Name: Naive Set Theory
ISBN: 9780387900926

Naive Set Theory

Every mathematician agrees that every mathematician must know some set theory; the disagreement begins in trying to decide how much is some. This book contains my answer to that question. The purpose of the book is to tell the beginning student of advanced mathematics the basic set theoretic facts of life, and to do so with the minimum of philosophical discourse and logical formalism. The point of view throughout is that of a prospective mathematician anxious to study groups, or integrals, or manifolds. From this point of view the concepts and methods of this book are merely some of the standard mathematical tools; the expert specialist will find nothing new here. Scholarly bibliographical credits and references are out of place in a purely expository book such as this one. The student who gets interested in set theory for its own sake should know, however, that there is much more to the subject than there is in this book. One of the most beautiful sources of set-theoretic wisdom is still Hausdorff's Set theory. A recent and highly readable addition to the literature, with an extensive and up-to-date bibliography, is Axiomatic set theory by Suppes.

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furyofantaresOct 2025

I had a professor who had taken courses from Paul Halmos, who authored one of my favorite math books, Naive Set Theory. I regarded the book so highly that I was somewhat star struck to learn my professor had taken courses from him.

I asked my professor what it was like, was he an incredible professor? He told me the lectures were useless! I was shocked and he said they were so well prepared they were effectively a recording of the material - and no advantage over reading the book. At that point I reflected and realized that, by far, my favorite professor was one who constantly made mistakes while solving problems on the board. He'd have to notice something wrong and track down the error, and it was wonderful and highly instructional to watch the process.

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nyrikkiOct 2024

As the Principia is pretty ugly and tortured at least for me let me offer:

Naive set theory. Halmos, Paul R. http://people.whitman.edu/~guichard/260/halmos__naive_set_th...

Note the first entry of "Ordered Pairs"

> What does it mean to arrange the elements of a set A in some order?

Also note how the earlier section on "Unordered Pairs" is more about building the axiom of pairing etc...to get to ordered pairs which gets to the Cartesian product, which outputs ordered pairs.

It doesn't matter if you go through Zermelo's theorem+Zorn, that states that every set can be well-ordered, or though Cartesian product's and/or AC. (Note: This is in FoL well-ordering and AC are the same, but not in SoL and HoL)

It is not that sets are expressly unordered, as a set of points in a line segment would very much have an order, but that you didn't actively arrange the elements in order to take advantage of properties that are useful to you.

Maybe I just hit mental blocks but IMHO it is important that when you make the assumption that "there exists a set." it is very important to realize that it is "unordered" because you haven't imposed one, but is not an innate property of an element of the set.

Hopefully that helps in addressing this from your original post.

> "ordered pair" is not part of set theory

While many creators of both naive and formal set theories may choose to not define (a,b) = {{a},{a,b}} explicitly, the output of the Cartesian product is the ordered pairs, so it doesn't matter, you don't have a useful set theory without them.

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seanhunterAug 2024

I'm assuming maximum good faith here but this only seems interesting/contradictory because you haven't defined the system rigorously. All the seeming contradictions are coming from the fact that you are applying Euclidian geometric constructions to set theory via this example involving covering things up with actual pieces of paper.

The parallel postulate doesn't deal with pieces of paper. It deals with lines in Euclidian space. These spaces have infinite dimension, and the lines have infinite length, so the very first deduction you are attempting to draw (about "half the logic" being represented) is literally meaningless. If you take an infinite Euclidian space and partition it into two halves with a line, they are each infinite in size. A Venn diagram is not a Euclidian concept, it is a diagramatic representation of set membership. So you can't say anything about logic purely from the idea you can partition a space in half.

Your next paragraph about the logic of camouflage is also meaningless for the exact same reason.

The section about an object on one side or the other of the line is simply a category error. If we have an object with non-zero area and a line in a Euclidian space then the object can be on either side of the line or the line can intersect with the object. Nothing about this changes the position of the object or the line and your "paper" construction simply adds some confusion to a very simple geometric system. If you're actually dealing not with a Euclidian space but with set membership then it's best to abandon this paper/line construction and just talk about set membership. If you have an object and two sets, the object can be in either set, both or neither. This is not a contradiction and the object has no "position" in this system. It's simply a question of definition. If you want to learn how set theory works I recommend you read Naive Set Theory by Paul Halmos.

The final paragraph is meaningless because all the things before are meaningless. If we're talking about Euclidian spaces these have nothing to say about logic (modified or otherwise) and if you are talking about sets then the parallel postulate is irrelevant because it is a geometric concept. I understand the Dedekind construction etc and really don't follow what you're saying about the parallel postulate being a statement about Cauchy completeness. From the fact that the coordinates of a point fully represent that point, the existance of lines in a Cartesian space (not the parallel postulate) is a statement about Cauchy completeness. That said, nothing you've put here affects the construction of numbers in any way.

If you want any of this to be actual mathematics you need to abandon your "paper" construction and define things properly.

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graycatJun 2024

One relatively simple but informal description of the problem is that the statement is self-referencing.

Related is the Bertrand Russell paradox:

X = {z | z is not an element of z}

So, X is an element of X if and only if it is not -- a contradiction.

The usual solution was to have two steps:

(1) Over here, say, on the left, we have everything we want to regard as elements.

(2) On the right, we have everything we want to regard as sets. The sets consist only of elements.

That's a good fix.

For more, there is the elegant, long popular, P. Halmos Naive Set Theory, now available in PDF, LaTeX, etc.

For more there is the appendix to J. Kelley, General Topology.

For more there is P. Suppes, Axiomatic Set Theory.

For more, sure you want more????

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hiAndrewQuinnNov 2023

Category theory is a pretty good one. But I think a lot of people would get a surprising amount of value from reading Halmos's slim _Naive Set Theory_, to see just how much work we can create using sets alone.

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stiffSep 2023

Here are a few popular science books:

  One Two Three . . . Infinity: Facts and Speculations of Science by George Gamow
  Infinity and the Mind: The Science and Philosophy of the Infinite by Rudy Rucker
  Yearning for the Impossible: The Surprising Truths of Mathematics by John Stillwell
  Roads to Infinity by John Stillwell

If you want actual textbooks then maybe pick up a set theory textbook like "Naive set theory" by Halmos.

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