I'm sure this is elementary but there is something I don't get. I thought that if you raise two to the power of aleph null you get a higher aleph. Raisng two to the nth power gives you the total number of ways of grabbing zero to n elements of the set of n distinct things.

So what if you raise two to the power of the number of prime numbers? That's two to the aleph null since there are aleph null primes. But each of these combos of primes you can multiply together and get a different composite number.(Far from all of them since no prime factor is repeated.) And the number of composite numbers is also aleph null. These composite numbers can also be put into numerical order. The first several are 6, 10, 14, 15, 21, 22, 26, 30, 33, 34, 38, 39, 42, 46, 51 .....

So where am I being led astray by these thoughts?

You still have just natural numbers, another set of them. They are countably infinite. It doesn't come up to aleph-one. https://en.wikipedia.org/wiki/Aleph_number

Naturally please correct me.

I’m having trouble understanding where the question is, the power set of the prime numbers should equal the power set of composite numbers because each can be mapped with 1-2-1 correspondence to the natural numbers which is what I understand plaaynde to be saying, they are both countably/listably infinite.

The power set of the prime numbers is made mostly (as any other infinite set) by infinite subsets. These infinite subsets all map to infinity.

You probably need to contemplate what one is actually saying when they "raise 2 to a cardinal number power." What you're calculating is the size of the power set.

It's best understood by analogy:

The size of the power set of {1,2,3} is 2^|{1,2,3}| because for each element in the set, you have to pick from two options to create a subset. Either the element is in the set or it's not.

So the fact that you might try to combine primes in some way to create other numbers literally has no meaning. You're thinking about the number of subsets of the original set that you can create, not some other arithmetic property about the elements in the set.

I think DS is perfectly aware that 2^N is the size of the power set. His reasoning is this:

- the size of the set of prime numbers is aleph null;
- so the size of the power set of the prime numbers should be aleph 1;
- but if I can map each element of the power set to a distinct natural number that would make the PS aleph null;
- and I can make this map, since I can map each element of the PS to the natural number obtained as the product of the numbers in the set!

What's wrong in the reasoning above, as I expressed in my earlier comment, is that aleph 1 elements of the PS are themselves of infinite size. Take for instance the set of all primes except 2. This set does not map to any natural number, according to the map proposed by DS.

I do not understand the words you are using.

What you don't understand?

The quoted passage is wrong. Take these two combos:

- the set of all primes except 2;
- the set of all primes except 13.

Multiply together the numbers in each of the two sets above. You don't get "different composite numbers".

For those that stumble into this thread without knowing anything about Cantor here is a primer.

The infinite set of numbers between for example 1 and 2 is an aleph-1, am I getting it right?

There is only one kind of infinity. The one that never existed in the finite past since it all came to be!

MdZ

Show me an example of Pi or 2^(1/2) in a natural process that is not an approximate theory of the world that eventually is shown to cut corners.

Show me an example in physics where infinity is validated as existing in a process that would otherwise fail without it.

Calculus to whom we owe the technological civilization world we see out there today is precisely born out of the real number line. It is just another level of approximation of the truth. It is not the truth.

nickthegreek is right. You are looking at the size of the Set of all subsets of all the primes. That Set includes infinite subsets. One example of an infinite subset of All the primes is the set of All the primes itself.

If you were interested in the size of the Set of all finite subsets of all the primes then your argument would be valid.


PairTheBoard

Come to think of it, David's argument would also work for the Set of all infinite subsets of the primes which exclude only finitely many primes. It's the subsets that both include and exclude infinitely many primes which are uncountable.


PairTheBoard

Are you saying that you think DS is mapping the particular subset to the natural number that is the product of the primes in the subset?

That is, given $S \subset P(\N)$, $S \mapsto \prod_{p \in S} p$?

That would make more sense than the thing I thought he was saying, which was quite a bit more confused.

I’m guessing you’re being led astray by general sloppiness. Keep calm and diagonalize.

2 3 5 7...
1 0 0 0...
0 1 0 0...
0 0 1 0...
1 1 0 0...
0 0 0 1...
1 0 1 0...
.
.
.
0001...

Since I don't get what the replies are saying I may not not be making myself clear.

The real numbers are uncountable because. I thought, if you name one, I can't name the next one, as I could with integers, prime numbers or even fractions.

I therefore thought that the combinations of numbers you got by raising 2 to the power of the n numbers you started with had the property that you couldn't name the "next one" when n is aleph null. I thought that because I'm told that 2 to the aleph null power is uncountable.

But since you can use this prime number trick to name the next combo after you name one, I thought that this means the combos are countable.

So which of the above words are inacurate?

Your incorrect word is "combinations" because you are using it to mean finite combinations. When the set is infinite, like the aleph null set of primes P, then the "combinations" in 2^P include infinite "combinations". Rather than "combinations", the more standard term is subsets.


PairTheBoard

What’s the next fraction after 1/2?

1/3


https://www.homeschoolmath.net/teach...-countable.php

that’s not what next means.

It's easy to see that the Collection of all subsets ("combinations") of all the primes is the same size as the reals. Order the primes as the 1st, 2nd, 3rd, .... prime
Then identify a subset ("combination") of primes by forming the binary decimal, for example:

0.1001010001...

where here the 1st, 4th, 6th, 10th, primes as well as infinitely many more indicated by ... are in the subset ("combination").

All such binary decimals then identify all subsets ("combinations") of the primes. At the same time, all such binary decimals identify all the reals between 0 and 1.


PairTheBoard

Are you kidding? How about 2/5?

Take care, David.

Timed out. You may think it's about the share numbers of combinations, but aleph-null and aleph-one is more a qualitative thing. About (in principle) countable and uncountable infinities, if I have understood it right.

Please, tell me which is the combo that comes after the one made by the 1st, 4th, 9th, 16th, ... , n^2th (n -> infinity) prime number. It comes before or after the one made by the first 50 primes, then the 101st-150th, 201st-250th and so on?

This is what I'm trying to say with little success. You think you ordered all the combinations. But you didn't (because you can't).