Sunday, July 12, 2009

My oversight, apologies, and thoughts

After last week's post concerning the Cantorian argument against possible worlds being maximally consistent sets of propositions posted by Dr. Vallicella, the Maverick Philosopher, I sent an email to Dr. Vallicella for his thought, and his initial response was that I missed the point of the argument entirely. He was kind and patient through out the exchange, and entirely correct instating that I had missed the point. Using "e" to represent the element relation, I had entirely missed the point that the truths being constructed were of the form t1 e {t1, t2} and ~(t1 e {t2, t3}. This means that Dr. Grim/Dr. Vallicella have created a set of size T X P(T) of truths from set T. Why I missed this, and what it means for my analysis of the proof, is below the fold.

As for why I missed it, it just seems to be a blind spot with me. Ii will probably not make this error again in the next month or two, but it will happen eventually, because I see a fundamental difference in the type of between saying "the earth orbits the sun" and "t1 e {t1, t2}", and I don't really see the latter belonging in T. The latter sort of statement I think of as being a formal truth, one that it true based on the initial definitions, definition that we select arbitrarily because of their usefulness. My understand of Dr. Vallicella's worldview (and if I mis-characterise him here, I apologize) is that there is no relevant difference between the two statements. As long as I carry the first view in my brain, I no doubt will fall prey to the same blindness about arguments founded in the second view in the future. I beg the patience of any readers in this regard.

Well, after being so kind and professorial in our exchange, Dr. Vallicella certainly deserves to have Grim's/his analysis validated, and it would have been my pleasure to do so. Unfortunately, I can't do this with honesty, because it turns out that the proof will still fail. One of the unfortunate side effects of mixing formal truths and non-formal truths is the formal truths have a tendency to grow past any reasonable size. When you include all the formal truths of the type "t1 e {t1, t2}", you wind up with T being a proper class, and that means |T| does not exist, and to the degree P(T) can be defined, |P(T)| does not exist either, so obviously you can't have |P(T)| > |T|.

This is easy to see when you consider what must be included in T. For simplicity sake, let's start with a universe with one non-formal truth, t0, and call the universe itself T0. We can use the very process describe to create R0 containing two truths: t1 saying t0 e {t0} and t2 saying ~(t0 e {}). Defining T1 as the union of R0 and T0, T1 = {t0, t1, t2}. Applying the same process to T1, you get 24 elements in T2, 402,653,184 elements in T3, etc. Then, you need to combine the contents of all the Tn into Taleph-0. From Taleph-0 you can build (assuming without loss of generality the generalized continuum hypothesis)Taleph-1, Taleph-2, etc., so that there will be a version of Tx greater than any specific cardinal number. This means that, when you finally union all these Tx to create T, T has the size of a proper class.

I want to thank Dr. Vallicella for his cooperation and gentlemanly behavior.


J said...

Start with Cantor's platonic assumptions (ie numbers and sets, including infinity, "existing" apart from the physical world), and you get.....more of Cantor's platonic assumptions (--and Cantor in the madhouse, shrieking about the Aleph or something).

Real analysis begins by offing infinity (or "sets of infinity"). Even if the set of all even numbers are in theory infinite, and the set of whole numbers are in theory infinite infinite (is that a bijection, as y'all say?), but not congruent, it like doesn't matter except as it helps to build bridges..... or bombs.

(the MavPhil himself belongs to the church of bad Platonism as well--)

One Brow said...


Welcome. I found your blog so interesting I added it to my roll. I hope to see your comments regularly.