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Topic: Infinity: The Story So Far
Replies: 298   Last Post: Apr 7, 2014 7:30 AM

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 Dan Christensen Posts: 8,219 Registered: 7/9/08
Re: Infinity: The Story So Far
Posted: Mar 1, 2014 11:58 PM
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On Saturday, March 1, 2014 11:23:53 PM UTC-5, Dan Christensen wrote:

> May I recommend something really, really simple like proving the existence of a number other than 0 using your "axioms" alone.

To show you how it is done, here is the proof using only the first 3 of Peano's axioms:

Axioms:

PA1: 0 in N (i.e. 0 is an element of the set N)

PA2: ALL(a):[a in N => S(a) in N] (i.e. S is a function on N)

PA3: ALL(a):[a in N => S(a)=/=0] (i.e. 0 has no pre-image under S)

Proof:

1. Applying PA3 for a=0, we have: 0 in N => S(0)=/=0

2. Applying PA1, we have: S(0)=/=0

3. Applying PA2 for a=0, we have: 0 in N => S(0) in N

4. Applying PA1, we have S(0) in N

5. Joining 2 and 3, we have: S(0) in N and S(0)=/=0

6. Generalizing, we have: EXIST(a):[a in N and a=/=0] as required.

Now, it's your turn, John Gabriel. How about it?

Dan
Download my DC Proof 2.0 software at http://www.dcproof.com
Visit my new math blog at http://www.dcproof.wordpress.com

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