Re: Goedel - interesting problem?

From: Acme Diagnostics (LFinezapthis_at_partpostmark.net)
Date: 06/16/04 

Date: 16 Jun 2004 03:23:12 -0500

Chris Menzel <cmenzel@remove-this.tamu.edu> wrote:
>On 15 Jun 2004 18:47:12 -0500, Acme Diagnostics <LFinezapthis@partpostmark.net>
>said:
>> Is it true that a consistent math theory as rich as ordinary arithmetic
>> will contain true statements that cannot be proved by it's axioms?
>
>Well, not if we take a theory in the usual sense (among logicians) to be
>the set of consequences of some set of axioms, and "theory T contains
>sentence S" in the usual sense to mean simply that S is a member of T.

Ok, I thought I was being more precise by using "theory" but I instead
I think I went out of scope. Forgetting what would be usual among
logicians altogether, *is it possible* to correctly say that ordinary
arithemetic and other axiomatic mathematical systems similar to
arithmetic, which are consistent, contain true statements that cannot
be proved by their axioms?
>
>> The word "contain" is important to me.
>
>Right. This is one of those points where it's a bit hard to avoid
>getting just a bit technical if Godel's Theorem is to be understood
>properly. The thing that contains the unprovable sentences of a theory
>is the formal *language* in which the theory is formulated, not the
>theory itself. The usual language for arithmetic contains the numeral
>0, a symbol for the successor ("plus one") operation, and the usual
>symbols for addition and multiplication. So what the Theorem says, a
>little more carefully (but still somewhat roughly), is that, for any
>given set of axioms written in the usual language of arithmetic, there
>are truths about the numbers that can be expressed in that language that
>are not provable from those axioms.

Ok. That is very helpful and informative. I always read "formal
language" as things like math definitions and terms, but now I will
read it as a separate thing including for instance symbols, rules, and
other explanation about the math theory. Is there any term that would
include both the theory and the formal language for it? How about
axiomatic system or mathematical system?

>> These things that cannot be proved, are they intrinsic to the theory?
>
>> You seem to indicate above that they are empirical truths, i.e. things
>> observed in the real world.
>
>No, again, they are truths about the numbers.

Thanks. Very important to me. Also what I had thought.

Larry

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