Of course aleph1 , since aleph1 * aleph0 = aleph1 .

What else? You know that it is consistent with set theory to assume that in between of c and 2^c there is no further cardinality whenever c >= aleph0.

It may well be that division of cardinals is considered pointless in mathematics since nothing interesting can be done with it. The careful old-fashioned presentation of set theory by Erich Kamke does not mention division of cardinals.

Interesting answer Ulrich, thanks

Actually there is an application - I'm working on right now.

Now - Is aleph2 times aleph1 = aleph 2 ?If yes - what is the proof ?

Of course. Since

aleph2 * aleph0 = aleph2 ,

aleph2 * aleph2 = aleph2,

aleph0 < aleph1 < aleph2

==>

aleph2 * aleph1 = aleph2

Just a brief aside. I get the conceptual idea of aleph0 and aleph1

can one of you please describe in (hopefully simple words if possible) what aleph2 actually is. What is meant by "The second uncountable cardinal" I thought i understood all four words in this sentence but find I am unable to fathom the phenomenological meaning of them put together. My rather naive thinking was leading me to, aleph2 = 2^aleph0 or something like that

Andrew, Aleph zero can be represented by the infinite number of natural and rational numbers, Aleph one by the number of mathematical points on any line segment, and Aleph two by the number of possible geometrical curves.

The issue you describe remains for higher numbered alephs though (which can be described by how they're built using diagonalizing, but are a bit hard to relate to real world items.)

Thanks HCR

It seems from your answer I was also under the mistaken impression that there were different alephs for natural numbers and real numbers. (The answer seems to me the set of reals MIGHT be aleph1 but nobody can prove or disprove it)

Anyway I have seen enough that while i am interested in it, I will leave this topic well alone as not being for a mere mortal like me just yet!

Actually there are different alephs for real and natural, it is for natural and *rational* numbers (not real but rational) that the aleph is the same at aleph naught :-)

Real numbers are already aleph one

The cardinality of the natural numbers set is aleph0 ... why the cardinality of the integer numbers set is not 2*aleph0 - 1 ???

@Samer

Since aleph0 = 2*aleph0 - 1 this seems to be no deep question.

Is (2*aleph0 - 1)/(aleph0) = 2 or 1 ?

Samer,

since 2*aleph0 - 1 = aleph0 = aleph0*aleph0 your optionlist 1,2 could be enlarged by appending aleph0. So, you see, division is not uniquely defined for cardinals in most cases. As I said, text books seem to have no division of cardinals. For good reasons!

Just to be provocative, there is an argument that the difference between Aleph 0,1 is the reason why matter does not completely fill up space in the universe. The argument goes as follows:

a) Space and time are a continuum on Aleph 1

b) Particles of matter in the Universe are countable by the number of degrees of freedom in the QM wave function of the Universe (Aleph 0)

It follows (I think) that there is always a large amount of space left over for even an Aleph 0 number of particles to move in.

Will aleph0 act differently than infinity in the shown example ?

Yes, it definitively will.

By intention, aleph0 is not a size of a number but the size of a set. It is unrelated to the concept of infinity in calculus.

So aleph0 is NOT a number but a counting and as a result, arithmetical operations could NOT be applied to it

Not quite Samer, because you can bind the size (or cardinality) of a set to a transfinite number describing something other than a set - for instance Aleph one is the number of mathematical points on any line segment, and Aleph two is the number of possible geometrical curves , and asking what the value is of the ratio of the number of geometrical curves over the number of points on a line segment is a legitimate question.

@Samer

Unfortunately you got a completely wrong impression. Cardinalities (also called cardinal numbers) are well defined objects and sums and products of pairs of cardinalities are well defined operations which give cardinalities as results. I refrain from reproducing these definitions here. They are standard and easy to find e.g. in Wikipedia. The only point about which one may have discussions since it seems to be not covered by standard knowledge and conventions is about division of cardinalities and this is what Chris' question is about.

What I am sure of it, is that Card(Z) / Card(N) = 2 and that is aleph0/aleph0 ... and Card(Q) / Card(N) > 2 and that is also aleph0/aleph0 ... so aleph numbers division gives strange answers ...

The two envelopes paradox is a paradox :) but to say that Z and N contain the same number of elements which is aleph0 is a paradox in my opinion because Z contains by construction twice the number of elements of N and here we know what each envelope contains ...

Both for aleph0 and the cardinality of the reals C (which may or may not be aleph1) one has

that the square of the infinite cardinal is equal to the cardinal itself: C^2=C. Since the possibility of division relies on the possibility of cancellation, if we could have division in infinite cardinals, this would have the unfortunate results that C=1, or aleph0 =1. So maybe this is in part the reason why this subject has not received much attention.

As far as I know the answer to your question is that the value is non-existing.

Mathematically, 'division' of x by y is defined as multiplication of x by y-1, where the latter is the multiplicative inverse of y. So before we can speak of division, we need to have a group structure in which every element x of a set has a multiplicative inverse x-1 so that x*x-1 = 1.

However, I have never seen the set of cardinal numbers being defined as a group under multiplication. That would be very strange, since the cardinals refer to sizes of sets. So e.g. the natural number 2 is a cardinal number, since (in the universe of sets) we have sets with two elements (such as {Ø, {Ø}}). But now consider the cardinals to form a group under multiplication. Then ½ would be a cardinal number too, since it is the inverse of the cardinal number 2. And ½ being a cardinal, that means that there would be a set with ½ element. That's nonsense - at least, according to me. So - to my knowledge - the cardinals cannot form a group under multiplication, they form a monoid or semi-group structure at best. And when there is no inverse of a cardinal, we cannot divide one cardinal by another. In other words: the division of aleph one by aleph null is undefined (nonexisting) as far as I know.

I don't know Marcoen. In narrow terms you may be right, when when you use real-world generators of the numbers your argument seems to not quite hold.

Samer had made a similar point, to which I had answered Quote asking what the value is of the ratio of the number of geometrical curves over the number of points on a line segment is a legitimate question Unquote, considering that the former number is Aleph-2 and the latter Aleph-1. There are many such "practical" examples, and Ulrich's answer seems to work in all such cases. 

It's not even necessarily contradictory with your answer, since an indeterminate outcome (such as that of ∞ over ∞ ) can take on any value including ∞ depending upon circumstances, and it is legitimate to work out / ascertain that outcome by other, independent means

Chris,

Set theory has no basis in reality: it is based on a number of abstract axioms and that's it. So sets are generated by the constructive axioms of set theory - not by "real-world generators". The cardinals are then the number of elements in these sets.

So yes, there will be a "number of geometrical curves": with the axioms of set theory we can construct a set of geometrical curves from the set of subsets of R3, and that set then has a cardinal number, and that is your number. Similarly, we can construct a set of points on a line segment: this set then has a cardinal number too.

However, if you then ask what the ratio between these two cardinal numbers is, then you have tacitly assumed that this ratio is defined. The point is, however, that all the beautiful properties of real numbers, with which one is used to work, do not necessarily apply to the set of cardinal numbers. In particular, while there is a ratio defined for any two nonzero reals, there is - to my knowledge - no such thing as a ratio of two cardinal numbers.

Of course, you are free to develop a new group, consisting of the set of cardinal numbers and the binary operation multiplication. But then you will have to solve the problem of the meaning of the new cardinal number ½.

Great answer Marcoen - developing a new group might be indeed an answer, you could even define a new set of binary operations to go with it (so that you could avoid ending up with cardinal numbers like 1/2), the issue being you'd have to be very careful to ensure consistency, which might not be trivial ....