This video is the second video in my series of videos covering the contents of the translations into English of Georg Cantor's famous 1895 and 1897 articles on the theory of aggregates and tranfinite numbers.
This video covers the material from section 2 of the 1895 article which is titled "Greater" and "Less" with Powers and which introduces the relation of 'less than' for cardinal numbers. This new relation is also shown to be a reflexive, symmetric and transitive relation between cardinal numbers. The theorem known as the Trichotomy of Cardinals is also stated along with some immediate consequences of this theorem.
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Welcome to this video, the second video in my series of videos on Cantor's 1895 and 1897 articles on the theory of aggregates and transfinite numbers.
Section 2 is titled Greater and Less with Powers.
There isn't a great deal for me to say about this section. Section 2 starts off with the conditions which define a particular kind of
relation on cardinal numbers.
We already have a notion of equality of cardinal numbers from section 1 and Cantor shows that two aggregates which satisfy the conditions A and B
cannot be equivalent and therefore cannot have the same cardinal number.
It's also shown that this new relation is not a symmetric relation between aggregates.
Unsurprisingly, this new relation is called less than or greater than depending on which way around we think of things and
point 2 is stating that the relation of less than is transitive.
To show transitivity it's useful to bear in mind a comment made at the top of page 87 that
given equivalent aggregates M and N then to every part M1 of M there corresponds then a definite equivalent part N1 of N and inversely.
An outline of the details of how transitivity may be proved will be shown on the screen.
Note that B is shown to hold for aggregates M and P first and then A is shown to hold.
As we've already seen, less than is not a symmetric relation and nor is it a reflexive relation.
Cantor makes an important observation on this page.
He says that each of the relations less than, equal to and greater than exclude one another
but we're not in a position at this stage to assert that any two cardinals must stand in one of these relations to each other.
That is, if A and B are cardinals
it's not possible to say that one of A is less than B, A is equal to B or A is greater than B must hold.
This is the trichotomy of cardinals which is equivalent to the principle in contemporary mathematics known as the axiom of choice.
Having said that, Cantor does state the trichotomy of cardinals as a theorem, theorem A
but without proving this theorem, stating on the same page that this theorem cannot be proved at this stage
possibly saying a little more than he realised at the time.
Cantor then gives a few theorems which follow from the trichotomy of cardinals.
Theorem B is mentioned also in the notes section, page 204 in the English translation
which says that Schroeder and Bernstein proved theorem B
without the supposition that one of the three relations of magnitude must hold between any two cardinal numbers.
Of course, Cantor doesn't prove theorem A, the trichotomy of cardinals, at any point in this work.
Anyone that's interested in knowing more about the trichotomy of cardinals, the axiom of choice, the continuum hypothesis and so forth
may find the book Zermelo's Axiom of Choice by Gregory H. Moore an interesting read.
Thank you for watching.
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