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The most incredible thing in this article is the assertion that there exists at least one math professor that is unaware of this theorem - I was taught all of this in the first algebra course at university...
Ah, he's just getting started on his journey into the set of real numbers! Eventually he will discover, "God made the integers. All else is man made.". In particular, man made the real numbers to be complete which means that every sequence that appears to converge, that is, meets, the Cauchy criterion, actually does converge. Really his discoveries are about the completeness property of the real numbers. So, in parti…
The clause "Aleph one, which is the infinity of the real numbers", is known as the continuum hypothesis, and has a fascinating background in itself.
First, the existence of Aleph one in axiomatic Zermelo-Fraenkel set theory depends (surprisingly) on the Axiom of Choice. If you reject AC, we can't show that there exists a unique Aleph one.
It gets weirder. If you accept ZFC (ZF set theory, plus the Axiom of Choice), we can prove that both Aleph one and the cardinality of the reals are greater than Aleph nought. However, Gödel proved in 1940 that Aleph one cannot be proven to be equal to the cardinality of the reals, given ZFC. In fact, none of the main ZFC axioms constrain the continuum hypothesis--there are some proposed axioms, like constructability, which imply CH, but nobody is really sure whether we should accept them.
This is very much a philosophical problem in mathematics: having proven we cannot decide on the basis of the axioms we widely accept, it's now up to us to choose which branch (or both) of mathematics is more useful or epistemologically satisfying--or find other axioms we can agree on that in turn constrain CH.
[edit] derp, just read to the bottom, and it's comment #1. Right then, carry on. :)
It is well proven that Aleph one, which is the infinity of the real numbers, is undeniably bigger than the infinity of the natural numbers. The clause "Aleph one, which is the infinity of the real numbers", is known as the continuum hypothesis, and has a fascinating background in itself. First, the existence of Aleph one in axiomatic Zermelo-Fraenkel set theory depends (surprisingly) on the Axiom of Choice. If you re…
Ah, he's just getting started on his journey into the set of real numbers! Eventually he will discover, "God made the integers. All else is man made.". In particular, man made the real numbers to be complete which means that every sequence that appears to converge, that is, meets, the Cauchy criterion, actually does converge. Really his discoveries are about the completeness property of the real numbers. So, in parti…
"God made natural numbers; all else is the work of man" - Leopold Kronecker. Possibly misquoted by Raymond Ayoub in "Musings of the Masters: An Anthology of Mathematical Reflections".
The most incredible thing in this article is the assertion that there exists at least one math professor that is unaware of this theorem - I was taught all of this in the first algebra course at university...
Yeah... I mean, am I the only one that thinks this article is kind of trite? Uncountability isn't a completely mind blowing concept to me.
This language really aggravates me. Thus far, there is no definition for what it means for one infinite series of numbers to be "bigger" than another.
> bear in mind that the set of real numbers is “even more infinite", and that’s the closest I can give you to an intuitive description.)
Again, there is no definition for what "even more infinite" means.
It seems like it's standard practice to talk to newbies about math without defining all your terms, and as a math newbie, that really turns me off. Sometimes it feels like math people are trying to "get away with" something, like politicians.
> It is well proven that Aleph one, which is the infinity of the real numbers, is undeniably bigger than the infinity of the natural numbers. This language really aggravates me. Thus far, there is no definition for what it means for one infinite series of numbers to be "bigger" than another. > bear in mind that the set of real numbers is “even more infinite", and that’s the closest I can give you to an intuitive desc…
If I defined all the terms, my article would be twice as long (and it's too long as it is.) Whoever wants precise technical terms is welcome to go on Wikipedia.
The interesting thing here is that it's much harder to put this problem properly into mathematical terms than it is to solve it. The whole insight here is that you a "description" of number is just some finite sequence of symbols from a finite alphabet. Now, if you understand why cardinality of continuum is greater than aleph null, it's totally straightforward to show that there are only countably many descriptions,…
Goedel and Cohen. Goedel proved it might be (constructible universe); Cohen proved it might not be (forcing). I think Cohen's on record as saying he suspects that with the "right" axioms, mathematicians might come to think that CH is obviously false.