From: "Hal Finney" <[EMAIL PROTECTED]> To: [EMAIL PROTECTED], [EMAIL PROTECTED] Subject: Re: Fw: Something for Platonists] Date: Mon, 16 Jun 2003 10:46:56 -0700
Jesse Mazer writes:
> Yes, a Platonist can feel as certain of the statement "the axioms of Peano
> arithmetic will never lead to a contradiction" as he is of 1+1=2, based on
> the model he has of what the axioms mean in terms of arithmetic. It's hard
> to see how non-Platonist could justify the same conviction, though, given
> Godel's results. Since many mathematicians probably would be willing to bet
> anything that the statement was true, this suggests a lot of them are at
> least closet Platonists.
What is the status of the possibility that a given formal system such as the one for arithmetic is inconsistent? Godel's theorem only shows that if consistent, it is incomplete, right? Are there any proofs that formal systems specifying arithmetic are consistent (and hence incomplete)?
Godel showed that if it's complete, a theorem about its consistency is not provably true or false within the formal system itself. We can feel certain that it *is* consistent nevertheless, by using a model that assigns meaning to the axioms in terms of our mental picture of arithmetic. For example, with the symbols for multiplication and equals interpreted the way we normally do in arithmetic, you can see that x*y=y*x must always be true by thinking in terms of a matrix with x columns and y rows and another with y columns and x rows, and seeing that one can be rotated to become the other. In the book "Godel's Proof", Douglas Hofstadter gives a simple example of using a model to prove a formal system's consistency:
"Suppose the following set of postulates concerning two classes K and L, whose special nature is left undetermined except as "implicitly" defined by the postulates:
1. Any two members of K are contained in just one member of L. 2. No member of K is contained in more than two members of L. 3. The members of K are not all contained in a single member of L. 4. Any two members of L contain just one member of K. 5. No member of L contains more than two members of K.
From this small set we can derive, by using customary rules of inference, anumber of theorems. For example, it can be shown that K contains just three members. But is the set consistent, so that mutually contradictory theorems can never be derived from it? The question can be answered readily with the help of the following model:
Let K be the class of points consisting of the vertices of a triangle, and L the class of lines made up of its sides; and let us understand Ďa member of K is contained in a member of Lí to mean that a point which is a vertex lies on a line which is a side. Each of the five abstract postulates is then converted into a true statement. For instance, the first postulate asserts that any two points which are vertices of the triangle lie on just one line which is a side. In this way the set of postulates is proved to be consistent."
As I think Bruno Marchal mentioned in a recent post, mathematicians use the word "model" differently than physicists or other scientists. But again, I'm not sure if model theory even makes sense if you drop all "Platonic" assumptions about math.
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