2

Sorry if this was asked before, I couldn't find.
So, I'm reading Hinman's Book "Fundamentals of Mathematical Logic" and I've got stucked in the beginning, more precisely at page 16, proposition 1.1.5. He's proving the unique readability for propositional sentences and during the proof he needs the following lemma:

For any finite sequences $\phi_0,\ldots,\phi_k$ and $\psi_0,\ldots,\psi_l$ of sentences, if the expressions $\phi_0 \ldots \phi_k$ and $\phi_0 \ldots \phi_l$ are identical, then $k=l$ and for all $i \leq k$, $\phi_i = \psi_i$.

The proof goes by induction on the length of the expression $\phi_0 \ldots \phi_k$, but I don't know why this is true and how to deal/see the "sequence of sentences". First of all I've tried to see the case where there's just $\phi_0$ and I think I shouldn't assume that these sequences are made of atomic sentences. So, couldn't I have $\phi_0 := (\to (\vee p_1 p_2) p_3)$ and $\psi_0 := (\vee p_1 p_2)$, $\psi_1 := (\to \psi_0 p_3)$? In this case the sequences $\phi_0$ and $\psi_0,\psi_1$ are identical as expressions, but the lemma doesn't hold, because I split the first sentence in two. I think that probably I'm not dealing well with the sequence thing, but it was not defined before on the book, so I'm in the dark here. I appreciate any help.

Xablau123
  • 156
  • Dont you need polish notation for this ? And your example dosent work. – Rene Schipperus Dec 19 '17 at 02:56
  • Yes, sorry. I'm dealing with polish notation in the book, but in my head I automatically translate to infix notation. I'll edit the question. – Xablau123 Dec 19 '17 at 02:59
  • Oh, now that I translated it to Polish notation back, I think that the problem was solved. The two expressions $\psi_0$ and $\phi_0$ are not identical anymore. Very counterintuitive thing. Was not expectating that. – Xablau123 Dec 19 '17 at 03:07
  • Yeah but you example is wrong cause $\psi_2$ contains $\psi_1$ in it. – Rene Schipperus Dec 19 '17 at 03:07
  • Then how should I see the sequence of sentences thing? I have no idea. – Xablau123 Dec 19 '17 at 03:10
  • Isnt there a proof in that book ? – Rene Schipperus Dec 19 '17 at 03:17
  • Yes, there is and after your comment about polish notation I could understand the proof. I think I should just assume that a sequence is the concatenation of sentences. Thanks for the help. – Xablau123 Dec 19 '17 at 03:22
  • In polish notation there are no parentheses; thus $\phi_0 := \ \to \lor p_1 p_2 p_3$. – Mauro ALLEGRANZA Dec 19 '17 at 07:19
  • 1
    I wonder why you are tackling Hinman's book? To be frank, you would do very significantly better to read good separate books on basic logic, model theory, computability, and set theory, rather than this book which doesn't give one of the best available treatments of any of those big topics. (Some of its chapters are singificantly better than others, but think I'd still only recommend them for supplementary reading.) See http://www.logicmatters.net/tyl/ – Peter Smith Dec 19 '17 at 08:20
  • I like very much the "modern" approach of Hinman's book, specially in the beginning of the subject. Besides, the book was recommended to me because he proves with details some important theorems that another books don't prove. (For example, I didn't like the approach with tableaux of Smullyan's FOL book.) – Xablau123 Dec 19 '17 at 18:59

0 Answers0