Hi,We keep going around and around because you keep dropping my distinction between two different cases . . . .
The statement that "The cat is red" is undecidable by arithmetic because it can't even be defined in terms of the axioms of arithmetic (i.e. it has *meaning* outside of arithmetic). You need to construct additions/extensions to arithmetic to even start to deal with it.
The statement that "Pi is a normal number" is decidable by arithmetic because each of the terms has meaning in arithmetic (so it certainly can be disproved by counter-example). It may not be deducible from the axioms but the meaning of the statement is contained within the axioms.
The first example is what you call a constructivist view. The second example is what you call a classical view. Which one I take is eminently context-dependent and you keep dropping the context. If the meaning of the statement is contained within the system, it is decidable even if it is not deducible. If the meaning is beyond the system, then it is not decidable because you can't even express what you're deciding.
Mark
----- Original Message -----
From: "Abram Demski" <[EMAIL PROTECTED]>
To: <[email protected]> Sent: Tuesday, October 28, 2008 9:32 AM Subject: Re: [agi] constructivist issues
Mark,You assert that the extensions are judged on how well they reflect the world.The extension currently under discussion is one that allows us to prove the consistency of Arithmetic. So, it seems, you count that as something observable in the world-- no mathematician has ever proved a contradiction from the axioms of arithmetic, so they seem consistent. If this is indeed what you are saying, then you are in line with the classical view in this respect (and with my opinion). But, if this is your view, I don't see how you can maintain the constructivist assertion that Godelian statements are undecidable because they are undefined by the axioms. It seems that, instead, you are agreeing with the classical notion that there is in fact a truth of the matter concerning Godelian statements, we're just unable to deduce that truth from the axioms. --Abram On Tue, Oct 28, 2008 at 7:21 AM, Mark Waser <[EMAIL PROTECTED]> wrote:*That* is what I was asking about when I asked which side you fell on.Do you think such extensions are arbitrary, or do you think there is a fact of the matter?The extensions are clearly judged on whether or not they accurately reflect the empirical world *as currently known* -- so they aren't arbitrary in thatsense. On the other hand, there may not be just a single set of extensions that accurately reflect the world so I guess that you could say that choosing among sets of extensions that both accurately reflect the world is(necessarily) an arbitrary process since there is no additional informationto go on (though there are certainly heuristics like Occam's razor -- but they are more about getting a usable or "more likely" to hold up under future observations or more likely to be easily modified to match future observations theory . . . .).The world is real. Our explanations and theories are constructed. For any complete system, you can take the classical approach but incompleteness (of current information which then causes undecidability) ever forces you intoconstructivism to create an ever-expanding series of shells of stronger systems to explain those systems contained by them.------------------------------------------- agi Archives: https://www.listbox.com/member/archive/303/=now RSS Feed: https://www.listbox.com/member/archive/rss/303/Modify Your Subscription: https://www.listbox.com/member/?&;Powered by Listbox: http://www.listbox.com
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