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#1 2008-12-02 10:59:23
- cadjeff
- Member
- Registered: 2007-05-08
- Posts: 26
number systems, natural, rationals, reals etc... & cardinality
cool ok, i'll do what i can first and then get back to you.
j
Last edited by cadjeff (2008-12-03 02:14:02)
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#2 2008-12-02 17:49:31
- Ricky
- Moderator
- Registered: 2005-12-04
- Posts: 3,791
Re: number systems, natural, rationals, reals etc... & cardinality
Yes, but we are here to help you. Start with one of the problems, and let us know what you are thinking, how far you got, what you're having trouble on.
"In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..."
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#3 2008-12-03 05:34:18
- mathsyperson
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- Registered: 2005-06-22
- Posts: 4,900
Re: number systems, natural, rationals, reals etc... & cardinality
You've got (i) and (ii) right as long as you're taking the natural numbers to include 0.
If they start at 1, then (i) is a model and (ii) isn't.
I'll use the version that includes 0 for the rest of this post, since that's what you're doing.
For (ii), you just need to briefly say why each of the axioms are satisified.
ie. e = 0 is a natural number.
x+2 = y+2 --> x = y
(P5) is a bit trickier. To show this, try to construct a system that satisfies the two conditions, and see what happens.
By the first one, 0 ∈ A.
But then the second one says that σ(0) = 2 ∈ A
Now we know that 2 ∈ A, so therefore σ(2) = 4 ∈ A.
Inductively, you can see that A is the set of even numbers (so A = N).
Do a similar thing with (P5) for question (iii), except this time you're trying to construct a set A that is not equal to N.
This will show that (iii) is not a model.
Why did the vector cross the road?
It wanted to be normal.
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#4 2008-12-03 17:32:23
- Ricky
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- Registered: 2005-12-04
- Posts: 3,791
Re: number systems, natural, rationals, reals etc... & cardinality
How do you show any set is equal to all of the integers? (Think of the Peano postulates)
"In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..."
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