Review

• We have witnessed the struggle with the limits of reason in the case of infinity, and seen both the emergence of a theory of infinite sets and also the Cantorian antinomy with the first version of such a theory.
• A natural inclination is to take a step back, and attempt to get some certainty by expunging conflicting intuitions and examining the formal nature of reason.
• This goal inspires both the project of formalizing (axiomatizing) as much as possible of mathemetical theory, and also attempt to prove that various such axiomatizations of math -- perhaps ideally all of math -- are consistent. That is, we would like to both create a powerful and clearly defined theory (of, say, infinite sets) and also prove that such a theory is consistent.
• Russell and Whitehead's Principia Mathematica was the most ambitious, if not the most rigorous or succinct, attempt to do axiomatize arithmetic. It is an axiomatization of arithmetic using type theory.

Background to Russell's Approach

• Russell recognizes that there are several contradictions which pose a specific problem for attempts to make reasoning, especially pure logical reasoning, clear and unproblematic. Here are just a few examples:
  
  
  1. The Cretan: "I am lying."
  2. The Liar: "This sentence is false."
  3. Russell's paradox: the set of all sets that are not members of themselves.
  4. Berry paradox: "the least integer not nameable in fewer than nineteen syllables"
  5. Burali-Forti paradox (this is quite similar to what we have called the Cantor Antinomy).
  
  
• What do these have in common? A kind of self-reference or reflexiveness! (For the Berry paradox, this is not so obvious, but it is because it refers to the number of syllables in our language, and so it refers both to numbers and to the language we are using to talk about these numbers).
• It is not enough just to rule out immediate self-reference: we can create looping versions of the paradoxes with as many steps as you like (Here's a two step Cretan: "The following sentence is false. The previous sentence is true." Try to make a three step!)
• Russell's solution is a theory of types.

  Types: Each statement has a type. (Roughly:) a statement just about numbers, our basic objects, is of the first type. A statement about collections of numbers is of type 2. A statement about collections of collections of numbers is of type 3, and so on.
  
  Type restriction: in the Principia Mathematica system, we only allow statements of type n to refer to things of type less than n.

• Solution: we don't have to go into the details to see intuitively how Russell's system works to avoid the paradoxes. Consider the first three listed above.

  1 and 2. The Cretan: "I am lying." If our system included something which allowed us to say, for example, statement P is true, we could use that to say something which meant, "This statement is not true," and create something like the Cretan paradox. But that would be impossible in type theory: such a statement would have to refer to its own type level (in order to say "this statment..."), which is forbidden. Thus any form of this paradox will not be well-formed.
  
  3. Russell's paradox: the set of all sets that are not members of themselves. Since in type theory a set can only collect things of lower types, no set can contain itself, and any claim about such an object is not well formed.
  
  4. Berry paradox: "the least integer not nameable in fewer than nineteen syllables." In a type theory formal system, any property analogous to number of syllables will refer to lower types; it cannot be used to describe its own statement.
  
  5. Burali-Forti paradox (this is quite similar to what we have called the Cantor Antinomy). (Cantor's antinomy is ruled out because we no longer have a set of everything -- such a set would also refer to its own type and higher types, since it would include itself and higher types.)

Success? Russell's system, perhaps with a little house cleaning, may give us everything we need in order to do arithmetic. Several questions remain:

• Is this system complete? That is, is every true well-formed formula of this system also provable in this system?
• (Hilbert:) Is this system consistent? That is, are we sure that we cannot prove some sentence P and also ~P? (Note that in this system, if it is inconsistent then it is trivially complete.)
• (Hilbert:) Is this system decidable? Can we have a certain mechanical test of any well-formed formula to decide whether it is a theorem or not?

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Next: Conventionalism.