Thursday, December 21, 2023
– Live Presentation with Peter Lynch–
Session at 12 Noon Atlanta time
(check your time here)
Please join us starting 10 minutes before this session using the following button.
For published CoM presentations please visit the G4G YouTube channel
After this virtual CoM presentation on Zoom, we will meet for an informal social session in a different Zoom space where we can all see each other (see the blue button below). That Zoom meeting will start around 1pm ET.
Please join the social using the following button:
Counting Sets with Surreals
How many odd numbers are there? How many even numbers? From Galileo to Cantor, the suggestion was that there are the same number of odd, even and natural numbers, because all three sets can be mapped in one-one fashion to each other. This jars with our intuition.Cantor defined transfinite numbers and showed how every set has a cardinal number. However, cardinality is a “blunt instrument”: the natural numbers, rationals and algebraic numbers all have the same cardinality, which fails to discriminate between them.The surreal numbers, discovered by John H. Conway around 1970, form the largest possible ordered field, with all the basic arithmetical operations, and sensible arithmetic can be carried out with them. A subclass of the surreals called the omnific integers plays the role of integers in the surreal context. The surnatural numbers are the class of non-negative omnific integers.Using the surnatural numbers, we define the “magnum” for subsets of the natural numbers that corresponds to our intuition about the size of these sets. The magnum of a proper subset of a set is strictly less than the magnum of the set itself, in harmony with Euclid’s axiom “the whole is greater than the part”. We find the magnums of a wide range of infinite subsets of natural numbers.
Peter Lynch is emeritus professor at University College Dublin. His interests include all areas of mathematics and its history. He writes an occasional mathematics column in The Irish Times, and has published three books of articles, entitled “That’s Maths”, Vols I, III and III. His blog is at http://thatsmaths.com.