Gödel, Tarski, & Logic. (for grace2u)

If they're saying the same thing, why use "or"? I wouldn't say "rich or thick" when referring to whipped cream. I wouldn't call God "all-powerful or omnipotent".Would it have simply been better to pick one of those terms, instead of both? My bad.

In this case, I think it does. Though I'm obviously no mathematician, Godel's proof seems to me to imply that axiomatic systems are not incomplete because they're weak or incapable - they're incomplete as a result of being strong enough to model number theory. That is, axiomatic systems don't become incomplete until they've reached a sufficient level of power and complexity.Now obviously the universe's logic cannot be weaker than our logic. Therefore we know that the universe's logic must be incomplete as well, not because it is weak, but because it is strong.Now, how could your perfect god radiate such imperfect logic? Now, you may think that god's logic is not incomplete, but if it isn't, it can't model number theory. That's pretty weak, don't you think? Why would god have weaker logic than we do?

I thought you were a mathematician not an engineer.... I will read your post in more detail after I respond to :ae:Thanks for the contribution.."Christe eleison"[This message has been edited by grace2u, 11-20-2003]

In this case the difference between a mathematician and an engineer is that the engineer would be satisifed with a finite series of '9's.From a mathematicians point of view we would say that :There is no e > 0 such that 1 - 0.9999... > e
Therefore 1 = 0.9999....(I won't try to get Greek text so you'll have to settle for an 'e' instead of an epsilon)

crashfrog responds to me:

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OR, crash. God is inconsistent OR incomplete. The two are just different ways of saying the same thing.

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If they're saying the same thing, why use "or"?

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Because when you make your choice as to which one you want, you don't get the other. A system can be formalized so that it is either incomplete or inconsistent. The system will generate statements such that they are either undecidable (incomplete) or contradictory (inconsistent) but not both. A single system will not generate one statement that is undecidable while at the same time generating another two statements that contradict each other.Take, for example, the question of the continuum hypothesis. The way we have set up our system, this question is considered undecidable (c = aleph-one is not contradictory and c < aleph-one is not contradictory). The system can be changed such that the statement is contradictory (c = aleph-one and c < aleph-one). What it cannot do is do both at the same time.So you get one or the other. The system will be either incomplete OR it will be inconsistent.

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I wouldn't say "rich or thick" when referring to whipped cream.

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Axiomatic set theories aren't whipped cream.

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Would it have simply been better to pick one of those terms, instead of both?

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No, because they don't mean the same thing. "Rich" and "thick," like "all-powerful" and "omnipotent," are synonyms. "Incomplete" and "inconsistent" are not. The former means that there will be some statements for which one cannot derive any truth value. The latter means that there will be some statements for which one can derive both truth values.Depending on how you deal with the system, incomplete can be turned into inconsistent and vice-versa, but the two are not the same thing.So which do you pick? A system where you land on "I don't know" or a system in which you land on "both"?------------------
Rrhain
WWJD? JWRTFM!

grace2u responds to me:

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0.999... = 1.

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I thought you were a mathematician not an engineer....

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I am. Why do you think the mathematicians never get blamed? The engineers tend to do the most sloppy things...like canceling the derivative symbols....------------------
Rrhain
WWJD? JWRTFM!

At least we can agree on one point. Perhaps there is hope...