Set Theory: Should You Believe?
web.maths.unsw.edu.au
web.maths.unsw.edu.au
I thought this was interesting, and the upvotes show that others here on HN agree. But this item is now ranked below 850 and falling - clearly it's being flagged.
I don't much like StackOverflow, MathOverflow and similar - I much prefer the style of HN. If I provided an alternative to HN that overcomes this problem, would you be interested?
cache: http://webcache.googleusercontent.com/search?q=cache:oyOudsA...
(there's quite a few annoying typos with the text "MATH" - is that a problem with the cache, or was the article converted from some other format, poorly? ah, thanks to the wayback link - those are cache errors)
summary: it's a rant (i'm not being critical here - it's quite interesting, but that seems the best description) against standard set theory as the basis for mathematics (more generally, a rant against an axiomatic approach at all). but it doesn't really offer much in the way of alternatives. no mention of constructivism or intuitionism.
"At no point do we or should we say, `Now that we have defined an abstract group, let's assume they exist'. Either we can demonstrate they exist by constructing some, or the theory becomes vacuous."
Doesn't that imply constructivism?
Also it is more powerful to abstract than it is to enumerate.
Let's suppose you write down all the syntactically valid 1 byte programs (in some language, say, Python). there aren't many of them, and you can certainly write them all down.
Now write down all the syntactically valid 2 byte programs, then 3, then 4, and so on. Now you have a list of all valid programs.
Some won't terminate, and some won't print anything, but any number that can ever be computed by any program will be printed by (at least) one of these.
But there are only countably many of them. That means that most of the numbers between 0 and 1 will not be printed by any program. Ever.
Are you now still happy talking about the collection of all the reals? Most of them (by which I mean uncountably many compared with countably many) can never be printed or calculated.
Seriously, I think that's part of the role of abstractions, and that's why we _currently_ can talk about and use reals.
Even though we can't write the entire Pi number, and the program used to calculate all its digits might never finish, we do have a symbol for it, and put it to good use. What do you think?
But there are only countably many numbers for which that is true. Uncountably many numbers cannot have symbols and cannot have programs to represent them. If we can never name them, not even in principle, does it make sense even to say that they exist?
(Note: Don't think you can extrapolate my own personal view from the above, I am merely pointing out that there are non-trivial questions here, questions that most people don't seem to notice or understand.)