jbn@GLACIER.STANFORD.EDU (John B. Nagle) (07/27/88)
Date: Sun, 24 Jul 88 12:42 EDT From: John B. Nagle <jbn@glacier.stanford.edu> Subject: Re undecidability To: AILIST@ai.ai.mit.edu Goetz writes: > Goedel's Theorem showed that you WILL have an > unbounded number of axioms following the method you propose. That is why most > mathematicians consider it an important theorem - it states you can never have > an axiomatic system "as complex as" > arithmetic without having true statements which are unprovable. Always bear in mind that this implies an infinite system. Neither undecidability nor the halting problem apply in finite spaces. A constructive mathematics in a finite space should not suffer from either problem. Real computers, of course, can be thought of as a form of constructive mathematics in a finite space. There are times when I wonder if it is time to displace infinity from its place of importance in mathematics. The concept of infinity is often introduced as a mathematical convenience, so as to avoid seemingly ugly case analysis. The price paid for this convenience may be too high. Current thinking in physics seems to be that everything is quantized and that the universe is of finite size. Thus, a mathematics with infinity may not be needed to describe the physical universe. It's worth considering that a century from now, infinity may be looked upon as a mathematical crutch and a holdover from an era in which people believed that the universe was continuous and developed a mathematics to match. John Nagle