Thursday, May 31, 2007

GEB Week 2: Chapter 4, Consistency

It's been a crazy week. Surprisingly, I found time to read this week's assignment. In the last few chapters, the author has tactfully introduced many of the important concepts in studying and understanding formal systems. One particular argument he has made is that the form of a formal system is completely disconnected from its meaning. Here's an excerpt which takes this further:

It now becomes clear that consistency is not a property of any formal system per se, but depends on the interpretation which is proposed for it. By the same token, inconsistency is not an intrinsic property of any formal system. (GEB20 94)

An alternative conclusion might have been that "all formal systems are consistent by definition," but the author disagrees with this characterization. Instead, he makes the stronger claim that it is completely improper to assign "consistency" or "inconsistency" to a formal system. What distinction is he making, and to what end?


Martin Harrigan said...
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Martin Harrigan said...

Completeness (with respect to some interpretation) is the other major theme of this chapter: Basically, a system is complete if every true statement which can be expressed in the notation of the system is a theorem.

We make an incomplete system complete by either:

1. adding new rules to the system making it more powerful, or

2. tightening up the interpretation.