0

I'm reading Axiomatic Set Theory by Suppes, and I'm having a bit of trouble understanding which rules of inference (logical system) he is using, here's an example (capital letters are used for sets):

Definition: $(y \text{ is a set})\leftrightarrow (\exists x) (x\in y)\vee x=0.$

Theorem 1. $x\notin0$.

Theorem 2. $(\forall x)(x\not\in A)\leftrightarrow A = 0$.

Proof. If $A=0$ then by theorem $1$ $x\notin A$. If $(\forall x)(x\notin A)$, then by definition $1$, $A=0$.

So my translation of this into a formal proof would be something like:

\begin{align} &\vdash \forall x\,(x\notin 0) &\text{Theorem 1.}\\ A=0&\vdash \forall x\, (x\notin A) &\text{Rule of replacement?}\\ \end{align}

Then by the deduction theorem (I'm guessing this is needed...), we get $\vdash A=0\rightarrow (\forall x)(x\notin A)$. Then a similar proof to get the converse, and lastly something like

\begin{align} 1.&\vdash A=0\rightarrow (\forall x)(x\notin A) &\text{Proof above}\\ 2.&\vdash (\forall x)(x\notin A) \rightarrow A=0 &\text{Proof above}\\ 3.&\vdash A=0\leftrightarrow (\forall x)(x\notin A) &\text{From 1,2.} \end{align}

But I'm not sure if this is what Suppes had in mind, and he doesn't seem to say so anywhere in the first chapters. Does anyone know the exact system he is using?

YoTengoUnLCD
  • 13,384
  • I think that you miscopied Theorem 2. – Asaf Karagila May 10 '16 at 22:49
  • @AsafKaragila whoops, of course, it's $A$ instead of $x$. Thanks Asaf. – YoTengoUnLCD May 10 '16 at 22:50
  • Should one use (logical) axioms, rules of inference, when doing mathematics? – André Nicolas May 10 '16 at 22:57
  • @AndréNicolas Well, his presentation looked like a very syntactical view of axiomatic set theory, so I believed that yes. In some cases, when he has e.g $p\wedge q,, \neg , q$ he says "from sentential logic, $p$", so it makes it seem more like that too... – YoTengoUnLCD May 10 '16 at 23:00
  • PS: your definition should be, $(Y\text{ is a set}) \leftrightarrow (\exists x)(x\in Y)\vee (Y=\emptyset)$ , should it not? – Graham Kemp May 10 '16 at 23:49
  • @GrahamKemp Yes, it's meant to be interpreted that way (I copied Suppes' style for reducing the number of parenthesis). – YoTengoUnLCD May 10 '16 at 23:50
  • 1
    Logical axioms and rules are left implicit. Details may be found into: Patrick Suppes, Introduction to Logic (1957). – Mauro ALLEGRANZA May 11 '16 at 06:14
  • The "sentential logic" part is based on: (i) every tautology is an axiom, as well as (ii) Rule T : we may introduce a sentence $S$ in a derivation if there are preceeding sentences in the derivation such that their conjunction tautologically implies $S$ and (iii) Rule CP : if we can derive $S$ from $R$ and a set of premises, then we can derive $R \to S$ from the set of premises alone. Thus, in a nutshell: modus ponens and Deduction Th. – Mauro ALLEGRANZA May 11 '16 at 06:21

1 Answers1

2

If the author is not providing an explicit set of logical rules to work with, the intention is most probably that you can imagine using your own favorite proof system for classical first-order-logic.

There are many such systems, all leading to the the same entailment relation -- that is, they differ markedly in what a proof looks like internally, but agree about what can be proved from which assumptions.

As long as you know any one of the proof systems that give rise to this common entailment relation, you're free to imagine using that to reason about the set theory being presented.

The particular problem you have with the reasoning you quote may be that it looks like your book is presenting a set theory with urelements where using a capital letter in a formula is an abbreviation for a formula that additionally asserts that the meaning of that letter is a set.

So the actual symbolic formula that represents your theorem $(\forall x)(x\not\in A)\leftrightarrow A = 0$ is $$(a\text{ is a set}) \to \bigl( (\forall x)(x\not\in a)\leftrightarrow a = 0\bigr)$$ in which you then need to unfold the "is a set" definition.

This means that during the symbolic proof you will have the "invisible" assumption that $A$ is a set available, and if you want to construct a formal proof, you'll need to have explicit proof steps when you're applying that assumption.

hmakholm left over Monica
  • 286,031
  • Thanks for your answer, I should've said it before, but the abbreviation part was not really a problem, it's really just about which inference rules are 'allowed'. I may pick up another book, as this may not be the best way to get into axiomatic set theory: There are exercises to prove some theorems, but I don't know what I should be using to prove them. – YoTengoUnLCD May 10 '16 at 23:52
  • @YoTengoUnLCD: From what you quote it looks like the book's favored style of proof is ordinary English mathematical prose. If you know a formal proof system, you can entertain yourself by trying to translate the book's proofs into it, but it does not immediately look like knowing formal proofs is necessary for using that book. – hmakholm left over Monica May 10 '16 at 23:56
  • 1
    @YoTengoUnLCD: I don’t recommend Suppes if you intend eventually to go beyond an introduction: it’s rather old-fashioned. A much better introductory text is Karel Hrbacek and Thomas Jech, Introduction to Set Theory, 3rd ed. It also presents proofs in ordinary mathematical prose rather than in some formal system, but I consider this a good thing: formalisms tend to get in the way of the actual ideas involved. – Brian M. Scott May 11 '16 at 15:19
  • @BrianM.Scott Thanks, I'll definitely check that out. Could you elaborate on why you don't recommend Suppes'? I mostly picked it up because it was available at my university's library (it hard to find real mathematics books in a engineering university :~( ). – YoTengoUnLCD May 11 '16 at 17:22
  • @YoTengoUnLCD: It’s been a while since I looked at Suppes, but my recollection is that the notation is quite old-fashioned in places. I also seem to recall that the style of argumentation is just different enough from what I’m used to in more recent mathematical writing to be a slight obstacle – only slight, and sensitivity to this probably varies a lot from one reader to another, but it’s still a potential nuisance. Finally, H&J covers a lot more material. In fact, the third edition covers noticeably more than the first, which was already very good. – Brian M. Scott May 11 '16 at 17:48