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        riddles >> medium >> Natural Number Challenge
        
(Message started by: Johno-G on Jan 10^th, 2003, 1:51pm)

Title: Natural Number Challenge
Post by Johno-G on Jan 10^th, 2003, 1:51pm

I'm not exactly sure how hard this problem is, as I was simply told how to do this in a lecture. (although I'm sure there must be more than one way...). It involves no actual mathematical working, and is supposed to be just a little bit of creative mathematical thinking!
Please, no spoil-sports letting everyone else in on the solution - make 'em work for it!

Challenge: Construct the set of natural numbers* from nothing. You are not allowed to use addition or multiplication.

* - By the set of natural numbers, I mean {0,1,2,3,...}.

Title: Re: Natural Number Challenge
Post by william wu on Jan 10^th, 2003, 2:11pm

What do you mean by "construct from nothing"?

Title: Re: Natural Number Challenge
Post by Johno-G on Jan 10^th, 2003, 2:21pm

I mean use nothing. Not the number zero, but the concept that there 'is nothing there'. The way I've learned it doesn't actually involve anything other than an abstract concept.
Once you know what this concept is, the rest shouldn't be overly difficult.
I hope that clears things up a little?

Title: Re: Natural Number Challenge
Post by Garzahd on Jan 10^th, 2003, 3:39pm

You're not making any sense.

Either give us a clue about this abstract concept, or please lecture us.

Title: Re: Natural Number Challenge
Post by Johno-G on Jan 10^th, 2003, 3:45pm

OK, sorry if I was a bit vague - I don't think I could've explained it without giving away exactly what you've got to do!
Think about constructing sets of different lengths. See if you can find a way of always creating another (remember - each element in a set must be different) set with exactly one more element in it.

Title: Re: Natural Number Challenge
Post by Johno-G on Jan 10^th, 2003, 3:46pm

Again, sorry if I'm not making myself too clear, but too much clarity leads to a far too easy problem!

Title: Re: Natural Number Challenge
Post by Jeremiah Smith on Jan 10^th, 2003, 4:08pm

I think he's talking about the abstract way of creating the naturals using sets. Using @ to represent the empty set, since I can't do the circle-with-a-slash:

0 is @
1 is {@}
2 is { @, {@} }, I think.
Each number is the set of all the numbers below it. Or something. I'm taking abstract algebra, next semester, and I'll learn all this fun stuff.

Title: Re: Natural Number Challenge
Post by redPEPPER on Jan 10^th, 2003, 5:26pm

Interesting. Although you started with something: the concept of Ø. It's as much nothing as zero is nothing, only in different fields.

Anyway, this demonstration rings a bell, but I don't remember the specifics. How does that construct the natural numbers? Does it have their properties? Can you add them or something? I can see the cardinal of each set is a natural number. But that involves the concepts of set, cardinality and ... natural numbers!

Title: Re: Natural Number Challenge
Post by Johno-G on Jan 11^th, 2003, 1:52am

To create the natural numbers, you simply need something that can be quantified, and that can only go up in descrete ammounts. The natural numbers are created by looking at how many elements are in a set. Using Jeremiah's notation of @ to represent {}, the empty set, we get:
@ is a set with 0 elements, giving you something you can quantify as being 'zero',
{ @ } is a set with 1 element,
{ @ , {@} } is a set with 2 elements,
{ @ , {@} , [@] } is a set with 3 elements, et cetera.
(think induction here:) As long as you have a starting point, (the empty set), you can take any set A with n elements, and if you then take A union {A} you will get a set with n+1 elements. Obviously, you can do this ad infinitum, so you have created an infinite number of sets, each of distinct lengths.

On a footnote, I can see your point about having to start with concepts, but I'd regard those as not actually being anything. Zero is a peculiar one, but you can have before you ZERO apples, you can't have zero sets before you, and you can't have any number of sets before you.

Title: Re: Natural Number Challenge
Post by Icarus on Jan 12^th, 2003, 8:34pm

Actually, there are several ways to model the natural numbers. All of them involve a certain amount of mathematical sophistication to do, including the set-theoretic version offered by Jeremiah and Johno himself. (When you get down to it, set theory involves very complex concepts.)

A different approach is the one used by Euclid, who effectively defined Natural numbers (actually all positive real numbers) as equivalence classes of pairs of line segments. This approach, which I am not going to describe here, could be considered less elementary than the set-theoretic one, since geometry requires set theory, but conceptually it is easier in some ways. After all, the Greeks came up with it more than 2000 years before Cantor provided the set-theory version.