Naive Set Theory (book)

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Naive set theory Naive set theory is any of several theories of sets used in the discussion of the foundations of mathematics. Unlike axiomatic set theories, which are defined using formal logic, naive set theory is defined informally, in natural language ...
for the mathematical topic.'' ''Naive Set Theory'' is a
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
textbook by
Paul Halmos Paul Richard Halmos ( hu, Halmos Pál; March 3, 1916 – October 2, 2006) was a Kingdom of Hungary, Hungarian-born United States, American mathematician and statistician who made fundamental advances in the areas of mathematical logic, probabi ...
set theory Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, ...
. Originally published by ''Van Nostrand'' in 1960, it was reprinted in the
Springer-Verlag Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 in ...
Undergraduate Texts in Mathematics Undergraduate Texts in Mathematics (UTM) (ISSN 0172-6056) is a series of college, undergraduate-level textbooks in mathematics published by Springer-Verlag. The books in this series, like the other Springer-Verlag mathematics series, are small ye ...
series in 1974. While the title states that it is naive, which is usually taken to mean without
axiom An axiom, postulate, or assumption is a statement (logic), statement that is taken to be truth, true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that whi ...
s, the book does introduce all the axioms of ZFC set theory (except the Axiom of Foundation), and gives correct and rigorous definitions for basic objects.Review of ''Naive Set Theory'', L. Rieger, . Where it differs from a "true"
axiomatic set theory Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics Mathematic ...
book is its character: there are no discussions of axiomatic minutiae, and there is next to nothing about advanced topics like
large cardinal In the mathematical field of set theory Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set ...
s. Instead, it tries to be intelligible to someone who has never thought about set theory before. Halmos later stated that it was the fastest book he wrote, taking about six months, and that the book "wrote itself"..

# Absence of the Axiom of Foundation

As noted above, the book omits the Axiom of Foundation (also known as the Axiom of Regularity). Halmos repeatedly dances around the issue of whether or not a set can contain itself. *p. 1: "a set may also be an element of some ''other'' set" (emphasis added) *p. 3: "is $A$$A$ ever true? It is certainly not true of any reasonable set that anyone has ever seen." *p. 6: "$B$$B$ ... unlikely, but not obviously impossible" But Halmos does let us prove that there are certain sets that cannot contain themselves. *p. 44: Halmos lets us prove that $\omega$$\omega$. For if $\omega$$\omega$, then $\omega$ − would still be a successor set, because $\omega$ ≠ ∅ and $\omega$ is not the successor of any natural number. But $\omega$ is not a subset of $\omega$ − , contradicting the definition of $\omega$ as a subset of every successor set. *p. 47: Halmos proves the lemma that "no natural number is a subset of any of its elements." This lets us prove that no natural number can contain itself. For if $n$$n$, where $n$ is a natural number, then $n$$n$$n$, which contradicts the lemma. *p. 75: "An ''ordinal number'' is defined as a well ordered set $\alpha$ such that $s\left(\xi\right) = \xi$ for all $\xi$ in $\alpha$; here $s\left(\xi\right)$ is, as before, the initial segment $\." The well ordering is defined as follows: if\xiand\etaare elements of an ordinal number\alpha, then\xi<\etameans\xi\in \eta\left(pp. 75-76\right). By his choice of the symbol < instead of \le , Halmos implies that the well ordering < is strict \left(pp. 55-56\right). This definition of < makes it impossible to have\xi\in \xi, where\xiis an element of an ordinal number. That\text{'}s because\xi\in \ximeans\xi<\xi, which implies\xi\ne \xi\left(because < is strict\right), which is impossible. *p. 75: the above definition of an ordinal number also makes it impossible to have\alpha\in \alpha, where\alphais an ordinal number. That\text{'}s because\alpha\in \alphaimplies\alpha= s\left(\alpha\right). This gives us\alpha\in \alpha= s\left(\alpha\right) =\, which implies\alpha<\alpha, which implies\alpha\ne \alpha\left(because < is strict\right), which is impossible.$

# Errata

*p. 4, line 18: “Cain and Abel” should be “Seth, Cain and Abel”. *p. 30, line 10: "x onto y" should be "x into y". *p. 73, line 19: "for each z in X" should be "for each a in X". *p. 75, line 3: "if and only if x ∈ F(n)" should be "if and only if x = ".

* List of publications in mathematics

# Bibliography

* Halmos, Paul, ''Naive Set Theory''. Princeton, NJ: D. Van Nostrand Company, 1960. Reprinted by Springer-Verlag, New York, 1974. (Springer-Verlag edition). Reprinted by Martino Fine Books, 2011. (Paperback edition)
2017 Dover reprint

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