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In a book about logic (propositional logic), author uses sets to describe propositional variables, operators (eg. $A=\lbrace \neg,\wedge,\vee,\Rightarrow,\Leftrightarrow\rbrace$)... How can those sets even exist? What are those operators exactly? (obviously not sets, althought in axiomatic set theory everything is set, so does it means that sets such as $A$ does not exist? What is it then, just a proper class?)

So, in other words, are mathematical sets and sets in logic different? (Mathematical sets made by axioms, and sets in logic made just intuitively?)

Another thing which I would like to see is a step by step construction of all basic mathematics such that you start by nothing. In other words, no use of anything which hasn't yet been defined. (e.g., if you start with logic, then you cant use sets in logic, or functions, or numbers because you haven't yet even defined them). So, is there some book written in that way, or maybe some internet resource?

Andrés E. Caicedo
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Ivan
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  • I do not understand your question. In analysis for example, we deal with sets that do satisfy the nine axioms of set theory and the axiom of choice, commonly known as ZFC. When we encounter a collection of items that is not a set, we do not use the term, eg equivalence classes (not sets). – christina_g Mar 18 '16 at 19:52
  • I think equivalence classes are sets, but they are just called this way for maybe historical reason. In most of "usual" mathematics, sets are OK (because everything is a set, numbers, relations, functions are sets), but what I'm asking is sets in logic, because propositional variable is certainly not a set, so how can we say that we have a set of propositional variables... – Ivan Mar 18 '16 at 19:58
  • By the way the notation $ A = { x,y,... }$ does not necessarily mean that A is a set, rather than collection of items that might as well be proper class. – christina_g Mar 18 '16 at 19:59
  • If "things" are defined then this is done by means of other "things" that have been defined allready. This however is a "well-founded" construction. So undefined things (primitive notions) are bound to show up at some time. – drhab Mar 18 '16 at 19:59
  • @Ivan The collection of all functions is a proper class [https://en.wikipedia.org/wiki/Class_(set_theory)] – christina_g Mar 18 '16 at 20:04
  • For your last paragraph you may find some of the answers to How to avoid perceived circularity when defining a formal language? helpful. – hmakholm left over Monica Mar 18 '16 at 20:15
  • Also see http://math.stackexchange.com/a/1334753 for an explicit explanation of what core part of mathematics is circular in the sense of being undefinable. – user21820 Mar 19 '16 at 17:19

2 Answers2

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First order logic may be encoded into ZFC, Zermelo Fraenkel set theory with the axiom of choice.

One of the first constructions of set theory is the set $\omega$ of the natural numbers. To define $\omega$, we first let $0= \emptyset$. Then $1=\{\emptyset\}$, $2=\{0,1\}=\{\emptyset, \{\emptyset\}\}$, and so on. Formally, the construction is carried out solely using the axioms, and so while we are not starting with "nothing" we are starting with as little as possible. From here we go on to define the rational numbers, reals, relations, cartesian products, etc.

After having constructed a sufficiently large amount of mathematics, we may begin encoding first order logic into ZFC. This is done using Godel numbering (see https://en.wikipedia.org/wiki/Gödel_numbering). Every logical symbol is assigned a natural number (and hence is associated with a pure set, since all the elements of $\omega$ are pure sets), and then doing first order logic within ZFC becomes a matter of arithmetic.

Of course, the symbols $\neg, \wedge, \to$ etc are part of the language of set theory in the first place, and so you may feel that the problem of forming the set $A=\{\neg, \wedge, \to, ...\}$ persists in ZFC. The answer to this of course is that the collection $A$ can only be a set if by the symbols "$\neg$," "$\wedge$," "$\to$" you mean their encoding in ZFC as elements of $\omega$.

But taken as symbols in the language of set theory, you are right, they are not sets, and hence do not exist from the perspective of ZFC. This is similar to saying that cows, lamps, and sandwiches cannot be sets, since they cannot be constructed using the axioms of ZFC.

For a good first book on formal ZFC set theory, see $\textit{Elements of Set Theory}$ by Herbert Enderton.

Exit path
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  • Well, before stating axioms of set theory, we need some logic, right? And Gödel's numbering actually means that logic operators $\neg$, $\wedge$, $\Rightarrow$ become numbers as well? – Ivan Mar 18 '16 at 20:12
  • @Ivan As far as I know set theorists consider set theory to be the foundation for all of mathematics, and this includes logic. You could say that set theorists use some form of "logic" in their reasoning, but if you want to establish a foundational hierarchy in mathematics then we don't typically assume anything beyond ZFC; you have to start somewhere! – Exit path Mar 18 '16 at 20:14
  • @Ivan As to your second question, yes the symbols are thought of as numbers. It is up to us though to understand what the numbers mean in any particular context. For example in analysis the element $2 \in \omega$ is just the number $2$, but in logic we may have assigned the set $2$ to $\to$. – Exit path Mar 18 '16 at 20:18
  • @Ivan: See my post that I linked to, which directly responds to your question, since indeed natural numbers are ultimately undefinable. Without something equivalent to natural numbers and the standard arithmetic on them, we cannot even talk about statements and proofs whatsoever and so cannot define any reasonable formal system at all. This is because having strings and basic string operations is actually equivalent to assuming consistency of $PA$! – user21820 Mar 19 '16 at 17:22
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When you write about putting things like $\Rightarrow$ and $\neg$ into sets, they are from a formal perspective "just ink" -- in other words "$\Rightarrow$" and "$\neg$" are, for this purpose, nothing more than strangely shaped variable letters, chosen so to remind you what it is you intend to be doing with them.

You're then supposed to imagine that each of these symbols stand for some mathematical object that is appropriately chosen not to conflict with any other set you need to use during the argument you're carrying out.

If you take care not to use any particular properties of these objects, other than they're all different from each other and from everything that already has a meaning, you don't need to choose particular set-theoretic objects right when you start reading. Instead, when you're done reading and reach a theorem whose validity you need to convince yourself of, at that point you can choose some sets that don't like anything you have seen during the proof. This will always be possible, which is why we don't really care about actually choosing formal meanings for the symbols.

hmakholm left over Monica
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