Stewart Shapiro. Colin McLarty. Andre Nies. Byunghan Kim. Michael Dunn. John P.
Giovanni Sambin. Laura Crosilla. Peter Clote. Grzegorz Malinowski. Erik Sandewall. Michael Hallett. Alexander Chagrov. Roman Kossak. Home Contact us Help Free delivery worldwide. Free delivery worldwide. Bestselling Series. Harry Potter. Popular Features. New Releases. Categories: Mathematical Logic. The Structure of Models of Peano Arithmetic. Description Aimed at graduate students and research logicians and mathematicians, this much-awaited text covers over forty years of work on relative classification theory for non-standard models of arithmetic.
With graded exercises at the end of each chapter, the book covers basic isomorphism invariants: families of types realized in a model, lattices of elementary substructures and automorphism groups. Many results involve applications of the powerful technique of minimal types due to Haim Gaifman, and some of the results are classical but have never been published in a book form before. Other books in this series. Category Theory Steve Awodey. Add to basket. Goedel's Incompleteness Theorems Raymond M. Foundations without Foundationalism Stewart Shapiro.
Recursion Theory for Metamathematics Raymond M. Computability and Randomness Andre Nies. Simplicity Theory Byunghan Kim. A little number theory then suffices to code sequences of numbers by single numbers. Consequently, well-formed formulas, as sequences of primitive symbols, are each assigned a unique number. Finally derivations, or proofs, of the system, being sequences of formulas, are arithmetized, and are also assigned specific numbers.
In this way, all the syntactic properties and operations can be simulated at the level of numbers, and moreover they are strongly representable in all theories which contain Q. Let us denote the formula which strongly represents this relation in F itself as Prf F x , y. Let us abbreviate this formalized provability predicate as Prov F x.
It follows that the latter is weakly representable though, it turns out, not strongly :. It has many important applications beyond the incompleteness theorems.
The Structure of Models of Peano Arithmetic - CERN Document Server
Such figures of speech may be heuristically useful, but they are also easily misleading and suggest too much. Thus, it can be shown, even inside F , that G F is true if and only if it is not provable in F. It is not difficult to show that G F is neither provable nor disprovable in F , if F only is 1-consistent.
For the first half, assume that G F were provable. However, because F in fact also proves the equivalence G , i. But this would mean that F is inconsistent. In sum, if F is consistent, then G F is not provable in F. For this first half, the assumption of the simple consistency of F suffices.
Then F cannot prove G F , for otherwise F would be simply inconsistent. In fact, in favourable circumstances, it can be shown that G F is true, provided that F is indeed consistent. This, however, would contradict the incompleteness theorem. Therefore, G F cannot be false, and must be true. This is why it is important to include the subscript F in G F. Such imprecise statements, however, should be taken at least with a grain of salt.
In , J. As it happens, if the formal system F under consideration is indeed consistent, Rosser's provability predicate is co-extensional with the ordinary provability predicate. It is illuminating to reflect on the first incompleteness theorem also from the model theoretic perspective—though the theorem itself does not in any way require this. Therefore, no natural number n can witness the formula.
Non-standard models have since then become a rich research area in mathematical logic see, e. Informally, the reasoning leading to the second incompleteness theorem is relatively simple. Let us abbreviate this formula by Cons F. The proof of the first part of the first incompleteness theorem i. This gives:. Consequently, Cons F cannot be provable in F either. But does this sentence really express that F is consistent?
Compare this with the remark above that G F does not, strictly speaking, express its own unprovability. Furthermore, might there not be other sentences which are provable and also express the consistency of F? Giving a rigorous proof of the second theorem in a more general form that covers all such sentences, however, has turned out to be very complicated. The basic reason for this is that, unlike in the first theorem, not just any, merely extensionally adequate provability predicate works for the formalization of the consistency claim.
The manner of presentation makes all the difference. One must thus add some further conditions for the provability predicate in order for the proof of the second incompleteness theorem to go through. Following Feferman , it is customary to say that whereas the first theorem and its relatives are extensional results, the second theorem is intensional : it must be possible to think that Cons F in some sense expresses the consistency of F —that it really means that F is consistent.
The proof of the second incompleteness theorem requires that the provability predicate in F satisfies a number of conditions which are used in the details of the proof.
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There are several different sets of conditions that will do. It uses a rather awkward set of conditions for the provability predicate. D1 is simply a restatement of the requirement from the proof of the first theorem that provability is weakly representable. Roughly put, D2 requires that the whole demonstration of D1 , for the candidate provability predicate Prov F , can itself be formalized inside F.
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Finally, D3 requires that the provability predicate is closed under Modus Ponens. If the arithmetized provability predicate indeed satisfies these conditions, the second theorem can be proved. It is not too difficult to show, using the derivability conditions, that:. This immediately yields the unprovability of Cons F , given the first incompleteness theorem.
Furthermore, Jeroslow demonstrated, with an ingenious trick, that it is in fact possible to establish the second theorem without D3. However, in some other cases e. Under the assumption that a provability predicate for a theory satisfies the derivability conditions or, by Jeroslow's trick, at least D1 and D2 it is relatively easy to prove the relevant case of the second incompleteness theorem. However, in practise one has to establish whether a proposed arithmetized provability predicate really satisfies the conditions case by case, and typically this is long and tedious.
This drawback, among other things see Feferman , led Solomon Feferman in the late s to look for an alternative line of attack to the second theorem see Feferman Feferman approaches the issue in two steps: First, he isolates the formulas Prov FOL x which arithmetize some standard notion of derivability in first-order logic in order to allow us to fix one chosen formula for provability in logic. How the set of non-logical axioms of the system at issue are presented is left open at this stage. Secondly, Feferman looks for a suitable constraint for presenting the axioms.
Among the formulas of the language of arithmetic, he isolates what he calls PR- and RE-formulas; the former correspond to the canonical primitive recursive PR definitions in arithmetic, and the latter to existential generalizations of the former.