Different Infinities-Series on Infinity Part 6

This video presents Cantor's argument that there exist different infinities by demonstrating a set that is uncountable. It will only make sense if you have watched the previous video, Part 5, which defines countable sets.

In essence, the size, or cardinality, of the rational numbers (fractions) is the smallest infinity: aleph null (or aleph zero). The cardinality of the real numbers, aleph one, is probably the next largest infinity, but that fact (the continuum hypothesis) has been shown to be impossible to prove under the current axioms of set theory.

Category:

Education

Tags:

• math
• infinity
• series
• infinitesimal
• Cantor
• countable
• sets
• diagonalization
• aleph
• uncountable
• continuum

[ILoveYouToThe9999999]

The probability that two distinct irrational numbers on the number line can be close enough to exclude any rational numbers is higher than the probability that two distinct rational numbers on the number line can be close enough to exclude any irrational numbers.

[Davidson1956]

To use probability to prove anything you would probably need to define a probability density function, prove it fits the situation, and then apply Chebyshev's Theorem.

Consider two arbitrarily close irrational numbers. One could find a sufficiently large integer N such that the product of N and each irrational produces two numbers with an integer in between them. Divide that integer by N and you've found the rational number between the two irrationals.

[ILoveYouToThe9999999]

The uncountability of the set of all irrational numbers means that the set of all irrational numbers is more numerous than the set of all rational numbers, and this means that there are at least two distinct irrational numbers on the number line such that there exist no rational numbers between them, but, this is a contradiction to what we know that between any two distinct real numbers on the number line there exists a rational number. Thus, the set of all irrational numbers is countable.

[Davidson1956]

Interesting argument, but the term "numerous" is vague or undefined in a mathematically precise way, too vague to claim that it would imply two irrational numbers could be close enough to exclude any rational numbers.

Consider that between any two irrational numbers there is a countable number of rational numbers, but between any two rational numbers there is an uncountable number of irrational numbers.

[ILoveYouToThe9999999]

I found a surjective function from the set of natural numbers to the set of irrational numbers, and this makes the set of irrational numbers countable.

[Davidson1956]

Please pardon my skepticism, but this would refute 140 years of well-established mathematics in the area of transfinite numbers. What is your function, or how could you describe it?

[tomcuchta]

The continuum does not have cardinality aleph-one unless you admit the continuum hypothesis.

[iateyourgranny]

Correction: the first number is assigned to 1, the second number is assigned to 2, etc. There is no "diagonalization" done here. What exactly is your point of assigning "1" to 1, "00" to 2, "011" to 3, etc? Get your method right before you set out to prove something.

[Thetarget1]

I've got a question: Does this mean that there is an infinite number of infinities? And are there then different kinds of infinite series of infinities, like there are different kinds of infinite series of numbers?

[ForeverWiked]

So why are one of them 'countable' but the other is not?

The argument of one was that you can always make another number by changing a digit. Well you can always change a real number by adding 1 to it, so in what way is it countable

[srfriggen]

So in the end you created a transfinite number, is that right? Is there no way to set up a scenario where this type of argument was repeated and repeated essentially creating (and counting) every transfinite number?

[tommyrjensen]

The explanation of the continuum hypothesis is not correct. What CH actually states is the equality of c and aleph1. Where c can be defined as 2^aleph0 and aleph1 can be defined as the smallest cardinal which is bigger than aleph0 (once one has a proof that such exists). Cantor may have driven himself crazy by trying to prove c=aleph1, sometimes thinking that he had proved it, sometimes that he had disproved it.

[rezkogitans]

@Davidson1956: You write "The cardinality of the real numbers, aleph one, is probably the next largest infinity".

No. Aleph one, by definition, is necessarily the next large infinity.

[StitchedMusik]

I don't understand. Isn't this the same as aleph-null?

[fonsidream]

It is a partial hypothesis to my senses.

Why not consider all possibilities like 00, 01, 10, 11 and then prove the hypothesis?

[theboombody]

@Davidson1956 I haven't found anything like that. I only found a diagonalization argument showing that even whole numbers can form that aleph 1 set. Of course the whole numbers have infinite digits, but at least there's nothing to the right of the decimal point, so I still call them whole. You can see my video "Does Counting Exist?" for further information.