From: Trivik Bhavneni (trivik@stwing.upenn.edu)
Date: Wed Apr 16 2003 - 23:27:30 BST
> How do you change the axioms of arithmetic?
that is the beauty of it. technically when you have a formal system the
symbols used are drained of all meaning. in a formal system (which is
what was wanted) you have the following:
1. the set of symbols used
2. a procedure to determine wether a sequence of symbols is an acceptable
string (sort of like a grammer checking thing so 1 = 2 is gramatically
correct while -+-12 is not gramatically correct) - tells you the formula
3. a set of rules which tell you how to get a new 'true' sequence from old
ones (transformation rules)
4. a set of axioms, or sequences of strings to start out with.
now to to formalize arithemitic one can chose from literally an infinite
number of posibilities for symbols(which is obvious), grammer rules,
transformation rules and axioms.
a formula is a theorem iff you can get it by performing a finite number of
transformation rules on the axioms to get it, else it is not a theorem
(the axioms are the theorems we are given to start out with).
once you have the system, you can interprete it from the outside to mean
something.
a popular example:
symbols used
'-', 'Q','P'
gramatically correct sentences
'xPyQz'
where x,y and z are (posibly empty) sequences of '-'
(so eg 'PQ--' is gramatically correct while 'P-PQ-' is not)
transformation rules
a. if 'xPyQz' is a theorem so is 'xPy-Qz-'
b. if 'xPyQz' is a theorem so is 'yPxQz'
axiom
'PQ'
fo now a list of theorems:
1. 'PQ'
2. 'P-Q-' (applying a)
3. '-PQ-' (applying b)
4. '-P-Q--' (applying a)
5. '-P--Q---' (applying a)
and so on.
this step by step process is called a proof of the final theorem produced.
now it must be clear by now what one could interprete this system as. that
is the addition of non negative integers, where P can be interpreted as
plus and Q as equles. so in a sence, above in steps 1 through 5 we
prooved that '1 + 2 = 3'. also you should be able to see (with some work)
that '--P-Q-' cannot be a theorem in the system (i.e. '2 + 1 = 1') so
while it is gramatically correct it is what we would interprete as being
false. now the whole thing about godels theorem is that asumeing we have a
consistent (that is there are no proofs for false statements)
formilization of arithemitic welll then it is imposible for every 'true'
statement to have a proof. what he did was basically tell us how to
produce such a gramatically correct sentence which is neither true or
false (using only numbers, '+', '*'); not proovable nor unproovable.
simple, good math thats all it is.
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