1. ∞ Infinity Week 10: Formal Mathematics

2. Week 7

3. Formalization

4. Infinities in Maths

5. Hilbert’s Programme “Real” mathematics is given to us in (Kantian) intuition. Facts like 2+2=4 are really facts and we can know and prove that they’re true from first principles (a priori). Is all mathematics “real” in this sense? In particular, what about maths involving infinity? Perhaps this lies outside the bounds of knowledge, in the pure play of reason where antinomies are possible. Hilbert hopes not, though! The way to draw this boundary is to reconstruct as much mathematics as possible formally, with great rigor and care, out of elements like basic arithmetic that are undoubtedly part of our immediate intuition. To proceed rigorously we need axioms for each field of maths. But in the more exotic fields, how can we tell our axioms are well-formed, i.e. that they don’t give rise to a contradictory theory? We must prove it. Mathematics owes us a proof of its own consistency.

6. Hilbert’s Programme 1.Express statements about infinite objects as finite strings of symbols in a precisely-defined formal language. 2.Express rules of inference as precisely-defined operations on symbol strings. 3.Derive theorems from the axiom strings by finitely many applications of rules of inference. 4.Prove that the rules of inference produce no contradictory sentence, by purely finite reasoning about strings of symbols.

7. (Hilbert, 1922)

8. The Axiomatic Method In modern terminology, every statement in Euclid’s Elements belongs to one of the following: An axiom – that is, a definition or something taken to be basic. A rule of reasoning that allows you to make deductions. A theorem – that is, something claimed to be true. A proof – that is, an argument showing that the theorem necessarily follows from the axioms. The axioms are as minimal as possible. If you don’t agree with them, nothing in the Elements works for you. A valid proof shows why you’re forced to accept the theorem if you accept the axioms and follow the rules of reasoning. The axioms and rules should therefore be “intuitively true” in some sense. A lot of high-powered C18 and C19 mathematics has abandoned this approach. It sets a very high standard but appears to be too difficult (or at least too slow) for advanced problems.

9. Contradiction A theory is consistent if it doesn’t prove any contradictions. If you can produce a proof of “P” and of “not P”, the theory is inconsistent. Assumption: if a theory is inconsistent, it has no model. That is, there are no “real” contradictions. It follows that if a theory demonstrably has a model, it’s consistent. Ex falso quodlibet It’s a feature of classical logic that if you can prove a contradiction, you can use that contradiction to prove any other statement you like, even if the two have nothing to do with each other.

10. Suppose we, who are working in a formalized language, have a theory that we interpret as being about sea creatures. Suppose this theory can prove “Whales are fish” and also, by a different line of argument, “Whales are not fish”. We may now prove “Humans are fish” as follows: “Whales are fish” is true, so “Either whales are fish or humans are fish” is also true. This is because “either X or Y” is true if X is true, regardless of whether Y is true. But “Whales are not fish” Well, “Either whales are fish or humans are fish” and “Whales are not fish”. So it must be that humans are fish! This is because when “either X or Y” is true and X is false, Y must be true. Nobody thinks this is a good proof that humans are fish. But it’s not so easy to point to one step that causes the trouble.

11. Formalized Arithmetic If we’re going to express mathematics formally, that had better include the basic system of whole-number arithmetic before trying to do all the fancy bits. This just allows us to do addition and multiplication. But it already includes non-trivial things like Goldbach’s Conjecture. There are various ways to capture basic arithmetic formally using axioms. Peano arithmetic Russell & Whitehead’s Principia Mathematica

12. Peano Arithmetic If we’re going to express mathematics formally, that had better include the basic system of whole-number arithmetic before trying to do all the fancy bits. This already includes non-trivial things like Goldbach’s Conjecture. First-Order Peano Arithmetic gives a simple formal theory that “obviously” has the arithmetic of the natural numbers as a model. That is, its axioms are explicitly chosen for that purpose. Most of it is uncontroversial, but we do need to add some machinery to do induction, which is how we get to prove a statement like “every even number is the successor of an odd number”. Without induction, we can prove for any particular even number, that it’s the successor of an odd number; we just can’t prove it for all of them at once. This is a direct consequence of the fact that there are infinitely many of them, and it’s where the trouble starts.

13. Axioms of Peano Arithmetic Zero is a number. If A is a number, so is the successor of A, written S(A). Zero is not the successor of any number. If S(A) = S(B) then A = B Suppose X is a set of numbers such that (a) X contains 0 and (b) for every A in X, S(A) is also in X. Then every number is in X.

14. Completeness? Question: Can arithmetic be formalized in a way that’s complete? That is, does it in principle decide the truth or falsehood of every sentence it can express? This means: for anything it can express, it must either prove that it’s true or prove its negation is true. If so, we have a firm basis on which to build “real” mathematics. If not, we have a problem – not as bad as inconsistency, but pretty disastrous for Hilbert’s programme. This is the subject of Gödel’s first incompleteness theorem.

15. Consistency? If we use a formal approach to build the rest of “real” mathematics, we must be sure it’s consistent. Otherwise it can prove any absurdity we like, and we’re back in the realm of the free play of pure reason. What does it mean to “be sure”? We must be able to prove it, of course. But in doing so we can hardly appeal to something more fundamental than our ultimate foundation! So the question is: can a formalized theory prove its own consistency? In the case of set theory, this is the subject of Gödel’s second incompleteness theorem.

16. Set Theory Basics

17. Set Operations The UNION of two sets A and B contains everything that is in either A or B, or both. The INTERSECTION of A and B only contains things that are in both A and B. If A contains all the elements of B (and perhaps more), we say B is a SUBSET of A. If A is a set, the set of all subsets of A (including the empty set and A itself) is called the POWER SET of A.

18. Power Sets To make progress we need to introduce a little more set theory. Suppose we have a set called S. Any collection of some or all of its elements is called a subset of S. Then the power set of S is the collection of all its subsets. For reasons we won’t get into this is written as 2 S. If S is “the things for sale in a shop”, 2 S is “all the possible baskets of shopping I could end up with”. Example: Let S = {a, b, c}. Then 2 S is { {}, {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, {a, b, c} } Notice the 2 S includes the empty set {} and S itself.

19. Week 8

20. Naïve Approach to Sets & Maps Sets as collections The Principle of Comprehension A map associates two sets in a certain way Example: diners at a restaurant (the domain) and dishes on the menu (the range) – see next slide. Injection, surjection, bijection (see next slide) The Schröder-Bernstein Theorem We saw the idea of counting as “pairing” in a previous session. We take this as our official definition of cardinality, which captures precisely the intuitive notion of “how many elements are in a set”.

21. Bijection Examples

22. Modelling the Natural Numbers We want to build some objects in pure set theory that “look like” the natural numbers. We do this with the Von Neumann construction: 0 = {} S(X) = X U {X} Addition has a simple inductive definition: A + B is done by repeating the successor operation on A once for each element of B (or vice versa)

23. Frege’s Hope Using a model like this, we can rebuild Peano arithmetic using just the very simple idea of a set. We can express everything in a formal language, which we can analyse rigorously so we’re sure our statements are precise and our proofs are valid. Essentially, maths would be “solved” – it could be reduced to a computer algorithm. This promises the removal of all doubt about arithmetic. If we can achieve this, we could expect to be able to do it for more advanced mathematics. Who knows, maybe we can do it for other areas of life too!

24. Russell’s Paradox Naïve set theory includes the Principle of Comprehension. Anything is a set that can be described by a suitable formal language that can be interpreted as making consistent statements about sets and their elements. The PoC licenses us to declare: “S is the set of all sets that don’t contain themselves” S is a contradictory object; any theory that allows this sentence to be proved is absurd: ex falso quodlibet. Yet we don’t need much “machinery” to do it: just the ability to refer to sets and what they include.

25. Axiomatic Set Theory Zermelo-Frankel Set Theory (ZF) is expressed in a formal language (predicate logic). It’s usually stated with about 9 axioms – statements that define what set theory is. There’s a version of the ZF axioms on the next slide, for those who are interested. But many of the details aren’t terribly important for us. Note to those already in the know: we’ve left out the Axiom of Choice, so this really is ZF not ZFC. We might talk briefly about this in this session. The most interesting axioms for us are existential – that is, they declare that some set or other exists in the theory. Without any existential axioms, one model of set theory would be complete “nothingness”. The only things it could prove would be things that are also true of “nothingness”. Not very useful!

26. A Version of the ZF Axioms Extensionality: Two sets that have exactly the same elements are the same set. Regularity: All sets are well-founded. Replacement: This is an axiom schema that allows us to “build up sets from other sets” by saying things like “X is the set of all x such that…” without falling into Russell’s Paradox. You’ll sometimes see an “Axiom of Separation” as an alternative. Null set: there exists a set of which nothing is an element Unordered pairs: given sets x and y, there exists a set {x, y} Union: given sets x and y, x union y exists. Power set: given x, its power set exists. Infinity: A specific infinite set exists.

27. The Axiom of Infinity This axiom guarantees the existence of a set that can be interpreted as “the set of natural numbers”. There are various ways to build sets that can be interpreted as natural numbers – the details aren’t important. This axiom plays a similar role to induction in Peano Arithmetic. Without it we have finite set theory, which would appear to be a theory of the potential infinite only. It can say “this even number is the successor of an odd number” for any even number you choose. It can say “in this set of even numbers, every one is the successor of an odd number” for any set of numbers you like, however large. But it can’t say “every even number is the successor of an odd number”. The point to note is that we can’t get our first infinite set just by juggling with finite sets. We must “import” or “assert” it.

28. Cantor’s Theorem The cardinality of 2 S is always strictly greater than that of S. This is easy to see when S is finite: if S contains n elements, 2 S contains 2 n elements (which partly explains the notation), and we always have that 2 n > n. When S is infinite it’s harder to see. Proof: 2 S contains, effectively, a copy of S (the “singleton” sets), so it can obviously never be of smaller cardinality. The tricky bit is to prove that it can never be equal either. We do it by reductio ad absurdum, so suppose S is infinite and we have a bijection from S to 2 S (meaning their cardinalities are equal). This means we have a list, indexed by elements of S, of subsets of S. Define D, a subset of S, such that D contains every element of S that indexes a subset of S that doesn’t contain it. Well, D is in 2 S, so D must be somewhere on our list. But it can’t be! For which element of S would index it? Not an element in D, for then it wouldn’t be in D. And not an element not in D, for then it would be in D! This absurdity tells us our initial supposition was wrong, and no such bijection can exist.

29. Number Systems

30. The Set of Natural Numbers We can think of counting as making a numbered list, where the numbers are natural numbers. If we run out of things to count, the number of the last element of the list is how many things there were – the cardinality of the set. If we don’t, it seems we should have a natural number for every element. Then as a first approximation we’ll say the things we’re counting are “infinite”. If we can list the elements of a set in this way – associating a different natural number to each one – we say the set is “countable”. It seems at this point as if every set should be countable.

31. Counting Sets of Numbers Here are some sets of numbers we can try to count. Remember, this means making a list of all the elements of each set, with a natural number for each element. Or, think of it as writing a computer programme. For any input (element in the set) the programme should be able to output the list-item-number for that element. Some sets we’ve already seen: The even natural numbers: 2, 4, 6, 8, 10, 12, … The square natural numbers: 1, 4, 9, 16, 25, 36, … The prime numbers: 2, 3, 5, 7, 11, 13, 17, 19, … More tricky ones: The integers (positive, negative and zero) The rationals (fractions) Binary sequences

32. Know Your Number System!

33. The Real Numbers

34. A Definition of the Reals

35. The Real Numbers

36. Cardinality of the Reals

37. Appendix: More on Formal Systems

38. Formal Languages A formal language isn’t much like a “natural” language like English or French. Nor does it look like the kind of mathematics human beings do. It looks more like the kind of code a computer might use. It consists of: a restricted set of symbols; A set of rules for combining them to make expressions; Another set of rules for combining expressions to make proofs.

39. Syntax vs Semantics What we’ve just described is syntax: mere symbol-shuffling. We like to think there’s another layer: the meaning of the symbols. This is their semantics. Without this, a formal language is just a kind of abstract game. To matter to us, it must be about something. Still, we can use it to state (purely abstract) axioms and proofs of theorems. Given a set of formally-encoded axioms, the set of all the theorems we can prove from them is called their theory. This is captured more formally by the idea of an interpretation of the pure symbolic syntax. We find a model – something outside the theory that it’s suppposed to be “about”. Then we map the symbols of the language, one by one, onto features of the model.

40. Propositional Logic This formal language uses letters to represent propositions, such as “Whales are fish”. The built-in symbols, besides letters, are ¬ (“not”), | (“or”) and & (“and”), along with parentheses “(“ and “)”. The rules for forming expressions are: Every letter on its own is an expression If X is an expression, so is ¬(X) If X and Y are expressions, so is (X | Y) If X and Y are expressions, so is (X & Y) Nothing else is an expression We then need a set of rules for combining expressions into proofs. These are usually pretty short but in the interests of space we won’t set this out here. Here is the proof we just did on the previous slide: P (P | Q) ¬(P) (¬(P) & (P | Q)) Q

41. Predicate Logic

42. A Version of the ZF Axioms Extensionality: Two sets that have exactly the same elements are the same set. Regularity: All sets are well-founded. Replacement: This is an axiom schema that allows us to “build up sets from other sets” by saying things like “X is the set of all x such that…” without falling into Russell’s Paradox. You’ll sometimes see an “Axiom of Separation” as an alternative. Null set: there exists a set of which nothing is an element Unordered pairs: given sets x and y, there exists a set {x, y} Union: given sets x and y, x union y exists. Power set: given x, its power set exists. Infinity: A specific infinite set exists.

43. The Axiom of Infinity This axiom guarantees the existence of a set that can be interpreted as “the set of natural numbers”. There are various ways to build sets that can be interpreted as natural numbers – the details aren’t important. This axiom plays a similar role to induction in Peano Arithmetic. Without it we have finite set theory, which would appear to be a theory of the potential infinite only. It can say “this even number is the successor of an odd number” for any even number you choose. It can say “in this set of even numbers, every one is the successor of an odd number” for any set of numbers you like, however large. But it can’t say “every even number is the successor of an odd number”. The point to note is that we can’t get our first infinite set just by juggling with finite sets. We must “import” or “assert” it.

44. Axiomatic Set Theory Zermelo-Frankel Set Theory (ZF) is expressed in a formal language (predicate logic). It’s usually stated with about 9 axioms – statements that define what set theory is. There’s a version of the ZF axioms on the next slide, for those who are interested. But many of the details aren’t terribly important for us. Note to those already in the know: we’ve left out the Axiom of Choice, so this really is ZF not ZFC. We might talk briefly about this in the last session. The most interesting axioms for us are existential – that is, they declare that some set or other exists in the theory. Without any existential axioms, one model of set theory would be complete “nothingness”. The only things it could prove would be things that are also true of “nothingness”. Not very useful!