Georg Cantor - Contributions to the Founding of the Theory of Transfinite Numbers - Section 6
5 Video Views·Jan 4, 2025
This video is the sixth 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 6 of the 1895 article which is titled The Smallest Transfinite Cardinal Number Aleph-Zero and which introduces the notion of transfinite cardinal number and transfinite aggregate. The transfinite cardinal number, aleph-zero, is introduced and it is shown that this is the smallest transfinite cardinal number. Various arithmetical properties of aleph-zero are also proved often by using some very clever correspondences between aggregates.
Theorem A of this section, which states that every transinite aggregate T has parts with the cardinal number aleph-zero is historically a very significant proposition. Despite the seemingly 'obvious' nature of this theorem it turns out to have been a very controversial theorem.
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Welcome to this video, the sixth video in my series of videos on Cantor's 1895 and 1897 articles on the theory of aggregates and transfinite numbers.
Section 6 is titled The Smallest Transfinite Cardinal Number, Aleph Zero.
Section 6 starts off with some definitions of finite aggregates, transfinite aggregates and transfinite cardinal numbers.
The totality of finite cardinal numbers, nu, is given as a first example of a transfinite aggregate.
We must be careful at this stage not to jump to any hasty conclusions that transfinite aggregates must be bigger than finite aggregates
or that transfinite cardinal numbers must be greater than finite cardinal numbers,
though of course this will turn out to be the case, we haven't actually shown this yet.
By section 1, every aggregate has a cardinal number, and so the cardinal number of the aggregate, nu, is denoted using the symbol Aleph Naught.
And then Cantor goes on to show that Aleph Naught cannot be equal to any of the finite cardinal numbers defined in section 5,
and thus showing that Aleph Naught is genuinely a transfinite cardinal number,
and therefore any aggregate with cardinal number Aleph Naught must be a transfinite aggregate.
Cantor shows, by means of an explicit correspondence, that the aggregates nu and the union aggregate of nu with the object E Naught are equivalent,
and by the results of section 3, we have that the cardinal number of the union aggregate of nu and the object E Naught
is equal to the cardinal number Aleph Naught plus 1.
The cardinal number of the aggregate nu is equal to Aleph Naught by definition.
However, by the results of section 1, two aggregates are equivalent if and only if they have the same cardinal number,
and so the cardinal number of nu, which is Aleph Naught, and the cardinal number of the union aggregate of nu and the object E Naught,
which is Aleph Naught plus 1, are equal.
Thus, Aleph Naught plus 1 is equal to Aleph Naught.
Therefore, Aleph Naught cannot be a finite cardinal number.
The results of points 3 and 4 of page 104, once proved,
will confirm that a transfinite cardinal number is greater than any finite cardinal number
by showing that Aleph Naught is both greater than any finite cardinal number
and is also the smallest transfinite cardinal number.
It would also seem then that 0 is ruled out at this point as a cardinal number
since 0 was not included in the finite cardinal numbers of section 5,
and all transfinite cardinal numbers, that is, those cardinal numbers which are not classified as finite cardinal numbers,
are all greater than any given finite cardinal number.
There's an interesting aspect to theorem A that's historically very significant.
The proof of theorem A, that is, that every transfinite aggregate T
has parts with the cardinal number Aleph Naught,
uses a form of the axiom of choice, sometimes called the denumerable axiom of choice.
This is indicated in the first line of the proof, which starts,
if by any rule we have taken away a finite number of elements, and so on.
That the elements of the aggregate T have been taken away by any rule
became a very philosophically hot topic in the early 20th century.
Note that the rule by which the elements are taken away is not stated.
It's simply assumed that such a rule exists,
but without the need to explicitly specify the rule.
In the eyes of a modern mathematician, this will mostly go unnoticed.
In modern mathematics, the axiom of choice is pretty universally accepted,
and making arbitrary choices is not considered in any way a problem.
For further details of this controversial topic, I direct you to the book
Zermelo's Axiom of Choice, by Moore,
to better understand some of the issues surrounding this aspect of mathematics.
The proofs of theorem A and B themselves don't present any real difficulties,
if this principle of arbitrary choices is accepted.
The only comment I will make here is, in the proof of theorem B,
when Cantor speaks of a definite law of correspondence between these two aggregates,
he's speaking of a one-to-one correspondence which must exist
since S is assumed to be a transfinite aggregate with cardinal number aleph-naught,
and so S is equivalent to the aggregate of finite cardinal numbers nu.
Skipping ahead to page 106, we see that
the cardinal number of the union aggregate of a-nu and b-nu
is equal to aleph-naught plus aleph-naught.
Cantor splits the aggregate of finite cardinal numbers,
which has cardinal number equal to aleph-naught,
into two separate aggregates,
the aggregate of odd finite cardinal numbers, 2-nu-1,
and even finite cardinal numbers, 2-nu,
each of which have cardinal number equal to aleph-naught.
Since the aggregates a-nu and b-nu
are indexed by the finite cardinal numbers,
then each of these aggregates has cardinal number equal to aleph-naught,
and therefore the aggregate 2-nu-1
is equivalent to the aggregate a-nu,
and the aggregate 2-nu is equivalent to the aggregate b-nu.
And since the cardinal number of a union aggregate
depends only on the cardinal numbers of the individual aggregates in the union aggregate,
then it follows that the union aggregate of 2-nu-1 and 2-nu
is equivalent to the union aggregate of a-nu and b-nu.
I'll spend some time explaining why the function in point 9 of page 107 works.
We see, using the diagram on the screen,
that the groupings of pairs of values of mu and nu,
as indicated on page 107,
account for all possible pairings of mu and nu,
and that on each grouping, rho, which equals mu plus nu, is constant.
We need to consider both the values of rho, which equals mu plus nu,
and the values of the expression mu plus nu minus 1 times mu plus nu minus 2,
divided by 2, the latter of which I'll henceforth refer to simply as kappa,
on each of the groupings of pairs of values of mu and nu,
as indicated on page 107.
We note that since the value of rho, and hence the value of mu plus nu,
is constant on the whole of each grouping,
then the value of kappa is also constant on each of the groupings,
and some of the corresponding values of kappa are indicated on the screen.
For each value of rho, the expression kappa
is the sum of the first rho minus 2 natural numbers.
Therefore, the sequence of values of kappa,
which starts 0, 1, 3, 6, and so on,
if continued, will simply be the sequence of triangular numbers.
We also note that on each grouping, the value of mu
takes all positive values in the range 1 less than or equal to mu
is less than or equal to rho minus 1.
So for rho equals 2, the function of 0.9 returns the value 1,
that is, 1 plus 0, where 1 is the only possible value of mu on this grouping,
and 0 is the value of kappa when rho equals 2.
For rho equals 3, the function returns the values 2 and 3,
that is, 1 plus 1, and 2 plus 1,
where 1 and 2 are the possible values of mu on this grouping,
and the value of kappa is 1 when rho equals 3.
For rho equals 4, the function returns the values 4, 5, and 6,
that is, 1 plus 3, 2 plus 3 and 3 plus 3,
where 1, 2 and 3 are the possible values of mu on this grouping,
and the value of kappa is 3 when rho equals 4.
In a similar way, for rho equals 5, we get 7, 8, 9 and 10,
that is, 1 plus 6, 2 plus 6, 3 plus 6 and 4 plus 6, and so on.
Notice that the smallest value on each grouping, rho,
is 1 greater than the rho minus 1th triangular number,
and the greatest value on each grouping, rho, is the rhoth triangular number.
The difference between the rhoth and the rho minus 1th triangular number plus 1
is rho minus 1, which is exactly the number of possible values of mu
on each grouping. So we see that varying mu
over the required range in each grouping fills in, so to speak,
the gaps between the rho minus 1th and the rhoth triangular numbers.
Each pair of coordinates corresponds to exactly one finite cardinal number,
and each finite cardinal number corresponds to exactly one pair of coordinates.
It's a one-to-one correspondence, and so
aleph-naught times aleph-naught is equal to aleph-naught as required.
On page 108, there's very little on which for me to comment.
Moving on to page 109, on this page, Kantor is just giving a bit of an idea
of where things are heading. The definite law by which the next greater cardinal number,
aleph-1, proceeds out of aleph-naught is not stated here,
but we're assured that there is such a law. Section 16 deals with cardinal number aleph-1.
The correction of a small typographical error on this page is shown on the screen.
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