W by the second condition. Hence by lemma 6. W forsomequordE. By the directed refinement theorem 2. W by theorem 6. Now by the directed reiinement theorem 2. If ii holds, then by theorem 6. By lemma 2. But then by corollary 3. PROOF by induction on n. Therefore, by the directed refinement theorem 2. In the latter case, by theorem 3. A co-ordinal quord A is said to be a principal.
Later on we shall show that A E3 - implies A is divisible by or divides Wwnfor every n. If A and B are isomorphic well-orderings then there is a unique isomorphism f such that every isomorphism between A and B is an extension off. We now prove a left cancellation law for co-ordinals using this classical theorem 8.
Later on we shall use the same technique to obtain a cancellation law for exponentiation for co-ordinals. Let p , be the map p with domain and range restricted to j ao,n :nENZ, then po is partial recursive. Further, if poj ao,x is defined, then its value is j u,, y for some y. Thus qo is partial recursive and one-one and also agrees with qe on C'B, i. Using theorem 8. Hence if w 5 vw, V and W are comparable by theorem 4. Otherwise, by lemma 8. Hence by the transitivity of Iand theorem 4. As in the classical case. Immediate from theorem 8. C and again using corollary 5. In the former case the assertion follows by the induction hypothesis.
Hence by theorem 8. Let p , be the map p with domain and range restricted to then p , is partial recursive. Further if is defined then its value is for some y. Now let q, be the map then clearly q, is partial recursive and agrees with qc on theorem 8. But by the definition of p,, any 2j oo. Thus we have shown that q, is a recursive isomorphism between B and C and the proof is complete.
PROOF as in the classical case. Hence it suffices, by lemma 4. Thus we see that the analogue of one of the classical laws for exponentiation breaks down in a very similar way to one of the multiplicative laws theorem 8. However, the similarity also extends to the cases where the analogues do go over. The assertion is trivial if A , B I 1. Otherwise, if E is a principal number for exponentiation, then A 8. By the transitivity of 5 and lemma 8. Now, classically, for ordinals a, fl, y, hence 8.
Hence by the classical theorem for ordinals ay we have IAI and hence A 8. We restrict our attention from now until the end of Part One of this monograph to co-ordinals, returning to the consideration of quords in Part Two. Since we are dealing with co-ordinals we shall repeatedly use the fact theorem 4. A co-ordinal P is a principal number for addition if, and only if, A 9. If A is a principal number for addition then the theorem follows at once from theorem 9.
We show that P, is a principal number for addition. Therefore PIis a principal number by theorem 8. This contradiction shows that Pz 2 PI and we conclude that A has a decomposition of the form 2. Qi Consequently, we have Q,. By theorem 2. Wis a principal number for addition. From theorem 4. The implication from right to left follows at once from corollary 5. By lemma 9.
Suppose the former holds, then by theorem 8. Thus for all A satisfying the hypothesis we obtain D, Q, R uniquely as required. As an immediate consequence we obtain our main theorem of this section. If O Ch. Shortly we shall show that this latter result holds for all infinite denumerable ordinals. However, we shall go on to give criteria for some collections of co-ordinals to contain precisely one representative for each member of a given collection of classical ordinals.
In this chapter we shall give simple criteria for sufficiently small well-orderings to be 'natural' in the sense that if two well-orderings of the same classical type are natural then they are recursively isomorphic. Clearly, it is sufficient to show that the co-ordinal associated with such a natural wellordering is uniquely determined. In fact, it will turn out that natural wellorderings are recursively enumerable. Here we shall only deal with addition and multiplication and shall treat exponentiation in chapter The results can be extended further by considering additional functions see e.
Immediate from theorems 4. We have the following strengthening of this result. If a is any infinite denumerable ordinal then there exist c mutually incomparable co-ordinals of ordinal a. Let B be a fixed co-ordinal of ordinal b. Such a co-ordinal exists since there exists a wellordering of type b and hence a well-ordering embeddable in R; though, of course, the embedding may not be partial recursive. Let V,, V2 be two incomparable co-ordinals of ordinal o then by corollary 5. Since, by theorem 5. By theorem 9.
In the same way one proves the following corollary; the details are left to the reader. W" By corollary By lemma Corollary However, not every principal number for addition is of the form W Aas we shall show later in chapter Every principal number for multiplication is a principal number for addition, but not conversely.
Now by lemma 7. The converse is false since W z is a principal number for addition by corollary Immediate from lemma The implications from right to left are obvious. Now suppose P ,. W by theorem 5. For any co-ordinal A , W W Ais a principal number for multiplication. Immediate from theorem That W is a principal number for multiplication follows at once from theorem 6.
The lemma now follows by induction. In exactly similar fashion one proves the following corollary; we leave the details to the reader. However, not every principal number for multiplication is of the form W w A. For every denumerable ordinal ct there is a principal number for addition for multiplication P such that IP I 2 u. Immediate from corollaries Later chapter 12 we shall strengthen these results and prove that there exist paths closed under addition and multiplication and exponentiation which are of length K, i.
As an immediate application of theorem If 1 As a corollary we get the result that the collection of principal numbers for exponentiation is strictly E,-unique. X exp cX. Suppose PEX exp , then by lemma 8. If A is a well-ordering of type a, then E A is a wellordering of type E,. That cp is one-one is clear from the uniqueness of the Cantor normal form.
By theorems 2. Let q be the partial recursive function defined by q e m i By transfinite induction as in the proof of theorem Similar to the above: left to the reader. A co-ordinal of the form E A is said to be an E-number. We first sketch the idea of the proof. Given such an expression then 1x1 1 is obtained by adding 2' and reducing the result so that we again get a unique expression. Clearly this addition can only affect the finite exponents and so the problem is to determine explicitly and effectively how "far in" the 2' has an effect on the exponents.
Using these two functions it easily follows that X N W - D where D is obtained from the numbers which do not represent successor numbers in X. Using a related sort of argument to that used to prove lemma Let and let q be the partial recursive function defined by and undefined otherwise. We prove this by induction on the maximum number n of applications of the cases 1 - 4 above in the computation of the q xi necessary to compute q x. The only difficulty is if cases 2 , 3 conflict. But e! Moreover, the xi are uniquely determined by the induction hypothesis. Suppose AEA, then let f be the partial recursive function defined by This requires a number of lemmata.
By lemma 7. This follows at once from corollaries By theorem Let A, B be as in the previous definition; then a pair Ch. Note that since we require p to be a partial recursive function of both variables the recursive isomorphisms in Recursive isomorphism of S. If A is a sequence well-ordering then d is said to be a sequence co-ordinal. In dealing with recursive isomorphism pairs we need not have assumed that Ixp i, x was one-one for all i and not just for the iEc'C where A , B , p , q are as in definition If p , q : A N B except that condition But this condition being satisfied for a recursive isomorphism pair p', q means that this pair determines a r.
Let us define, for A a S. We leave the reader to check that C p , 9 satisfies iii , iv of dehition 1. Theorem We shall say that a C. There are two ways at least! Pointwise addition seems not to be very useful so we only treat the other kind of addition. The pure theory of addition of C.
By hypothesis there exist p, q:A and r. Then the reader will readily complete the proof. We shall omit proof of most of these analogues: the essential difference is already contained in the proof of theorem There the definition of u mirrors exactly the definition of u.
We shall not need refinements until we deal with theorems which essentially involve at least one of the Ai. In this section we state some of the theorems on addition of C. We conclude this section with a theorem which is not obtained in the same way as the above but is useful. In TARSKI , infinite sums are considered and although these do not hold in full generality we can make translations of them using Z. Clearly, d iis well-defined if C is given. However, when we do not specify the C it will be clear either how C is to be chosen or that the assertion is purely existential and therefore independent of the particular choice of C from the various possibilities.
This is the case in the next theorem and lemma. Before we prove this theorem we prove the following: By the separation lemma 2. That Do D' then follows from lemma 2. We now have two cases to consider. In case i it is readily verified cf. Then we leave the reader to check again compare the proof of 2. This completes the proof of the lemma and theorem I50 [Ch. If B IE d , and each or equivalently, some A E d contains no zeros then there is an dosuch that for some dl The results on bounds apply almost without exception puce the mention of classical ordinals to quords and details of those results which can be readily adapted to the present framework may be found in CROSSLEY A co-ordinal A is said to be an upper bound for a collection of co-ordinals d if BEd implies B By the anti-symmetry of least upper bounds are unique if they exist at all.
Also, it is clear that all least upper bounds are minimal, though the converse is certainly not true as we shall show. In fact we shall prove that all least upper bounds are trivial and determine when minimal upper bounds exist. Further we shall show that if one minimal bound exists then so do 2 " O mutually incomparable minimal bounds.
Constructive Type Theory
We can define lower bounds in the obvious way but we have no interesting results concerning them cf. If a collection of co-ordinals d has an upper bound then any two members of the collection are comparable. A collection of co-ordinals has a least upper bound which is a co-ordinal only if it is countable. Suppose rd is a collection of co-ordinals with least upper bound A then IAl is a countable ordinal, call it a. Hence I I is a one-one function from d into the ordinals - Clearly s is partial recursive and one-one everywhere it is defined. Thus, as required, we have s: C N D.
We claim that B w and all the B ' are minimal upper bounds for d. There exist 2' long paths. Let Pcu be constructed exactly as in the proof of lemma We now show that PCU E a? The former assertion follows from the latter since C. C'W has 2N0infinite, and therefore cofinal, subsets. Let U be such a set with the ordering induced by W. We repeat the construction of theorem We choose the B l s see the proof of lemma Clearly this can be done whilst still ensuring B,G R; for example, we can take B: as in the n proof of lemma It follows that u : W-U which is a contradiction.
We leave the reader to check that each such W Ais a minimal upper bound for d. Throughout this chapter we shall give references to the order analogues in this monograph and when the proofs required here differ only slightly or not at all from those in the order analogue cases we shall omit them leaving the reader to look them up and make the necessary amendments. Since some of these will be references to the next two chapters the reader who is not familiar with isols and R. So the R. Definition However, since Sp and pp are r. A set A is said to be isolated if it is finite or immune. If A e B and finite finite then B is immune immune Ais 1 The following statements are equivalent.
Every infinite r. Then g is clearly partial recursive and everywhere defined since S is infinite. Therefore g is recursive and enumerates a subset T of S in order of magnitude so T is a recursive set. The other implications are trivial. We can define cardinal addition of R. As in chapter 2 we can show that any two R. We also have a separation lemma and refinement theorem but although the separation lemma is virtually identical with that for C.
This theorem justifies definition The proof of this theorem is rather more complicated than that of lemma 6. Then there exist disjoint r. In fact, sincef" y is defined for all n ifyeB and since f i s one-one x, s always exist it is clear that j s , x eR. Iis reflexive, antisymmetric and transitive on R. Reflexivity and transitivity are clear. The following statements are equivalent for an R.
Clearly the negation of ii implies the negation of i. Consequently there exist disjoint r. Restrict domain of p to B, u C1 and range of p to B , u D,. We leave the reader to fill in the details to show that the map, q, defined below is such that q:C-D. The above theorem illustrates the significance of the R. A is said to be an is01 if it satisfies any of the conditions in theorem An is01 contains a recursive set if, and only if, it is finite. There are in general KO partial recursive, one-one maps from an infinite isolated set onto itself - just exchange two points in the set - but we shall show in the next chapters theorems However, we cannot map isolated sets into proper subsets of themselves so isols are effective analogues of Dedekind finite numbers.
If A is finite the lemma is trivially true. The proof is virtually given as the proof of theorem However, we shall not prove any theorems about exponentiation of R. As in theorem We now turn to cancellation laws for isols. We shall not give proofs for these are, in general, of three kinds: 1 for theorem In the theorems below all R. The following statements are false. There exists an is01 X and R.
We state only a very weak version which parallels our theorem If P X is a function of X alone constructed by finite composition from functions of the forms x f A , X. The question naturally arises as to what are the order analogues of isols? Originally, Kreisel proposed quords to the author, but PARIKH , showed that quords are not closed under exponentiation and it was not until the Leicester Logic Colloquium in that Nerode and the author came to the conclusion that what are the most natural analogues are IosoIs.
We shall introduce these in the next chapter. However, there is a larger class of C. Classically, finite ordered sets may be defined as linearly ordered sets which contain no descending or ascending chains. So we now consider cf. As in chapter 3, for sets embeddable in R by a recursive isomorphism, this condition is equivalent to every non-empty subset having a minimum and a maximum element. Immediate from theorem 3. The following statements are equivalentfor a C. Suppose A is not quasi-finite.
Definition of 'Constructive Dismissal'
However, there are recursive quasifinite linear orderings. This result, proved below as theorem We recall that a C. If A is quasi-finite and r. Clearly we may assume without loss of generality that A E R. Now by the proof of theorem 3. Hence the order type of A is as stated in the theorem. We first sketch the main lines of the proof. Since R has type 1 q we can identify the elements of Seq which are non-zero with rational numbers and we can identify real numbers with Dedekind sections of R. For the proof we first construct a real number A such that neither the upper nor the lower Dedekind We then show that this real number A is the limit of a recursive sequence of rationals in fact it is the unique limit.
Finally we show that the intersection of this sequence with the lower upper Dedekind section of A cannot contain a recursive ascending descending chain. We now proceed to the details. Let wo, wl, Since Seq is a recursive predicate such an enumeration is readily obtained from any of the usual enumerations of sets of natural numbers, cf.
We shall also use the notation Ei, 17,,to express that a predicate is expressible in prenex form with i alternations of number quantifiers the first being existential, universal. XEO, isin El- U l. A predicate is in A, if, and only if, it is recursive in predicates in El or ITl. Main construction. We add the 1at the end simply in order to avoid violating the definition of sequence number.
Stage 1. Suppose a,, For each n, a,, is eventually determined and no a,, is ever changed once determined. If a,, Thus a,, The first part of the lemma now follows by induction and the second assertion is trivial. A,, A,, are not r. If A, is r. Thus o,, contains three distinct sequence numbers u say. Similarly A,, is not r. Now the procedure for constructing A is clearly recursive in El hence by lemma Since A,, A,, are not r,e.
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For if C had a recursive ascending chain then C would have a limit point distinct from A. But if 2 2 1 then C would have a limit point distinct from A which we have shown is impossible. Recursive Quasi-finite C. Recursive Quords are closed under addition and multiplication so the theorem follows from definition There exist No recursive quasi-finite C.
Since C'A is infinite such a B exists. B is clearly a r. But if B contained a recursive ascending or descending chain then so would A hence 6 is r. But B A by theorem By theorems 3. It is not a linear ordering or a partial well-ordering. The former follows from theorem The other properties are obvious.
Constructive Order Types
By the proof of theorem We now turn to losols which we consider are nice and reasonable. This chapter is devoted to examining their basic properties: some of the proofs become very simple but with more complicated proofs we obtain some more far reaching cancellation laws. However, we leave over the general problem of the theory analogous to the Myhill-Nerode theory of combinatorial functions to another occasion.
Aczel has shown that the obvious analogue is not fruitful, basically because there is no obvious correspondence between numbers and order types but Nerode and the author have now found methods of circumventing these difficulties. Since losols are special cases of C. We do, however, stress again that all linear orderings we consider are to be embeddablein R by means of one-one, partial recursive, order preserving maps. See theorem The converse is false. In theorem A is said to be a losol if it contains an isolated linear ordering.
It has the following motivations: it suggests linearly ordered isol, which is not strictly accurate but losols are C. A recursive C. A is said to be an initial middle, final segment of A is said to be a weak predecessor of B if A E B. The first part is obvious and the third follows from the second. Since A is infinite any AEA contains countably infinitely many elements in its field.
By part i of the present theorem and theorem Thus there are exactly KO distinct initial segments of A. As we have argued many times before these subsets give rise to 2"O C. If A is a quord and A. Such a set exists by the classical version of theorem 1. Further, since A, is countable let a,, a,, Let cpo, pi, q2, We could restrict ourselves to order preserving cp to prove the theorem but this is clearly not necessary.
We now define a full binary tree whose nodes lie in Seq'. The A, will be the branchesof this treeand i willbe satisfied automatically. We define our tree in stages. Bourbaki, N. Meldrum trans. Bridges, D. Cederquist, J. Constable, R. Coquand, T. Zalta ed.
2. The Constructive Interpretation of Logic
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Schuster eds. Johnstone, P. Joyal, A. Julian, W. Kushner, B. Lietz, P. Loeb, I. Lombardi, H. Lorenzen, P. Lubarsky, R. Markov, A. Istituta imeni V. Marquis, J. Rose and J. Shepherdson eds. Jonathan Cohen ed. Menger, K. Mines, R. Moerdijk, I. Moore, G. Myhill, J. Mathias and H. Rogers eds. Naimpally, S. Palmgren, E. Palmgren, K. Segerberg, and V. Stoltenberg-Hansen eds. Petrakis, I. Picado, J. Richman, F. Riesz, F. Sambin, G. Skordev ed. Schuster, P. Schwichtenberg, H. Simpson, S. Specker, E. Steinke, T. Troelstra, A. Barwise ed. Brouwer Volume 1 , Oxford: Clarendon Press. Brouwer Volume 2 , Oxford: Clarendon Press.
Vickers, S. Waaldijk, F. If an unwritten change occurs, which would have been enforceable as a directed change under the contract had it been in writing, then it is a constructive change. Conduct of the Owner's representative that has the effect of requiring a Contractor to perform work in a manner different from that stipulated under the original contract may also be a constructive change. If the constructive change is instigated, but is not formalized in writing as a directed change, how then does the change occur?
The constructive change need not be a direct order. In fact, constructive change is frequently the indirect result of the conduct or actions of the Owner's representatives, such as:. Types of Constructive Change Constructive change may be difficult to identify and may involve controversy over the interpretation of the Contract documents. We list below several types of constructive changes that have generally been recognized in the industry, with a brief definition of each.
Constructive Acceleration is a constructive change created by the following series of events:. Deficient and Defective Contract Documents produce a constructive change when errors or omissions in the Plans and Specifications lead to changes in work scope or methods that could not have been anticipated from the contract; or, when the Contractor performs the work in accordance with the Plans and Specifications but cannot produce the desired result, despite his conformance with the Plans and Specifications. Interference and Disruption cause a constructive change that arises from acts or omissions of the Owner that interfere with the Contractor's schedule or methods of work.
Owner-Furnished Items lead to a constructive change when material or equipment that the Owner furnishes to the Contractor is late, defective, or different in nature from that which was anticipated. Superior Knowledge and Misrepresentation produce a constructive change that occurs when the Owner fails to disclose information to the Contractor that would have influenced the Contractor's bid price or planned method of operation.
Differing Site Conditions can cause a constructive change that arises from a variation in physical conditions which, had it been known before bid, would have materially affected the Contractor's bid price or planned method of operation.