Tuesday, July 27, 2010

Putting the Grim Philosopher argument to sleep

Now that we understand just what sort of thing a maximal collection of consistent truths must be, we can get around to discussing the attempt to form a contradiction based upon its possible existence, which argument The Maverick Philosopher (Dr. Vallicella) seems determined to support.

In fact, The Maverick Philosopher is so intent on preserving this argument that my comments pointing out a summarized version of my last post are no longer present. That is, I posted the first comment on 7/14, it was gone by 7/15, I posted another comment on 7/18 in the morning, I checked back on 7/18 in the afternoon to see it there, it was gone on 7/19, I sent an email on 7/19 asking why my comment was removed, and so far I have received no response to that email. I should not be surprised The Maverick Philosopher prefers to deal the disproof of his argument by covering up the evidence. After all, this is a man who, despite the presence of a dictionary definition that specifies "A strong fear, dislike, or aversion" as the definition of 'phobia', insists that "A phobia is an irrational fear. If you use the word in any other way you are misusing it. " Apparently avoiding the inconvenient is one of his preferred methods of dealing with reality. So, I'll have to produce the reality regarding his proof on my blog below the fold, where he can avoid it by not coming here at all. For someone who shakes in his boots at the idea of Muslims and thinks it rational, discretion is undoubtedly the better part of valor.

Addressing the 4 points in order:
1. Cantor's Theorem states that for any set S, the cardinality of the power set P(S) of S > the cardinality of S. ... But the proof needn't concern us. It is available in any standard book on set theory.

Quite accurate, assuming the power set can be formed. That's part of the issue with this proof. When you get to a certain size, a collection goes from being a set to a proper class. Depending on the particular set theory, this means either that, for some proper class V, either P(V) cannot be constructed (NBG and ZFC) or it cannot be larger than V (MK). Either way, Cantor's Theorem does not apply.

2. Suppose there is a set T of all truths, ... But according to Cantor's Theorem, the power set of T is strictly larger than T. So there will be more of those truths than there are truths in T. It follows that T cannot be the set of all truths.

As I demonstrated in the last post, the size of T is the size of the ordinal numbers, which is a proper class. So, there cannot be a greater cardinality in the class of all subclasses of T, even if such a subclass were to exist.

3. Given that there cannot be a set of all truths, the actual world cannot be the set of all truths. This implies that possible worlds cannot be maximally consistent sets of propositions. I learned the Cantorian argument that there is no set of all truths from Patrick Grim. I don't know whether he applies it to the question whether worlds are sets.

Since we have no disproof of the notion that there can be a class of all truths (such a class would exist under any typical rendition of set theory), it turns out that worlds can indeed be maximal collections of consistent truths.

4. As far as I can see, the fact that possible worlds cannot be maximally consistent sets does not prevent them from being maximally consistent conjunctive propositions.

This position requires at least some reason to differentiate between the category of consistent sets and the categroy of consistent conjunctive proposition, in that there are member of one which are not naturally a part of the other. However, there is an obvious and natural one-to-one relationship between the category of consistent sets and the category of consistent conjunctions. Namely, {pi}i ε I <---> &i ε I(pi). In fact, if you define a partial order on the category of consistent conjuction by sayin that &i ε I(pi) <= &j ε J(pj) whenever, for each i ε I, pi ε {pj}j ε J, this relationship even preserves the partial order of the set category. This means any maximal consisent conjunction would map to a maximal collection of consistent propositions. So, if you say the maximal collections of sets does not exist, perforce the maximal consistent conjuction does not exist, either.


J said...

Vallicelli shut his site down to any dissent a few years ago, though he lets a few tame skeptical remarks appear once in a great while.

Really, I question his sort of pseudo-platonic (or whatever it is) analysis of theology. He routinely suggests that all sorts of "truths" hold which have no relation to...empirical reality for lack of a better term. And like Feser, the MavP's opposed to evidentiary reasoning in any form. So what are his premises? The Ontological Argument's about the only a priori sort of theological argument (and not a very sound one, as Kant showed).

Someone may say there's no evidence for some particular religious claim (e.g. miracles..if not historical reading of the Bible) and he says...well, justify "evidentialism." It's not really logic, but logical tricks...the priest's stock in trade.

J said...

Oh, I should have read through the set theory jazz. Alas, Cantor was wrong--it's a sophisticated (or is it..mad) type of reification. For that matter, infinity doesn't..."exist" (which is to say, claiming there are different-sized infinities..integers, vs even numbers, seems about like saying...Pegasus glides around Sirius...Infinity is merely a potential, ie, countability...from my tentative conceptualist definition {with a slight hint of nominalism} of "cardinality"...).



One Brow said...

I'm still think on the whole notion of whether there is an actual infinite. Do positional values count as real things? If you can be one inch, then another inch, etc. and without limit, then there are an infinite number of positions you can hold.

J said...

you're probably aware of the ancient problem of zeno (ie travel half the distance of the arrow to the target, and then half that, and half that etc)...that does not mean the arrow never hits the target. It means that one can apply the divisor to the distance--but it's still just one magnitude, right (the distance from archer to target). It can be summed. 99.999999 will work for one....

And similarly I read n+ 1 as...you can always add a number. But that doesn't create infinite sets. It means something like the mathematical model can always be expanded (via exponents, say, factorials, etc)...to fit like engineering or physics problems, particles, molecules, etc. But the actual sets do not "exist" like molecules do. IT's a model. (needs a better proof, but that's sort of the conceptual basis for something like constructivism...).

And for that matter, I think infinity, whether numerical or...temporal helps the believers. It's so vast and overwhelming that they call it God (Descartes uses an argument like that somewhere).

One Brow said...

Zeno wasn't aware of infinite sums, certainly, nor of infinitesmals times. Although, it length and width are quantized, you eventually can't go just half the distance.

I agree merely making a model does not make an infinite set. My question was, if a location is a real thing, and you can then overlay the infintie model onto distinct locations, would that be a real instantiation of the the infinite model?

J said...

Zeno's Dichotomy does offer an infinitesimal of a primitive sort--so how does the series end? by the time the arrow hits the target, a calculator with the 1/n function (1/2, 1/4, 1/8, 1/16, etc) would still be clicking away, and would click..infinitely. So, the ancient skeptic says...motion doesn't exist. We probably can't accept that, but even with limits we choose to set parameters and sum, really halt the series, right, at 10 decimal points (or 1000, whatever). That doesn't really solve the problem, logically speaking (tho will do for practical purposes).

One Brow said...

I agree that limits we choose to set for parameters don't solve Zeno's paradoxes. However, if length and/or time are quantized, that's not a limit we choose to set.

I'm stil curious if you consider a location to be a real thing, and if so, would that make an infinite number of locations into an instantiation of a model for infinity?