Naive Set Theory, Halmos

22nd Dec 2022

4Viliam

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(I haven't read this specific book, but I have read other books about sets.)

For starters, what exactly does the word "naive" mean, in the context of *Naive Set Theory*? It seems that different authors use this word a little differently, but generally it means: "I allow myself to skip the technical details when they become boring, and also the difficult topics because this is a book for beginners".

Sets in set theory are not generally

assumedto exist -- with the sole exception of the axiom of infinity, our existential starting point. Sets in set theory are insteadconstructedin steps, from the axiom of infinity, using the existential conditionals of the remaining axioms of set theory.

I think the class of sets whose existence can be proved from axioms is called "constructible universe". This is the minimum that must exist. But the axioms are not saying that *other* sets do *not* exist.

So if you can imagine other sets, whose existence is neither provable from the axioms, nor does it *contradict* them... then such sets may or may not exist. More precisely speaking, now we have multiple possible meanings of the word "set". (Of course, if you add some new set, you also need to add all those sets whose existence can be proved from the axioms *and* the existence of *this* set.) This is basically what it means that some statements are not decidable from the axioms: depending on what meaning you choose, such statements can be either true or false.

I guess a specific example would be useful here. We have a set of all natural numbers. Now suppose that God flips a coin for each of these numbers, and creates a set "G" of those where the coin came up heads. (There were infinitely many coin flips. The coin is fair, so there were infinitely many heads, infinitely many tails, and there is no finite text that could describe the results exactly.) Does the set "G" exist? Your intuition might say yes, but it is not provable from the axioms, because all *proofs have finite length*, and therefore cannot specify objects that do not have a finite definition.

Observe that there is some method in this apparent madness; the number of elements in the sets , , or … is, respectively, zero, one, or two

Each natural number is represented by a set containing representations of all *smaller* numbers. This is convenient because it allows to express "*a < b*" as "the representation of *a* is a subset of the representation of *b*". And it makes the definition of a successor very simple. Furthermore, this also works for (some?) infinite sets; for example ω is the smallest ordinal *greater than* all integers, and simultaneously a set of all integers (i.e. the smallest set *containing* all integers). -- In other words, for certain sets, you can use "smaller" and "subset" as synonyms; prove one statement, get the other statement for free.

the role of the axiom [of choice] is to guarantee that possibility in infinite cases...

Generally, it happens many times that something is obvious for finite sets, but for infinite sets it might be tricky to prove, or not true at all. You could see an infinite chain of arguments where each step is a correct proof and yet it does not help you, because proofs cannot be infinitely long.

Example, as an intuition pump: Imagine that there is a function that receives a set of natural numbers, and returns a natural number. We know that *f(empty set) = 42*, and *f(any set with one element) = 42*, and also for each two sets A and B *if f(A) = 42 and f(B) = 42 then f(A union B) = 42*. At this moment you may be tempted to say "okay, obviously the function returns a constant 42 no matter what". But actually, all we know is that it returns 42 for finite sets; for infinite sets it could be anything.

Epistemic status: Unorganized thoughts of mine generated after reading this book -- not a concerted attempt to convey any single particular mathematical insight.Set theory is the mathematical study of collections. These collections --

sets-- can be either the collection of nothing (theempty set), or collections... of other collections. Countries, for example, are sets of people, making the United Nations aset of sets of people.Somewhat surprisingly,

just about all the rest of mathematics can be framed as the study of sets (of sets of sets of sets... ).Knowing some set theory lets you translate the myriad objects and claims that you encounter elsewhere in the various areas of math into more set theory, putting math on one common intellectual foundation.## Contents and Notes

## 1. The Axiom of Extension

## 2. The Axiom of Specification

Reading this book, I picked up the idea that sets with arbitrary properties aren't simply assumed to exist in math. As Halmos puts it,

Another related "correctly orienting towards mathematics" maxim along these lines is: "assume nothing, prove nothing." You don't do math without any premises, because nothing interesting follows from no premises at all. Nor do you freely assume arbitrary premises and see what follows deductively from them. Sets in set theory are not generally

assumedto exist -- with the sole exception of the axiom of infinity, our existential starting point. Sets in set theory are insteadconstructedin steps, from the axiom of infinity, using the existential conditionals of the remaining axioms of set theory.## 3. Unordered Pairs

## 4. Unions and Intersections

## 5. Complements and Powers

## 6. Ordered Pairs

Sets do not natively track the ordering of their elements: {x,y,z}={z,y,x}. We can

introducethe notion of ordering of elements, though, in the above fashion.You can begin to see from the above treatment how something like a shape in the Cartesian plane can be formalized as an unwieldly set.

## 7. Relations

## 8. Functions

Functions can be understood as sets too. Think of a function as a database (potentially infinitely large) of ordered pairs, with a representing elements in its domain and b elements in its range. a and b can be anything, so so long as a and b themselves can be formalized as sets so can the functions defined on them.

Thus, natural numbers are constructable as sets, and so are straightforward functions from the naturals to the naturals.

I wish the terms

injection, surjection,andbijectionmade an appearance in the book, in place ofone-to-one,onto,andone-to-one correspondence,respectively.I tripped over the latter terms a bit ("Wait a second... was that supposed to be aninjectionor abijection?").## 9. Families

The above "unacceptable" notation for a family did indeed trip me up a few times. It's easily the worst piece of notation in the book. (You'd guess that a symbol like {Ai} would equal the set containing only Ai, but this is not at all the case!)

## 10. Inverses and Composites

This chapter got me to actually think somewhat carefully about associativity:

Let f be a function from X to Y, and f−1 its inverse.

Associativity seemed unobjectionable and uninteresting in high-school algebra. What was missing then, I think, was an appreciation of how so many functions

irreversibly throw away information about the object in question. Associativity means something like "no information about the object we're discussing is discarded by these functions, so you may evaluate the functions in any order and always come to the same terminal conclusion."If you want the set of all people on the planet, it doesn't matter whether you pool all the countries first into the UN,

thencount the UN's population, or you first count every country's individual population, then sum those population counts. Neither approach "changes the subject," so you get to the same figure in the end.## 11. Numbers

The

x+=x∪{x}.successorx+ of a set x is the set of all the elements of x together with x itself:Incidentally, there's a lot of that pattern of construction in this book.

First, establish that there exists some set with some property.Then, use the axiom of specification (or some other means) to pinpointjustits subset with the desired property (excluding potential misc. elements).## 12. The Peano Axioms

## 13. Arithmetic

## 14. Order

Remember, a

partial orderon a set X is a reflexive, transitive, andantisymmetricrelation in X, while anequivalence relationon X is a reflexive, transitive, andsymmetricrelation in X.## 15. The Axiom of Choice

Starting here, we turn our studies to infinite set theory.

## 16. Zorn's Lemma

## 17. Well Ordering

## 18. Transfinite Recursion

## 19. Ordinal Numbers

Ordinal numbers are the rigorous take on the elementary-school game of counting higher than infinity.

Imagine (N,<): all the naturals, ordered by <. Now imagine (N,R): all the natural numbers, ordered by <

except for0, which is just defined to be higher in the relation R than any other natural. You can see that there's a similarity^{[5]}between the naturals beginning at 0 in (N,<) and the naturals beginning at 1 in the second ordering. (There exist infinitely manybijectionsfrom N to N, of course, but that's not enough to get a similarity here -- we're specifically after amonotonebijection.) Since 0∈(N,R) is left out of the similarity, the ordinal number similar to (N,R) is larger than the ordinal number similar to (N,<).^{[6]}In ordinal numbers, we express this

ω<ω+1<ω+2<⋯<ω+ω=ω2<ω2+1<⋯<ω3<ωω=ω2<⋯(Please note that addition here isn't quite addition in the familiar sense: 1+ω=ω<ω+1.)

## 20. Sets of Ordinal Numbers

## 21. Ordinal Arithmetic

## 22. The Schröder–Bernstein Theorem

Here's the most beautiful proof in the book:

For whatever subjective reason, I really aesthetically enjoy the exhaustion-by-sequence-tracing character of this proof!

## 23. Countable Sets

## 24. Cardinal Arithmetic

## 25. Cardinal Numbers

Cardinals are least ordinals equivalent to some set X.

Note the connection to the discussion of bijections and similarities above, in §19: (N,<) and (N,R) could be equivalent to the same cardinal number but similar to distinct ordinal numbers.

^{^}The symbol ∅ is defined to mean the empty set, {}.

^{^}In the sense of there existing an equivalence relation in that set.

^{^}A

well orderon a set X is (1) a partial order that (2) connects any two elements in X (is total) and (3) has a least element for each subset S⊂X.Thus, partial orders are less demanding than total orders are less demanding than well orders.

^{^}Anof a partially ordered set X is the subset of X

{x∈X:x<a},a∈X.initial segments(a)^{^}A

f(a)≤f(b) if and only if a≤b.similarityis a bijection f(x) between sets X and Y such that when a,b∈X, thenNote that the former expression is evaluated in Y and the latter in X.

^{^}Similarities between two well-ordered sets are unique (p. 72).

^{^}Two sets X and Y are

equivalent, X∼Y, when there exists a bijection between X and Y. If X and Y are subsets of Z, then equivalence in this sense is an equivalence relation in the power set P(Z) (p. 52).^{^}You can easily make overlapping sets disjoint by creating a bijection between each element x∈X,y∈Y and (x,0)∈X×{0},(y,1)∈Y×{1}, respectively.