Re: [math-fun] Tarski and Multiplication
Let me know if you find anything. You (& Tarski) now have me intrigued! At 12:22 PM 7/7/2014, Scott Fenton wrote:
Hi Henry,
That citation's the one I found already. I'll try the editor though. Thanks for the idea.
-Scott
On Mon, Jul 7, 2014 at 1:39 PM, Henry Baker <hbaker1@pipeline.com> wrote:
You might try the book that is referenced:
Tarski, Alfred; Jan Tarski (24 March 1994). Introduction to Logic and to the Methodology of Deductive Sciences (4 ed.). USA: OUP. ISBN 978-0-19-504472-0
Or you might try to contact the wikipedia article editor.
At 10:15 AM 7/7/2014, Scott Fenton wrote:
Hi all,
In Tarski's "Introduction to Logic", he gave a nice little set of eight axioms that he says uniquely define the real numbers ( http://en.wikipedia.org/wiki/Tarski%27s_axiomatization_of_the_reals). He further said that, using less than, cuts, and addition, you could define multiplication in this system. Does anyone here know if he published anything else on this system? I can't seem to dig anything up, and I'd like to know how he defined things.
Thanks, -Scott
OK, I can't find anything other than a mention of it being related to Eudoxus' definition of magnitude. From Euclid's Elements, that goes as follows (rephrased into modern notation): If A, B, C, D are positive reals, then A/B = C/D iff: forall n, m in NN : ( ( n*A < m*B iff n*C < m*D ) and ( n*A = m*B iff n*C = m*D ) and ( n*A > m*B iff n*C > m*D ) )
From total ordering, this can be simplified to:
forall n, m in NN : ( n*A < m*B iff n*C < m*D ) With cross-multiplying, setting B to 1, and renaming variables, we get this: If A, B, C are positive reals, then A*B = C iff forall n, m in NN: ( n*A < m iff n*C < m*B ) Since n, m are naturals, the multiplication there can be replaced by summation: forall n, m in NN: ( sum(A,k,1,n) < m iff sum(C,k,1,n) < sum(B,k,1,m) ) We can generalize this to all real numbers as follows (I think): Given A, B real, A*B is defined as follows: If A or B is 0, then A*B = 0 Else, A*B is the unique real number "x" such that: ( (0 < A iff 0 < B) iff x < B ) and forall n, m in NN: ( sum(abs(A),k,1,n) < m iff sum(abs(x),k,1,n) < sum(abs(B),k,1,m) )
From a modified version of density, I'm pretty sure I can prove uniqueness of x. The problem now is to prove existence and the field properties. Anyone got any ideas? Remember, we've only got addition, limits, and comparison to work with.
-Scott
Sorry, replace "x < B" with "0 < x" above. On Tue, Jul 8, 2014 at 10:53 AM, Scott Fenton <sctfen@gmail.com> wrote:
OK, I can't find anything other than a mention of it being related to Eudoxus' definition of magnitude. From Euclid's Elements, that goes as follows (rephrased into modern notation):
If A, B, C, D are positive reals, then A/B = C/D iff:
forall n, m in NN : ( ( n*A < m*B iff n*C < m*D ) and ( n*A = m*B iff n*C = m*D ) and ( n*A > m*B iff n*C > m*D ) )
From total ordering, this can be simplified to:
forall n, m in NN : ( n*A < m*B iff n*C < m*D )
With cross-multiplying, setting B to 1, and renaming variables, we get this:
If A, B, C are positive reals, then A*B = C iff
forall n, m in NN: ( n*A < m iff n*C < m*B ) Since n, m are naturals, the multiplication there can be replaced by summation:
forall n, m in NN: ( sum(A,k,1,n) < m iff sum(C,k,1,n) < sum(B,k,1,m) )
We can generalize this to all real numbers as follows (I think):
Given A, B real, A*B is defined as follows:
If A or B is 0, then A*B = 0 Else, A*B is the unique real number "x" such that:
( (0 < A iff 0 < B) iff x < B ) and forall n, m in NN: ( sum(abs(A),k,1,n) < m iff sum(abs(x),k,1,n) < sum(abs(B),k,1,m) )
From a modified version of density, I'm pretty sure I can prove uniqueness of x. The problem now is to prove existence and the field properties. Anyone got any ideas? Remember, we've only got addition, limits, and comparison to work with.
-Scott
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Henry Baker -
Scott Fenton