The best answers are voted up and rise to the top. Home Questions Tags Users Unanswered. Recommendations for reading in Constructive Mathematics Ask Question. Asked 5 years, 6 months ago.
- Foundations of Constructive Mathematics.
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Active 5 years, 6 months ago. Viewed times. Mathmo Mathmo 1 1 silver badge 6 6 bronze badges. To get a modern idea I strongly recommend to read the introduction of the HoTT book: ncatlab. I second the HoTT suggestion.
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HunanRostomyan - there is an english translation published by American Mathematical Society in All of them deal with the "mathematical side" and not with the philosophical. About this one, see : Michael Dummett, Elements of Intuitionism 2nd ed, The SEP article on Constructive Mathematics is written by Douglas Bridges, who is a very reputable source on the current state of the field.
He's also given a good introductory lecture on the subject, with slides available at masfak. Mauro Many thanks - the SEP articles seem to be very thorough. Sign up or log in Sign up using Google. Sign up using Facebook. Alternatively, we could have allowed multiple outputs as long as the witness follows through and is accountable for each output. Any such witness can be converted into a witness with unique outputs.
The semantic content of an input-output pair for a statement is the assertion in classical logic that the pair is correct. Equivalently, witnesses can be described in computational terms. Witnesses are given as a blackbox or an oracle and need not be constructive. When given a witness, you may give it input and will receive appropriate output; you will get no other information. Multiple instances of a witness can be run, and any instance can be copied.
When requested a witness as input to a witness , you simply accept input and provide appropriate output; you may give false output, but that may cause the instance of the witness to enter an infinite loop or give arbitrary output. By running multiple instances of a witness W for A, we obtain the response tree for W, from which we can obtain W as a real number. Note that W has to meet the requirements for every path through the tree. The response tree specifies the behavior for every possible input. For a different notion, see Section 5 Narrow Constructive Truth.
Constructive truth for intuitionistic analysis is defined in Section 6.
Recursive analysis can be interpreted in arithmetic and is not considered separately in this paper. The requirement on witnesses for implication is precisely such that key properties of implication are constructively true. Because the whitespace in witnesses can absorb time complexity, every constructively true statement technically has a polynomial time computable witness.
The transformations on witnesses corresponding to the connectives and quantifiers are also recursive and hence polynomial-time computable. Like in the classical logic, every occurrence of a subformula is either positive or negative. Every statement A has the associated set of witnesses S A. Conjunction and disjunction are not strictly necessary since with some modifications to the formula, they may be replaced by the universal and existential quantifier, respectively.
Constructive Mathematics without Choice
Recursive realizability, or just realizability, is like constructive truth except that witnesses must be given as standard codes for partial recursive functions. A statement is realizable iff it has a witness in this sense. The set of realizable sentences is a consistent completion of some constructive formal systems; it has the same Turing degree as the truth predicate for arithmetic.
Intuitionistic predicate logic, Heyting arithmetic, and Markov's principle are realizable. Some realizable statements are false; an example is "not every partial recursive function is either defined or undefined at zero". Extended Church Thesis is the realizable assertion actually, a schema that truth equals realizability. A constructive proof provides a number and demonstrates that the number codes a constructive witness.
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However, even when the constructive proof is short, an actual computation of say which disjunct is true may be unfeasibly long. A true constructive formal system is a formal system in which only constructively true statements are derivable, and in which derivations can be constructively converted into witnesses for constructive truth. When one is more interested in the witnesses than in the truth, using a constructive formal system may be a good way to manage complexity. For example, constructive formal systems have applications in computer science since algorithms can be "read off" constructive proofs.
When one is more interested in realizability than in truth, a constructive formal system that proves false statements but does not prove unrealizable statements may be useful.
However, the savings have to be balanced against the difficulty of not using some logical truths; and to avoid contradiction, one must always note when "A" is used as a shorthand for "A is realizable". Constructive truth acts as a necessity operator in intuitionistic modal logic.
Constructive arithmetical truth satisfies the theory. In the above, we have assumed that every element is constructive and given as a construction ; see Section 6 for a treatment of non-constructive elements. However, intuitionistic logic has natural extensions. However, this principle can fail if the domain consists of say codes for total recursive functions or if we impose time limits on computations.
Such x exists by standard results in recursion theory.
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For example, every real number Cohen generic over recursive sets can be split into infinitely many mutually generic predicates x n. Let T be a recursive tree of integer sequences under extension. If T is well-founded, then we simply wait until incompatible s and t are given, and then copy the witnesses for P s and P t. If T is not well-founded, then a "bad" witness for the antecedent will start to provide an infinite path through T and then once we commit to incompatible s and t, provide witnesses for every P except for P s or for P t.
Proof Complete. Then, fulfill all requests for witnesses for P m as witnesses become available. By induction on the rank of n, for every n, a witness for P n will be provided. For a witness S, every such combination included must be true. The same idea is used to prove the next theorem. Proof: Let T be a recursive tree of integer sequences under extension. If T is well-founded, then the antecedent has a recursive witness, and hence the full statement is not constructively true. Now, that statement is not a tautology and hence witnesses S n. In contrast to theorems 1 and 4, we have the following theorem, which shows that in terms of the number of implications, the complexity of the formulas used in the proofs of the theorems above is optimal.
Negations do not count as implications. The above formula is arithmetical, which completes the proof. We conjecture constructive arithmetical truth to be complete for second order arithmetic and have the following partial results:. Proposition 6: The relation of being a witness is complete for second order arithmetic. Use the witness for the antecedent to guess an infinite descending path, and when that fails, give a witness to the consequent. Proof: We build on the proofs of Theorem 4 and Proposition 6.
Now, consider a witness X to B. Every witness to A will have to respond to every infinite path through T corresponding to X. A "bad" witness Y for A will require correct answers to P and will delay giving away incompatible elements until the path used as input into A is past the point where some P 2n is not given by X, and then give out P s and P t for incompatible s and t that are independent of the path.
For every witness Z to D, the input-output pair corresponding to X and Y is not tautological. Thus, a non-tautological input-output pair can be extracted from Z through complete search, which completes the proof. We expect that the proof of Theorem 7 can be generalized to prove that constructive arithmetical truth is complete for second order arithmetic. However, there are combinatorial difficulties in actually stating a correct C n , so we do not know how to prove the conjecture. Narrow constructive truth is a particular modification of constructive truth in which witnesses are not completely given.
A witness for narrow constructive truth provides a code for a recursive interactive function such that for every possible interaction, the witness requirements are met. The meeting of requirements is defined inductively on the complexity of the formula. If A is false, then the requirements for A are immediately violated. A new instance may copy the input and output but not the internal state from any point of an existing instance it is unclear if omitting this provision would change which statements are narrow constructively true. For other connectives, the meeting of requirements is as usual.
Failure to choose violates the requirements at infinity but not at any finite time. Narrow constructive truth is similar to constructive truth, with the difference limited to statements with implications inside antecedents. Narrow constructive truth implies constructive truth for statements without antecedents inside antecedents inside antecedents, and perhaps for all arithmetical statements.
The meeting of requirements is hyperarithmetically definable. The reasoning can be used to compute the expressive power of the full language, which allows nesting of the narrow constructive truth operator. A hyperjump of X is essentially the set of standard codes for well-founded relations that are recursive in X.
The proof is by induction on n , and is analogous to the proof of Theorem 8. Before defining constructive analysis, we note that analysis can be interpreted in second order arithmetic. Analysis deals not with arbitrary functions but continuous and other reasonable functions, and these can be represented by real numbers.
Thus, analysis can essentially be interpreted through constructive truth in arithmetic. Philosophically, one can argue that natural numbers and constructive arithmetical truth are basic, so the high complexity of constructive arithmetical truth provides an additional ontological basis for analysis. We use the language of second order arithmetic. Capital letters represent sets of natural numbers.