Advanced Logic for Applications by Richard E. Grandy (auth.)

By Richard E. Grandy (auth.)

This e-book is meant to be a survey of crucial ends up in mathematical common sense for philosophers. it's a survey of effects that have philosophical value and it really is meant to be available to philosophers. i've got assumed the mathematical sophistication bought· in an introductory good judgment path or in studying a easy good judgment textual content. as well as proving the main philosophically major leads to mathematical good judgment, i've got tried to demonstrate a variety of equipment of facts. for instance, the completeness of quantification thought is proved either constructively and non-constructively and relative advert­ vantages of every kind of facts are mentioned. equally, positive and non-constructive models of Godel's first incompleteness theorem are given. i'm hoping that the reader· will improve facility with the tools of facts and likewise be attributable to examine their adjustments. i guess familiarity with quantification concept either in lower than­ status the notations and to find item language proofs. Strictly talking the presentation is self-contained, however it will be very tricky for somebody with out historical past within the topic to persist with the cloth from the start. this is often beneficial if the notes are to be obtainable to readers who've had various backgrounds at a extra basic point. in spite of the fact that, to cause them to obtainable to readers with out historical past will require writing one more introductory common sense textual content. a number of workouts were incorporated and lots of of those are critical elements of the proofs.

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We now must consider the form which our trees 2 can take. Consider a branch of a tree B and let Bn be the portion of the branch generated in Tn. e. for all k > n Bk = Bn. For some n, all Bk for k > n are the result of writing permutations of the top node at Bn. This can occur if we have a sequent q, ~ 8 where all formulas in q, are atomic or are universal formulas (v)B such that all instances with terms in the relevant list have already been added 32 CHAPTER III and all formulas in 8 are atomic or are existential formulas (3 v)B such that all instances with terms on the list L have already been added.

By induction on the order of A. EXERCISE 6. Prove Lemma 1. Since IE(Ff) is the identity relation and (DE, IE) satisfies the sentences of T F, we can easily construct a model (DE, IE> for T= by letting I Ebe like IE except that it is not defined on Fr COROLLARY. T= f- A = iff TF f- A F. EXERCISE 7. Prove this corollary. CHAPTER VI GODEL'S INCOMPLETENESS THEOREMS: PRELIMINARY DISCUSSION In this section I will try to outline the ideas behind the proofs and indicate what facts need to be established about first order theories which are intended to formalize number theory in order to prove the theorems.

FO, q,2~ (Jt. FO, thus: q,(, q,2, (J2 which is derivable FO~Fo, (JI, (J2 q,IFo, q,2 ~ FO, (J(, (J2 FO, q,2~ (J(, FO, (J2. q,(, r LEMMA 3. Let T be the completed tree for ~ Ll, if the top nodes of T are derivable, then ~ Ll is derivable. Proof. By examining the ways in which T is generated. r LEMMA 4. If T is a tree generated by our definition for a sequent then if T falsifies a node of T T falsifies r ~ Ll. Proof. Again by inspecting the rules used in generating the tree. r ~ Ll EXERCISE 8. Show Lemma 4 in detail in case the procedure is operating on a LPF of the form (A ::l B).

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