NialScorva

I'm pretty sure that ZF Set theory isn't first order, it's second order, or rather is sufficiently complex to represent second-order logic. My understanding of the issue is that first order logic is a system where variables can range over objects, and a second-order logic is one in which variables can range over properties as well. For example, the statement "There is a property that holds for all prime numbers" is a statement of the second order. Doesn't Godel's proof do this very thing? It takes a first order computational language and encodes it into itself, masquerading the properties of objects within the objects themself. Isn't that what the Prov(x) function does? Show that for all X there is no Prov(x)+1 (or however the diagonal is phrased)?

As to the provability question, I'm certain that Euclidean geometry has been proven consistent, and I was pretty sure that at least some forms of propositional calculus have been shown consistent, though I can't find any references at the moment (connection sucks, browsing is painful, will get back to you when my cable comes back).
I've always considered mathematical proof to me symbol shuffly according to strict rules, and nothing more. The fact that it corresponds quite reliably with reality is a happy bonus.

Skepticism. Consistently is roughly "there does not exist Prov(x) and Prov(~x)". Contrary to inuition, consistency is the skeptically superior position, because inconsistency can be shown evidentially, whereas consistency cannot.

CardinalMan

Your distinction between first-order logic and second-order logic is correct. However, I've always seen ZF set theory done in first-order logic. For example, the book "Set Theory: An Introduction to Independence Proofs", develops ZF in first-order logic and proves the independence of the Axiom of Choice and the Continuum Hypothesis from the others in this setting. One can use first-order logic to talk about first-order properties, but you can not quantify over all of them (such as saying there exists a property, or for all properties). Thus, you can not technically say that there exists a property which is true of all sets, but maybe you can exhibit one. This isn't really a hindrance, and pretty much all of mathematical practice can be performed in first-order set theory. Once you develop set theory in first-order logic, you can then go and develop first-order logic, second-order logic, etc. within set theory, just like the rest of mathematics.

Yes, they have been shown to be consistent. However, the proofs are necessarily mathematical (after all, what else could they be?). Hence, if you want to formalize these proofs, you would have to do them in a mathematical system, say within set theory. Thus, in order to trust them, you would have to trust the consistency of set theory. However, we know that set theory can not prove its own consistency.

This raises another question that I've often thought about. Suppose a system can prove its own consistency. Should we trust it? After all, any inconsistent system in which you allow proof by contradiction (which nearly all mathematicians do) is able to prove its own consistency.

Then why should we trust mathematical proofs more than experimental evidence? Is the square root of 2 really absolutely irrational, or may we one day overthrow this discovery like any other scientific theory? I struggle with these questions myself, so I'm curious what others think about them.

I'm not sure I fully understand what you are saying here. Are you saying that consistency is skeptically superior simply because it is a statement of the form "For all x, ___", while one could demonstrate inconsistency by simply giving an example?

CardinalMan

jpbrooks

I'm sorry that I missed your reply. I usually don't have time on weekdays to scan through all of the threads that I have posted comments in for replies, so I often miss them.

My reply in my last post wasn't directed at any particular standard. It was merely a reply to your comment, "That depends on what standard of 'truth' you adopt".

However, now that you have mentioned materialism, I would say that it could only throw doubt on our ability to confirm truth when it is accepted as an ontological assumption that purports to provide a complete account of reality; not as an assumption to guide epistemological inquiry. I don't find much to object to concerning the application of ("materialistic") scientific methods to the study of the material world. In that case, other methods of inquiry that are discovered would not be ruled out of the question a priori.

What I find problematic is how "ontological materialism" (i.e., materialism as an ontology) can account for how the materialist knows (with certainty) that "ontological materialism" is true. (For example, how is the materialist to account for how he or she knows that no "immaterial" entities exist anywhere in reality?) Since any other truth claim that is consistent with materialism relies on materialism's truth, not being able to be certain about the truth of materialism itself appears to throw doubt on any of its other "truths".

[ February 10, 2002: Message edited by: jpbrooks ]</p>

NialScorva

Pretty much. Consistency can only be proven by showing that there does not exist a contradiction, and there's no mathematical way to show that one cannot exist. Like any other unprovable existential claim, we presume falsehood until such a time as it's show empirically. Consider how many proofs rely on the Axiom of Choice or Reimann hypothesis, but are useful none the less. It's not mathematically rigorous, I know, but we're dealing with a problem that is necessarily extra-mathematical.

I guess it doesn't bother me so much because I'm primarily a software engineer, and as such absoluteness is far less important than utility. I don't have to do things exactly, there's always a tolerance. From that, my life is implicitely about accepting the Church-Turing thesis, whether or not it's strictly correct. Everything I do is about taking real world systems and making them isomorphic to the machine I'm working on, which in turn is isomorphic to a very simple boolean arithmetic. Again, it's not mathematically rigorous, but I don't deal in mathematically rigorous things. Even when I'm working on a project that is mathematically rigorous, I'm not necessarily mathematically rigorous *about* it. Consider that the decision to even use mathematics to describe a problem cannot be a mathematically rigorous choice.

Sorry for the rambling.