asg@pyuxf.UUCP (08/04/88)
From: pyuxf!asg To: bcr!ailist-request@stripe.sri.com Date: Tue, 2 Aug 88 10:43 EDT Subject: Re: undecidability Responding-System: pyuxf.UUCP Path: pyuxf!asg From: asg@pyuxf.UUCP (alan geller) Newsgroups: comp.ai.digest Subject: Re: undecidability Summary: Infinity IS natural Message-ID: <375@pyuxf.UUCP> Date: 2 Aug 88 13:51:14 GMT Article-I.D.: pyuxf.375 Posted: Tue Aug 2 09:51:14 1988 References: <19880727030404.9.NICK@HOWARD-JOHNSONS.LCS.MIT.EDU> Organization: Bell Communications Research Lines: 55 In a previous article, John B. Nagle writes: > 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 Actually, infinity arises in basic set theory, long before any notion of 'finite space' is introduced (viewing mathematics as an inverted pyramid, from lowest-level set theory and logic up). Two axioms suffice to introduce infinity: the axiom of the null set, which says that there exists a set 0, which is empty; and the axiom of construction (or of union, or whatever you prefer to call this axiom), which says that if a and b are sets, then so is {a, b}. These two axioms allow one to construct 0, {0}, {{0}}, etc., which is an infinite series. In fact, it is possible to create models of set theory which are constructed using only sets of this form. In physics, 'quantization' does not mean 'granularization', despite the popular understanding that this is so. While there are physicists who work on theories of granular space, mainstream quantum physics interprets space as a continuum. Indeed, even quantized measurables such as energy levels are seen as selected values 'chosen' out of a continuum by being the eigenvalues of some operator. Also, the notion that the universe is finite is still contraversial; while most cosmologists seem to believe that the universe is closed (i.e., finite), there is still no experimental evidence to support this view (this is why cosmologists talk about the 'missing mass', which is needed to close the universe gravitationally; nobody's found it yet). Alan Geller Bellcore Nobody at Bellcore takes me seriously.
steve@hubcap.UUCP (""Steve" Stevenson") (08/07/88)
Path: hubcap!steve From: "\"Steve\" Stevenson" <steve@hubcap.UUCP> Newsgroups: comp.ai.digest Subject: Re: undecidability Date: Thu, 4 Aug 88 09:09 EDT References: <19880803191849.8.NICK@HOWARD-JOHNSONS.LCS.MIT.EDU> Organization: Clemson University, Clemson, SC Lines: 23 From a previous article, by asg@pyuxf.UUCP: > In a previous article, John B. Nagle writes: >> Goetz writes: >>> [Goedel's incompleteness ... unbounded number of axioms] >> Always bear in mind that this implies an infinite system. >> There are times when I wonder if it is time to displace infinity from >> its place of importance in mathematics.... > Actually, infinity arises in basic set theory, ... But isn't this the point? The nominalists/finitist won't let you get to that step. Take for example your mythical perfect(?) computer programmer. To such a person, the discussion of infinity in any guise is lost: there just aren't any infinite processes (by almost anybody's) definition. Intuitionist as a little better - only countable infinity allowed. The foundational issue is whether or not it is legit to propose successor and related things as legit bases for mathematics. That's John's point: The canon of infinity may not be all that good an idea.-- Steve (really "D. E.") Stevenson steve@hubcap.clemson.edu Department of Computer Science, (803)656-5880.mabell Clemson University, Clemson, SC 29634-1906