This lecture was primarily about course organisation, the history of logic, and the idea of metamathematics; that is, the shift from examining ordinary objects to examining the process of examination itself.
Prologue
- Berry’s Paradox
- Consider the number
described as “the least positive integer not definable in under eleven words”. Finitely many short descriptions cannot describe every positive integer, so such a least number appears to exist. - Yet the quoted description has ten words and appears to define
. The contradiction exposes the difficulty of unrestricted, self-applicable definability, rather than a contradiction in arithmetic.
- Consider the number
- Gödel numbering
- Encode expressions, proofs and programs as natural numbers, with distinct objects receiving distinct codes. Questions about syntax can then be expressed as questions about numbers.
- Infinity
- Potential infinity: a process can always continue, for example by passing from
to , without treating all its stages as a completed collection. - Actual infinity: an infinite collection, such as
, is treated as a completed mathematical object.
- Potential infinity: a process can always continue, for example by passing from
- Finitism
- Ground mathematical reasoning in finite constructions and effective operations, rather than completed infinite totalities. Potential infinity need not be excluded: finite construction does not impose a fixed largest natural number.
- Zeilberger was discussed in connection with criticism of infinite mathematics.
- Tertium non datur (law of excluded middle)
- Classical logic validates
, even when neither alternative has been proved. - Intuitionism, associated with Brouwer, requires constructive justification and does not accept excluded middle unrestrictedly. A proof of a disjunction must establish an alternative; an existence proof must provide a witness.
- Continuum Hypothesis
- If ZFC is consistent, it neither proves
nor proves . Nevertheless, classical logic proves . This is not a proof of either alternative, nor an assertion that holds in every model of ZFC.
- If ZFC is consistent, it neither proves
- Classical logic validates
- Axiom of choice
- Every family of non-empty sets has a function selecting one element from each set. This asserts existence even for infinite families without specifying a selection algorithm.
- Banach–Tarski paradox
- Using choice, a solid ball can be partitioned into finitely many sets and reassembled by rotations and translations into two balls of the original size. The pieces cannot all be measurable, so this does not describe physical cutting or contradict ordinary volume conservation.