How many real numbers exist? New proof moves closer to an answer
quantamagazine.org
How many real numbers exist? New proof moves closer to an answer
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Re: How many real numbers exist? New proof moves closer to an answer
#2I don’t get for Cantor’s diagonalization proof, why do we need to use the diagonal digits to form the new number?
Would the proof work the same if we instead used the first digit of every number in the list?
Re: How many real numbers exist? New proof moves closer to an answer
#3I don’t get for Cantor’s diagonalization proof, why do we need to use the diagonal digits to form the new number? Would the proof work the same if we instead used the first digit of every number in the list?
There are only 10 possible first digits of the new number, so you can't choose a number that will differ from all other numbers in the first digit. If you do it diagonally, you'll always have 9 other options to choose from, since you just have to make it different from that one number!
Re: How many real numbers exist? New proof moves closer to an answer
#4Great article. Since it looks like a lot of folks are interested in this article, some extra background.
First, what is forcing? The article actually has a great description of ultrapowers (a key part of the construction) but it goes by a little fast, so you might like Tim Chow's "A beginner's guide to forcing" [1] which does a good job not only laying out the mathematical details at a high level, but also really clearly explaining the philosophy of how you prove something independent of the axioms. I found it very illuminating.
Second, the article has some interesting notes on how mathematicians go about what axioms to select and which not to. Penelope Maddy's "Believing the Axioms" [2] is the classic on this topic (it has two parts). It is focused on set theory, so it has a nice description of Martin's axiom and the arguments for and against. It was nice to read because it is a deeply technical argument (set theory is hard!) but at the same time there is no "right answer"—all of these axioms, after all, are independent of ZFC, and it's "ok" to add any of them. The arguments are sometimes aesthetic ("rules" versus "surprises"), sometimes pragmatic (what they can and cannot prove), and sometimes involve deep values of what the universe should be like (should higher cardinalities be like lower ones? weirder? simpler?).
It might all seem abstract, but if your day job is programming, imagine an argument over how to architect a large and complex system. Perhaps both architectures are possible, but which one is "right"? What arguments would you deploy? Set theorists are also building a large and complex system (the universe of sets) and are having arguments over how it should be built, which things it should make easy and which hard, which technologies should be supported natively (forcing?) and which should not.
[1] http://timothychow.net/forcing.pdf [2] https://www.cs.umd.edu/~gasarch/BLOGPAPERS/belaxioms1.pdf
Re: How many real numbers exist? New proof moves closer to an answer
#5I don’t get for Cantor’s diagonalization proof, why do we need to use the diagonal digits to form the new number? Would the proof work the same if we instead used the first digit of every number in the list?
That wouldn't work because n-th digit differing from 1st digit of n-th number doesn't guarantee that the new number isn't accounted for already.
Re: How many real numbers exist? New proof moves closer to an answer
#6I don’t get for Cantor’s diagonalization proof, why do we need to use the diagonal digits to form the new number? Would the proof work the same if we instead used the first digit of every number in the list?
There are only 10 possible first digits of the new number, so you can't choose a number that will differ from all other numbers in the first digit. If you do it diagonally, you'll always have 9 other options to choose from, since you just have to make it different from that one number!
Makes sense, thanks
Re: How many real numbers exist? New proof moves closer to an answer
#7I don’t get for Cantor’s diagonalization proof, why do we need to use the diagonal digits to form the new number? Would the proof work the same if we instead used the first digit of every number in the list?
To fill in some details:
Two real numbers are different if in the same place they have different digits. [1] So the idea is to take your infinite, hypothetical list of all real numbers A[] and make a new real number X, where for all i, there's some digit where A[i] and X differ. Easiest way is to make X differ from A[i] at digit i.
[1] There's a subtlety here about repeating nines at the end of a number, but it is inessential.
Re: How many real numbers exist? New proof moves closer to an answer
#8I'm not trying to be flippant, although it may come off that way: why does any of this matter?
Re: How many real numbers exist? New proof moves closer to an answer
#9It depends on what you define as a number. For example reals and complex have different properties from integers. Is there a mathematical reason why both are considered numbers? You can count (with) integers but not with reals. Thus one appears to be a number while the other appears to be a measure.
Re: How many real numbers exist? New proof moves closer to an answer
#10I don’t get for Cantor’s diagonalization proof, why do we need to use the diagonal digits to form the new number? Would the proof work the same if we instead used the first digit of every number in the list?
Then you cannot guarantee that when you change the first digit of the second number, what you get is not the first number.