The Continuum Hypothesis keep anyone else up at night?

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In short: there are an infinite number of integers (those which can be expressed as a fraction, 3/1 = 3, etc). You can demonstrate this by considering first the match up of odd to even numbers.

1 and 2
3 and 4
5 and 6
7 and 8
...on to infinity.

Then consider the match up of even numbers to all numbers.

1 and 2
2 and 4
3 and 6
4 and 8
also extending to infinity, giving the conclusion that the even numbers are infinite, and the same for all numbers in general!

Thus the idea of cardinality was introduced, so the integers are an ordered infinity, you can write a list of them.

The real numbers on the other hand.

Say you write a list out:

a: a1, a2, a3, a4, a5
b: b1, b2, b3, b4, b5
c: c1, c2, c3, c4, c5

You put those in order, then go through and mark off the diagonals:

a1, b2, c3, d4, e5

Then produce a new number from that

x1, x2, x3, x4, x5

It is in between numbers on your list, yet is not on your list, and thus can be done for any list.

So you can not order the entire list of real numbers, because you can always produce a new statement from them in between the "slots" of your list, similar to Godel's argument.


The question then is, is there an order of infinity between that of the integers (aleph nought) and that of the real numbers (c)?

That's very interesting. Too complicated for me though, because it contains numbers.

Etymological puzzles keep me up at night though. If you let yourself think by association, starting with one innocent word and tracing it's cognates through history and different languages... feels so illuminating and fascinating. To me.

Yeah, following words was always fun, and it can be related to math as well!

Remember: math doesn't necessarily mean arithmetic!

Watch out. Some have come crazy while working on the continum hypotesis.

I've often said, Cantor was nuts, and be careful when pondering infinity for madness lurks along that path.

nothing keeps me awake at night...hardly anything gets me up in the morning... except

It is caffeine that sets my mind in motion. It is from the juice of the bean that my thoughts acquire speed, my hands begin to shake, the shakes become a warning...

Ah, the infamous Diagonal Argument. We learned of this in one of my classes, but it wasn't until I saw it presented in a book (Journey Through Genius, i believe) that it - and the whole idea of infinity begun to make sense.

That could be, though, because the college professors tend to skip over the ideas and focus on making sure you know how to prove the function between the evens and the odds is indeed a bijection - because it's not as important that we marvel at Cantor's genius and the ideas of mathematics as that we know the theorem "a function is a bijection iff it has an inverse".

The weird thing about infinity has always been that you can take the integers and partition the integers into two sets that have the same size as the integers. You've got to be a mathematician if you love those kind of puzzles.

I'm an aspiring mathematician, once I realized I could let go of arabic numerals completely it was like everything about math became beautiful.

Thinking in arabic numerals is jarring, turned me off of math for a long time.

Numbers have shape, structure, information, elegance.

(0, 1, 2, 3, 4, 5, 6, 7, 8, 9) do not.



I've been pondering if the set of all spaces is possible to list in an ordered fashion, as it shouldn't REALLY suffer from the diagonal argument... yet fractal dimensions introduce ways I could potentially see it arising.

I know there are more possible spaces than integers, because you can assign an integer to any space, then specify a new space with more coordinates from that one.

Though you could also argue that you can assign fractional values to dimensions, pushing the cardinality higher than that of the reals... perhaps.

Apparently Paul Cohen proved a few years ago that the truth or falsehood of the continuum hypothesis is undecidable.

When Godel did his work on undecidable propositions, mathematicians did not really think it would be applicable to everyday maths, so to discover that a question regarding how many real numbers there are was "undecidable" was quite a shock.

I'll admit I only have a vague idea of what I'm talking about here.

Amir Aczel wrote a good book about Cantor: "The Mystery of the Aleph".

It's a bit trickier than that, he proved that you can't decide it or the axiom of choice from ZF axioms alone, making it independent of them entirely.

It's still an open issue, and not one which will be readily solved until a new more powerful form of logic is produced, which is kinda what I'm pondering.

P.J.Cohen proved that CH (continuum hypothesis) and its negation are both consistent with the rest of set theory, so the question is closed. CH cannot be proven from the rest of set theory.

ruveyn