling that there was a time before Cantor when much of mathematical thought was conducted in rather different terms. It took centuries for the modern concept of a function to emerge, for instance. Indeed, Cantor was led to his discovery of set theory in the first place by his investigations of Fourier analysis. Dirichlet showed that the Fourier series converges for continuous functions, but just what larger classes of functions can be thus represented was, and still is, an open question. Cantor was interested in the study of points of discontinuity. This focused his attention on the collection [Menge] of points where the given function becomes discontinuous. Only over the course of investigations such as these did the mathematicians of the mid-to-late nineteenth century move from a concept of a function as a rule for assigning values to the function given a variable lying in its domain of definition (which would mean piecewise analytic, in modern terms) to the more general concept of an arbitrary mapping between sets (which of course comprehends analytic functions as a special case). Halmos says nothing about all this, though.
The second half of Halmos’ text gets into more advanced topics, such as Peano’s axioms, order relations, the axiom of choice, Zorn’s lemma, ordinal and cardinal arithmetic, the Burali-Forti paradox and the Schröder-Bernstein theorem. He concludes with remarks on limit numbers and the continuum hypothesis. To this reviewer, the nicest sections in this second half are the ones on transfinite recursion and the ladder of countable ordinals. Readers of this text will have seen proof by ordinary induction (over the natural numbers) in introductory analysis. The transfinite case is a little more involved and worth contemplating. Regrettably, Halmos’ concise format does not permit the inclusion of a complete derivation of any interesting non-trivial result with the aid of transfinite recursion.
In all, a fine text that accomplishes the limited objectives it sets out to achieve. What does not come out in the above discussion is Halmos’ inimitable style. Perhaps the primary value in a mathematical textbook lies in what the author chooses to say in order to motivate, delineate and explain the concepts he introduces. Most students would find the bare-bones style of definition-theorem-proof hard to take (which is why Bourbaki’s courses, though comprehensive and definitive, are not really a good resource for learning a subject the first time). Halmos strikes about the right balance between saying enough to help the reader follow the exposition and putting in too much, so that the text would become tedious (many authors fall into the trap of writing too much in the hope of keeping things approachable for beginners—for instance, Pugh in his text on real analysis; their efforts would be better spent on thinking more carefully about how to put what little would be sufficient to a reader who knows how to ponder a statement and is capable of thinking for himself to a certain degree. Walter Rudin is master at this kind of crafted exposition, which is why his textbooks enjoy their deserved reputation. What we have said is true, of course, of good English style in general; cf. Fowler in his Modern English Usage).
Every upcoming student of mathematics ought to be acquainted with naïve set theory. Simple things such as it treats are normally the most profound. For this reason, a text such as the present one by Halmos can mislead a reader who lacks that hard-to-define but essential quality known as mathematical maturity. Though labeled an undergraduate text, it really belongs at the threshold separating undergraduate from graduate-level mathematics. In principle, a college freshman could easily work his way through Halmos (at the formal level), but, without enough maturity, he would all but certainly miss out on most of its import. Halmos does a creditable job of conveying the basic facts of set-theoretic life which, once in one’s life at least one ought to have gone through a clear presentation of, even should one never recur to them later in the course of more advanced studies. This is all we have the right to ask at this level, but it is, in itself, quite a lot. Halmos is to be commended for his service.
In most ordinary mathematical practice, however, all this is irrelevant. For the purpose of everyday work, one settled upon a concise list of set-theoretic axioms proposed by Zermelo and Fraenkel, which include an axiom of comprehension which allows one to introduce infinite sets in a controlled enough manner. The Zermelo-Fraenkel axioms, possibly supplemented with the axiom choice or the continuum hypothesis (these are controversial among the experts!), are quite sufficient for the great bulk of contemporary mathematics, including analysis, geometry, topology, algebra and so forth. Only a small band of specialists in the foundations of logic much care about alternative schemes of axioms upon which to base set theory, or for foundational approaches that go outside the realm of traditional set theory such as homotopy type theory and higher category theory. Somehow, no one else using ZF has ever encountered any jarring contradiction in the real applications of mathematics, despite the paradoxes that are suspected to be lurking in the theory. Thus, the everyday practicing mathematician can be content to pursue his daily work of proving theorems using set theory more or less along the lines Cantor would have, without being too much concerned about the thorny foundational issues. Set theory as it was practiced in Cantor’s day now goes under the name of ‘naïve set theory’. The great Hilbert quipped in 1926, ‘Aus dem Paradies, das Cantor uns geschaffen, soll uns niemand vertreiben können’.
This much by way of preface to this review of Paul Halmos’ little textbook entitled Naïve Set Theory in the Springer undergraduate mathematics series. Clearly, here is where one has to start and first to gain experience, before graduating to the forbidding complexities of modern research into the foundations of formal logic and set theory. Halmos has tailored his textbook to cover what he judges every aspiring mathematician ought to know about basic set theory before specializing.
The treatment starts out very elementarily, with an introduction to the notation and basic concepts in terms of which professional mathematicians now view their subject: unions, intersections, powers and complements of sets; ordered pairs, functions as mappings of sets and relations as a certain subset; von Neumann’s approach to defining the natural numbers etc. The presentation of these ideas can be deceptive, because the reader might have no inkling that there was a time before Cantor when much of mathematical thought was conducted in rather different terms. It took centuries for the modern concept of a function to emerge, for instance. Indeed, Cantor was led to his discovery of set theory in the first place by his investigations of Fourier analysis. Dirichlet showed that the Fourier series converges for continuous functions, but just what larger classes of functions can be thus represented was, and still is, an open question. Cantor was interested in the study of points of discontinuity. This focused his attention on the collection [Menge] of points where the given function becomes discontinuous. Only over the course of investigations such as these did the mathematicians of the mid-to-late nineteenth century move from a concept of a function as a rule for assigning values to the function given a variable lying in its domain of definition (which would mean piecewise analytic, in modern terms) to the more general concept of an arbitrary mapping between sets (which of course comprehends analytic functions as a special case). Halmos says nothing about all this, though.
The second half of Halmos’ text gets into more advanced topics, such as Peano’s axioms, order relations, the axiom of choice, Zorn’s lemma, ordinal and cardinal arithmetic, the Burali-Forti paradox and the Schröder-Bernstein theorem. He concludes with remarks on limit numbers and the continuum hypothesis. To this reviewer, the nicest sections in this second half are the ones on transfinite recursion and the ladder of countable ordinals. Readers of this text will have seen proof by ordinary induction (over the natural numbers) in introductory analysis. The transfinite case is a little more involved and worth contemplating. Regrettably, Halmos’ concise format does not permit the inclusion of a complete derivation of any interesting non-trivial result with the aid of transfinite recursion.
In all, a fine text that accomplishes the limited objectives it sets out to achieve. What does not come out in the above discussion is Halmos’ inimitable style. Perhaps the primary value in a mathematical textbook lies in what the author chooses to say in order to motivate, delineate and explain the concepts he introduces. Most students would find the bare-bones style of definition-theorem-proof hard to take (which is why Bourbaki’s courses, though comprehensive and definitive, are not really a good resource for learning a subject the first time). Halmos strikes about the right balance between saying enough to help the reader follow the exposition and putting in too much, so that the text would become tedious (many authors fall into the trap of writing too much in the hope of keeping things approachable for beginners—for instance, Pugh in his text on real analysis; their efforts would be better spent on thinking more carefully about how to put what little would be sufficient to a reader who knows how to ponder a statement and is capable of thinking for himself to a certain degree. Walter Rudin is master at this kind of crafted exposition, which is why his textbooks enjoy their deserved reputation. What we have said is true, of course, of good English style in general; cf. Fowler in his Modern English Usage).
Every upcoming student of mathematics ought to be acquainted with naïve set theory. Simple things such as it treats are normally the most profound. For this reason, a text such as the present one by Halmos can mislead a reader who lacks that hard-to-define but essential quality known as mathematical maturity. Though labeled an undergraduate text, it really belongs at the threshold separating undergraduate from graduate-level mathematics. In principle, a college freshman could easily work his way through Halmos (at the formal level), but, without enough maturity, he would all but certainly miss out on most of its import. Halmos does a creditable job of conveying the basic facts of set-theoretic life which, once in one’s life at least one ought to have gone through a clear presentation of, even should one never recur to them later in the course of more advanced studies. This is all we have the right to ask at this level, but it is, in itself, quite a lot. Halmos is to be commended for his service.
June 30, 2020
I am of the opinion that anyone with an interest in mathematics should at some point in their career study some (mathematical) logic and some (formal) set theory (the appropriate amount of each may be difficult to determine). For the latter i highly recommend this book. Halmos guides the reader through each of the axioms of set theory, explaines the reasonings behind them, and their immediate consequences. One might need to look elsewhere for a more formal introduction, but even so this book will provide a great basis on which to expand one's knowledge of the subject.
May 10, 2017
تنها توصیفی که برای این کتاب میشه آورد اینه که این کتاب زیباست! هر صفتی بیشتر استفاده کنم احساس میکنم تخیل رو محدود کردم در توصیفش.
May 15, 2021
I got this book to understand how the theory of sets is developed and how one can define naturals, rationals, their arithmetic etc using only set theory and first-order logic! I just skimmed after the well-ordering chapter (includes ordinal and cardinal arithmetic) as I felt further efforts have diminishing returns.
This is perhaps the best exposition I could have asked for! I think I have a good understanding of how set-theoretical principles are used to develop the theory of relations, naturals, order and the theorems of induction, recursion etc. It helped to work through the examples and attempt the exercises. Took longer than expected but it was worth it. I probably can appreciate it better sometime in the future. But for now, I feel sated.
Maybe I will pick it up again sometime.
This is perhaps the best exposition I could have asked for! I think I have a good understanding of how set-theoretical principles are used to develop the theory of relations, naturals, order and the theorems of induction, recursion etc. It helped to work through the examples and attempt the exercises. Took longer than expected but it was worth it. I probably can appreciate it better sometime in the future. But for now, I feel sated.
Maybe I will pick it up again sometime.
October 11, 2021
I decided to read this book because it is a recommended reading on the guide to MIRI’s AI alignment research
. I've enjoyed reading the more pop-scientific books on AI like Human Compatible: Artificial Intelligence and the Problem of Control but wanted to get into the weeds and read technical papers.
This book made me truly appreciate the elegance and beauty of set theory. I studied discrete mathematics at university but I was too busy worrying about test scores to have the capacity for appreciation.
This book is short and simple and imperfect. It lists out all the axioms of set theory in a not-super-rigorous way in the span of ~100 pages.
My problem with this book is that Halmos is trying too hard to use English. English is a terrible language for describing something as subtle as set theory. Whenever I googled things to try and understand them more rigorously, I was sent down an endless rabbit hole because everything is much more subtle in the axiomatic set theory world (e.g. ZFC set theory) when compared to the naive set theory world. A lot of my time reading this book was spent trying to figure out how to interpret the English sentences that Halmos used to describe the axioms.
. I've enjoyed reading the more pop-scientific books on AI like Human Compatible: Artificial Intelligence and the Problem of Control but wanted to get into the weeds and read technical papers.
This book made me truly appreciate the elegance and beauty of set theory. I studied discrete mathematics at university but I was too busy worrying about test scores to have the capacity for appreciation.
This book is short and simple and imperfect. It lists out all the axioms of set theory in a not-super-rigorous way in the span of ~100 pages.
My problem with this book is that Halmos is trying too hard to use English. English is a terrible language for describing something as subtle as set theory. Whenever I googled things to try and understand them more rigorously, I was sent down an endless rabbit hole because everything is much more subtle in the axiomatic set theory world (e.g. ZFC set theory) when compared to the naive set theory world. A lot of my time reading this book was spent trying to figure out how to interpret the English sentences that Halmos used to describe the axioms.
May 27, 2016
Good introduction to set theory. Pretty terse in terms of proofs, leaving a lot of steps up to the reader, which I like. Each section is only about 2 pages, but manages to cover a good amount of intuition. Focuses largely on how set theory is used as a basis for modern mathematics, and in particular how to build up a structure with the properties of the natural numbers, including order. Set theory is vital to know in modern mathematics, but you almost certainly don't need the level of depth this book goes into for day-to-day work. Still, it is interesting and a fun read for the curious.
October 18, 2014
This text shows its age -- it's heavily wordy and pretty light on presenting things in mathematical notation. Although I have never formally studied set theory, I didn't get much out of it, though it did serve to reinforce my knowledge of some of the algebra behind sets.
July 20, 2019
Don't be fooled by the size of this book, it is quite dense and certainly not to be skimmed through. It is also the chattiest math textbook I have encountered, which is not always a good thing. Also, some of the proofs are kind of sketchy.
December 15, 2016
This is one of my favorite books, ever, even among nonmathematical books. Paul Halmos is often held up as a paragon of mathematical writing, and reading this book one can see why. It has a laid-back, and even humorous style, which makes it a great pleasure to read. The Axiom of Choice is introduced in the way that the founders of set theory first saw it, as a guarantee that the Cartesian Product of two nonempty sets is nonempty, and is developed into its modern form. The book ends simply with the statement of the Continuum Hypothesis.
This is one of the smallest books I have, at only 104 pages. Instead of chapters it has sections, each between three to five pages long, with 25 sections in total. While it may seem small, it can take a surprising amount of time to read it, due to the confusing nature of set theory itself. It has some intimidating exercises, which include proving the equivalence between The Axiom of Choice and The Well-Ordering Theorem.
For further reading, I would recommend Set Theory and the Continuum Hypothesis(which is available for under $10 online) by Paul Cohen, who won the Fields Medal for discovering the method of forcing.
This is one of the smallest books I have, at only 104 pages. Instead of chapters it has sections, each between three to five pages long, with 25 sections in total. While it may seem small, it can take a surprising amount of time to read it, due to the confusing nature of set theory itself. It has some intimidating exercises, which include proving the equivalence between The Axiom of Choice and The Well-Ordering Theorem.
For further reading, I would recommend Set Theory and the Continuum Hypothesis(which is available for under $10 online) by Paul Cohen, who won the Fields Medal for discovering the method of forcing.
June 27, 2019
Did not totally finish. In the final few chapters, the theories got a little too complex for my current mathematical abilities. Perhaps one day I shall revisit this and be able to understand the manipulate the content more.
However I walked away with the basics of set theory: sets, inclusion, specification, relations, functions, numbers, unions, intersections, powers. The foundations of mathematics. Bears an eerie resemblance to LISP.
The author's style was stylish, so to speak. Some of the contents definitely were covered in the dust of a professor, but there were occasional diversions into an area more casual. Sometimes the author would complain about the lack of consistency between mathematicians, like the logic reasoning for picking one operator symbol or expression over another.
One final note: the edition I read, the one with the pair of dice on a black background, DO NOT buy it. The printing and type setting looks like it was done by an intern. No margins on the pages, the front title is pixelated, each page has been printed at a slight rotation. Avoid it, spend the extra money to get the hard back copy or another paperback.
However I walked away with the basics of set theory: sets, inclusion, specification, relations, functions, numbers, unions, intersections, powers. The foundations of mathematics. Bears an eerie resemblance to LISP.
The author's style was stylish, so to speak. Some of the contents definitely were covered in the dust of a professor, but there were occasional diversions into an area more casual. Sometimes the author would complain about the lack of consistency between mathematicians, like the logic reasoning for picking one operator symbol or expression over another.
One final note: the edition I read, the one with the pair of dice on a black background, DO NOT buy it. The printing and type setting looks like it was done by an intern. No margins on the pages, the front title is pixelated, each page has been printed at a slight rotation. Avoid it, spend the extra money to get the hard back copy or another paperback.
March 16, 2018
I was going to give the book a 4 stars rating,but since in the book cover the book contain itself,it totally made it a hard 5.
December 10, 2022
Food for thought, an essential book for any Philosophically inclined Mathematician. The author not only exposes the basic tenets (and conflicts) of Set Theory, but does so in a clear and often entertaining way.
If anything, he seems to use the expression "one-to-one" to mean both an injective and a bijective mapping, which leads to confussion in some proofs.
If anything, he seems to use the expression "one-to-one" to mean both an injective and a bijective mapping, which leads to confussion in some proofs.
June 4, 2017
This book is brilliant. Simply brilliant. It is so much more than a math textbook. It is a glimpse of how mathematics, and mathematicians, work.
The operative word in the title is “Naïve”. As the author explains in the introduction, it means that he takes a somewhat informal approach to axioms and proofs, but as also stated in the introduction, the book is axiomatic in that he does state axioms and use them in subsequent proofs. The way it is unlike other formal axiomatic books is that axioms and proofs are not simply stated distilled down to their final concise and often incomprehensible form. Instead, Halmos lets you in on the motivation of why things proceed the way they do, even at the expense of formality.
For example, he spends the second last chapter giving you the rules of cardinal number arithmetic before even defining them in the last chapter – that comes in the last chapter, but not before he explains why we chose that definition among other alternatives.
These motivating passages are actually less frequent than I would have liked, but they do enough to motivate not only specific definitions, but to motivate what the axiomatic set theory approach is all about. What I got out of it is that set theorists aim to find definitions of intuitive, self-evident concepts using the bare minimum of new constructions. I’ll let the author explain on page 25:
The reason this is done, as is hinted to later in the book in the chapter on the axiom of choice is that mathematicians want to know if the existing set of axioms force an obvious conclusion, or whether a fact that seems self-evident can be dropped and the remaining axioms remain consistent and permits a more general mathematical system.
This book is by no means easy, but the author’s tone is just relaxed enough to relieve some of the intimidation that comes with studying a formal mathematics textbook. He can even be quite humorous at times – I laughed out loud reading a passage on page 45:
How many math books are this much fun?
The operative word in the title is “Naïve”. As the author explains in the introduction, it means that he takes a somewhat informal approach to axioms and proofs, but as also stated in the introduction, the book is axiomatic in that he does state axioms and use them in subsequent proofs. The way it is unlike other formal axiomatic books is that axioms and proofs are not simply stated distilled down to their final concise and often incomprehensible form. Instead, Halmos lets you in on the motivation of why things proceed the way they do, even at the expense of formality.
For example, he spends the second last chapter giving you the rules of cardinal number arithmetic before even defining them in the last chapter – that comes in the last chapter, but not before he explains why we chose that definition among other alternatives.
These motivating passages are actually less frequent than I would have liked, but they do enough to motivate not only specific definitions, but to motivate what the axiomatic set theory approach is all about. What I got out of it is that set theorists aim to find definitions of intuitive, self-evident concepts using the bare minimum of new constructions. I’ll let the author explain on page 25:
“The concept of an ordered pair could have been introduced as an additional primitive, axiomatically endowed with just the right properties, no more and no less. In some theories this is done. The mathematician’s choice is between having to remember a few more axioms and having to forget a few accidental facts; the choice is pretty clearly a matter of taste. Similar choices occur frequently in mathematics…”
The reason this is done, as is hinted to later in the book in the chapter on the axiom of choice is that mathematicians want to know if the existing set of axioms force an obvious conclusion, or whether a fact that seems self-evident can be dropped and the remaining axioms remain consistent and permits a more general mathematical system.
This book is by no means easy, but the author’s tone is just relaxed enough to relieve some of the intimidation that comes with studying a formal mathematics textbook. He can even be quite humorous at times – I laughed out loud reading a passage on page 45:
“The slight feeling of discomfort that the reader may experience in connection with the definition of natural numbers is quite common and in most cases temporary."
How many math books are this much fun?
July 24, 2020
The book is a quick walk through the fundamentals of Set Theory. It's comprehensive and it was written with the idea to cover as much as possible in 25 sections of approximately 4 pages each. I suppose that because of this, some explanations are too abstract, at least for me which I'm trying to rediscover math after 3-4 years of inactivity. If you haven't been very good at math or you are not working currently with math concepts, then I would not recommend this book as a quick reminder about Set Theory.
May 30, 2021
This book helped me immensely with improving my mathematical foundations. I read this just before my real analysis and abstract algebra classes and it helped me a lot with structuring my proofs and understanding of mathematical arguments.
Even for non mathematicians, the first few chapters are a fantastic way to commence your exploration of logic and the base of mathematics!
Even for non mathematicians, the first few chapters are a fantastic way to commence your exploration of logic and the base of mathematics!
July 1, 2021
Nice and short introduction to Set theory. It does not have enough visual content, intuitive examples and good exercises. For all of that I was using https://www.math24.net/topics-set-theory
Anyway, the book was written by mathematician. And it feels like you speak with mathematician and somehow touch philosophy of math.
Anyway, the book was written by mathematician. And it feels like you speak with mathematician and somehow touch philosophy of math.
January 23, 2021
An excellent introduction for someone just coming to the ideas of set theory. Conversational and light hearted, it makes learning of the basics of set theory a joy. A good first math book to read.
April 3, 2021
A great comprehensive text book
September 21, 2022
Pretty good exposition. I feel like the level of complexity was very inconsistent though and made for an at times annoying read. Some of the notation is a bit dated.
January 13, 2023
Acabei chegando no livro "Naive set theory" por indicação do canal "A ciência da estatística" e foi uma grata surpresa descobrir que havia essa tradução do livro. Comprei o livro para estudar sobre teoria de conjuntos (mais num nível de curiosidade do que no nível acadêmico) e achei muito bom. O livro é bom mesmo, aborda diversos assuntos e abre muito a nossa cabeça (mesmo que a gente não entenda tudo).
Entretanto, eu arrependi de ter comprado a tradução. Primeiro porque a tipografia dessa edição é horrível. Além disso, logo no início do livro você percebe que a tradução é bem ruim e eu acabei tendo que acompanhar também a versão original pra tirar algumas dúvidas. Há alguns pontos em que a gramática está errada mas há também frases cujo sentido é totalmente diferente do original.
E, pra piorar, há erros em equações que mudam tudo e você acaba perdendo muito tempo tentando entender até que desiste e vai checar o original. Por exemplo, na página 11 existe esse trecho abaixo (∈ e ∈' significam, respectivamente, pertence e não pertence):
Se B ∈' B, ...., a afirmação B ∈ A acarreta B ∈' B, uma contradição.
Aí você se pergunta: se B ∈' B, como que B ∈' B é uma contradição? Aí você checa o original e está assim:
If B ∈' B, ..., the assumption B ∈ A yields B ∈ B - a contradiction again.
E aí sim as coisas fazem sentido.
Além disso, a qualidade da edição é sofrível. Textos apagados, folhas descolando da lombada etc.
Resumindo, o conteúdo do livro original é excelente. Se por possível, fique com o original.
Entretanto, eu arrependi de ter comprado a tradução. Primeiro porque a tipografia dessa edição é horrível. Além disso, logo no início do livro você percebe que a tradução é bem ruim e eu acabei tendo que acompanhar também a versão original pra tirar algumas dúvidas. Há alguns pontos em que a gramática está errada mas há também frases cujo sentido é totalmente diferente do original.
E, pra piorar, há erros em equações que mudam tudo e você acaba perdendo muito tempo tentando entender até que desiste e vai checar o original. Por exemplo, na página 11 existe esse trecho abaixo (∈ e ∈' significam, respectivamente, pertence e não pertence):
Se B ∈' B, ...., a afirmação B ∈ A acarreta B ∈' B, uma contradição.
Aí você se pergunta: se B ∈' B, como que B ∈' B é uma contradição? Aí você checa o original e está assim:
If B ∈' B, ..., the assumption B ∈ A yields B ∈ B - a contradiction again.
E aí sim as coisas fazem sentido.
Além disso, a qualidade da edição é sofrível. Textos apagados, folhas descolando da lombada etc.
Resumindo, o conteúdo do livro original é excelente. Se por possível, fique com o original.
January 7, 2020
An accessible introduction to set theory going well beyond what my college Fundamentals of Mathematics course covered. Contains some mind-blowing content for the amateur/beginning mathematician. The book also served to clarify several concepts that were not clear from my algebra and analysis classes (for example, why we can talk about the inverse of a function without knowing it is bijective--the set-theoretic construction of a function always has an inverse).
In my opinion, the best part of the book is the derivation of the natural numbers and their arithmetic properties from the set theory axioms. Also very cool are the proof that a universe of discourse cannot exist, and the discussions on ordinal and cardinal numbers.
In my opinion, the best part of the book is the derivation of the natural numbers and their arithmetic properties from the set theory axioms. Also very cool are the proof that a universe of discourse cannot exist, and the discussions on ordinal and cardinal numbers.
June 12, 2023
Not that good. The author openly asserts in his preface that its all trivial and without much value but lets get it over and done with so we can get back to the deep important stuff ! Groups, integrals, manifolds! One wonders why he even bothered writing the book in the first place.
The treatment is glib and superficial. Though he does do a pretty good job explaining the axiom of choice. When he gets to the good stuff - ordinals and cardinals - he caves completely and his treatment is rushed, incomplete and not much better than useless.
The treatment is glib and superficial. Though he does do a pretty good job explaining the axiom of choice. When he gets to the good stuff - ordinals and cardinals - he caves completely and his treatment is rushed, incomplete and not much better than useless.
December 12, 2019
Though I did end up abandoning it, that's just because it got a little above my level of understanding. Maybe the style of explanations is not really for me, because I was a little lost while trying to understand the new stuff, but it did very much help me gain a more in-depth knowledge on the things I knew already.
Also, the sense of humor was a cherry on top.
Also, the sense of humor was a cherry on top.
August 14, 2021
This short book does what the author intended as outline in the preface, it's a series of short articles explaining concepts from set theory, someone unfamiliar with thses concepts can profit from a simple overview of the ideas as well as an outline of the important concepts of the field, and it can also serve as a refersher for someone who is already familiar with subject.
February 19, 2023
Classic intro to set theory - no real technical expertise is required as he begins with basic foundational axioms and builds on them. It’s pretty short can be finished relatively quickly. Halmos was famed for his ability to discuss complex topics in math and had a style of pedagogy that encouraged non-specialists to take in interest in the major topics of his day.
March 27, 2024
Really good book. I wish I had read this in school, but I'm glad to finally learn about ordinal numbers. Two caveats: 1) The notation gets quite confusing in the sections on Zorn's Lemma and Transfinite Recursion, I had to use some auxiliary sources to help understand. 2) It's definitely a bit dated since the continuum hypothesis is mentioned at the end as an unsolved problem!
October 20, 2021
This book is more like {Set of elements} :) Language of book is really easy to understand (exception last chapters), one big plus it's that chapters are not protracted how in many other math books, it's give a way to undst more easy even for book from 1960
October 19, 2023
Should be taught early in high school. This is one of the first texts anyone approaching mathematics should pick up, it will teach the level of abstraction you are required to allow when doing pure mathematics, and it is clear and essential subject matter.
December 12, 2024
Didn't want to add too many sorts of textbooks but feel obliged to say that this treatment of set theory (engaging, while still displaying the proper skeptical attitude towards formalism) is harder to do than I realized (having tried other texts)
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