Gödel incompleteness theorems and the limits of their applicability. I
... The central point of Gödel’s proof was the theorem on the decidability in P of all primitive recursive relations (however, Gödel proves this theorem only schematically).4 In particular, this enables him to express an independent statement for the theory T in the form ∀x ϕR (x), where R is primitiv ...
... The central point of Gödel’s proof was the theorem on the decidability in P of all primitive recursive relations (however, Gödel proves this theorem only schematically).4 In particular, this enables him to express an independent statement for the theory T in the form ∀x ϕR (x), where R is primitiv ...
The Foundations: Logic and Proofs
... A lemma is a ‘helping theorem’ or a result which is needed to prove a theorem. A corollary is a result which follows directly from a theorem. Less important theorems are sometimes called propositions. A conjecture is a statement that is being proposed to be true. Once a proof of a ...
... A lemma is a ‘helping theorem’ or a result which is needed to prove a theorem. A corollary is a result which follows directly from a theorem. Less important theorems are sometimes called propositions. A conjecture is a statement that is being proposed to be true. Once a proof of a ...
(A B) |– A
... 1. A, B are not formulas, but meta-symbols denoting any formula. Each axiom schema denotes an infinite class of formulas of a given form. If axioms were specified by concrete formulas, like 1. p (q p) 2. (p (q r)) ((p q) (p r)) 3. (q p) (p q) we would have to extend the set o ...
... 1. A, B are not formulas, but meta-symbols denoting any formula. Each axiom schema denotes an infinite class of formulas of a given form. If axioms were specified by concrete formulas, like 1. p (q p) 2. (p (q r)) ((p q) (p r)) 3. (q p) (p q) we would have to extend the set o ...
4 The Natural Numbers
... Suppose we want to show something of the form (S), which says (roughly, at least) that every natural number has property F. There are two cases to consider – F has a proper extension, or it doesn’t. If F has a proper extension, which is to say that the set {x:F[x]} is legitimate, then we need merely ...
... Suppose we want to show something of the form (S), which says (roughly, at least) that every natural number has property F. There are two cases to consider – F has a proper extension, or it doesn’t. If F has a proper extension, which is to say that the set {x:F[x]} is legitimate, then we need merely ...
Teach Yourself Logic 2017: A Study Guide
... Of course, those are just two possibilities from very many. This is not the place to discuss lots more options for elementary logic texts (indeed, I have not in recent years kept up with all of the seemingly never-ending flow of new alternatives). But despite that, I will mention here two other book ...
... Of course, those are just two possibilities from very many. This is not the place to discuss lots more options for elementary logic texts (indeed, I have not in recent years kept up with all of the seemingly never-ending flow of new alternatives). But despite that, I will mention here two other book ...
071 Embeddings
... it stands because this disjunction gives the 0 of the lattice. We must treat each member of the list ...
... it stands because this disjunction gives the 0 of the lattice. We must treat each member of the list ...
Logic and the Axiomatic Method
... We need some basic information about sets in order to study the logic and the axiomatic method. This is not a formal study of sets, but consists only of basic definitions and notation. Braces { and } are used to name or enumerate sets. The roster method for naming sets is simply t ...
... We need some basic information about sets in order to study the logic and the axiomatic method. This is not a formal study of sets, but consists only of basic definitions and notation. Braces { and } are used to name or enumerate sets. The roster method for naming sets is simply t ...
![overhead 7/conditional proof [ov]](http://s1.studyres.com/store/data/001382039_1-0b1da7da92f361d09e7b75df5e92d0f1-300x300.png)
overhead 7/conditional proof [ov]
... - the arrow and vertical line marking the scope of the assumption is called a SCOPE MARKER - note that on line 7., the justification includes the line numbers for the WHOLE SUBPROOF ...
... - the arrow and vertical line marking the scope of the assumption is called a SCOPE MARKER - note that on line 7., the justification includes the line numbers for the WHOLE SUBPROOF ...
Peano and Heyting Arithmetic
... This means that even though the underlying sets X and Y might be different, we can find a copy one of these orderings inside the other. In particular, this allows us to induce an ordering on well-orderings themselves: (X, ≺) is less than or equal to (Y, ≺0 ) if there is an order-preserving bijection ...
... This means that even though the underlying sets X and Y might be different, we can find a copy one of these orderings inside the other. In particular, this allows us to induce an ordering on well-orderings themselves: (X, ≺) is less than or equal to (Y, ≺0 ) if there is an order-preserving bijection ...
Logic Programming, Functional Programming, and Inductive
... In practice, though, this is rarely possible: logic procedure sets usually have less information content than the specifications to which they conform, even though they may be complete. ...
... In practice, though, this is rarely possible: logic procedure sets usually have less information content than the specifications to which they conform, even though they may be complete. ...
