Topics include the basic concepts of set theory, cardinal numbers, transfinite methods, and a good deal more in 25 Naive Set Theory By: Paul R. Halmos. PATRICK SUPPES—Axiomatic Set Theory. PAUL R. HALMOS—Naive Set Theory. JOHN L. KELLEY —Introduction to Modern Algebra. R. DUBISCH—Student. Halmos Naive Set Theory is a classic and dense little book on axiomatic set theory, from a "naive" perspective. Which is to say, the book won't.


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The Journal of Symbolic Logic 34 2 It is the purpose of this purely expository book to tell the beginning student of advanced mathematics the basic principles and facts of set theory. The treatment is called naive because the language used is the same as that of ordinary informal mathematics.

But as a matter of fact the treatment is axiomatic because the proofs are based on the axioms which are commonly used in the well-known halmos naive set theory.

A comparative study of alternative sets of axioms is not contemplated. The style is almost conversational and makes for pleasant reading.

It halmos naive set theory the drawback that some proofs are not as complete as might be desirable. Only the essential theorems are displayed and the reader should unearth the other ones from the informal explanations.

Monthly 68 4 Having worked both in formal logic and in plain mathematics, Halmos is admirably qualified to write this book.

Book Review: Naive Set Theory (MIRI research guide)

It is possible that he was first prompted to set pen to paper when as a Van Nostrand editor he saw the manuscript of Suppes's Axiomatic Set Theory. In any event the two books form a halmos naive set theory contrasting pair, Suppes's for the careful logician investigating say the independence of the axiom of choice, Halmos's for the mathematician-in-the-street who just wants to stay out of trouble when he does say measure theory.

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It seems halmos naive set theory the reviewer that Halmos should have relaxed his "naive" principles occasionally, at least in his examples. 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.

In some theories this is done. He can even be quite humorous at times — I laughed out loud reading a passage on page As to set theory applied to machine learning, it halmos naive set theory be that what is needed differs from the content of Halmos' book.

I like the book by E.

Naive Set Theory

Axiom of choice The axiom of choice lets you, from any collection of non-empty sets, select an element from every set in the collection. The axiom is necessary to do these kind of "choices" with infinite sets. In finite cases, one can construct functions for the job halmos naive set theory the other axioms.

Though, the axiom of choice often makes the job easier in finite cases so it is used where it isn't necessary.


Zorn's lemma Zorn's lemma is used in similar ways to the axiom of choice - making infinite many choices at once - which perhaps is not very halmos naive set theory considering ZL and AC have been proven to be equivalent.

Well ordering A well-ordered set is a totally ordered set with the extra condition that every non-empty subset of it has a smallest element.

This extra condition is useful halmos naive set theory working with infinite sets. The principle of transfinite induction means that if the presence of all strict predecessors of an element always implies the presence of the element itself, then the set must contain everything.

Why does this matter?

Naive Set Theory (book) - Wikipedia

It halmos naive set theory you can make conclusions about infinite sets beyond w, where mathematical induction isn't sufficient. Transfinite recursion Transfinite recursion is an analogue to the ordinary recursion theorem, in a similar way that transfinite induction is an analogue to mathematical induction - recursive functions for infinite sets beyond w.

In modern lingo, what Halmos calls a "similarity" is an "order isomorphism".


Ordinal numbers The axiom of substitution is called the axiom schema of replacement in modern use.