Diagonalization Fixed Point Exercises

This is the second of three sets of fixed point exercises. The first post in this sequence is here, giving context.

  1. Recall Cantor’s diagonal argument for the uncountability of the real numbers. Apply the same technique to convince yourself than for any set , the cardinality of is less than the cardinality of the power set (i.e. there is no surjection from to ).

  2. Suppose that a nonempty set has a function from to which lacks fixed points (i.e. for all ). Convince yourself that there is no surjection from S to , for any nonempty . (We will write the set of functions from to either as or ; these are the same.)

  3. For nonempty and , suppose you are given a surjective function from the set to the set of functions from to , and let be a function from to itself. The previous result implies that there exists an in such that . Can you use your proof to describe in terms of and ?

  4. Given sets and , let denote the space of total computable functions from to . We say that a function from to is computable if and only if the corresponding function (given by is computable. Show that there is no surjective computable function from the set of all strings to .

  5. Show that the previous result implies that there is no computable function from which outputs if and only if the first input is a code for a Turing machine that halts when given the second input.

  6. Given topological spaces and , let be the space with the set of continuous functions from to as its underlying set, and with topology such that is continuous if and only if the corresponding function (given by ) is continuous, assuming such a space exists. Convince yourself that there is no space which continuously surjects onto , where is the circle.

  7. In your preferred programming language, write a quine, that is, a program whose output is a string equal to its own source code.

  8. Write a program that defines a function taking a string as input, and produces its output by applying to its source code. For example, if reverses the given string, then the program should outputs its source code backwards.

  9. Given two sets and of sentences, let be the set of all functions from to defined by substituting the Gödel number of a sentence in into a fixed formula. Let be the set of all sentences in the language of arithmetic with one unbounded universal quantifier and arbitrarily many bounded quantifiers, and let be the set of all formulas with one free variables of that same quantifier complexity. By representing syntax using arithmetic, it is possible to give a function that substitutes its second argument into its first argument. Pick some coding of formulas as natural numbers, where we denote the number coding for a formula as . Using this, show that for any formula , there is a formula such that .

  10. (Gödel’s second incompleteness theorem) In the set , there is a formula such that holds iff the sentence is not provable in Peano arithmetic. Using this, show that Peano arithmetic cannot prove its own consistency.

  11. (Löb’s theorem) More generally, the diagonal lemma states that for any formula with a single free variable, there is a formula such that, provably, . Now, suppose that Peano arithmetic proves that for some formula . Show that Peano arithmetic also proves itself. Some facts that you may need are that (a) when a sentence is provable, the sentence is itself provable, (b) Peano arithmetic proves this fact, that is, Peano arithmetic proves , for any sentence and (c) Peano arithmetic proves the fact that if and are provable, then is provable.

  12. (Tarski’s theorem) Show that there does not exist a formula with one free variable such that for each sentence , the statement holds.

  13. Looking back at all these exercises, think about the relationship between them.

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