Registered: Sep 2011
There is a stir in the math logic community, and even a rare bit of press (http://www.newscientist.com/article...and-beyond.html)
for this most ivory-tower-ish of disciplines. It is due to a claim of Hugh Woodin, among others, that they may have found a formal theory which unifies all of set theory into a single framework, called "Ultimate L", for answering whether mathematical statements should be considered true or false.
Prof. Woodin really believes in the infinite. He has devoted a career to studying it, and is one of the most respected living mathematical logicians. Recently, at the Harvard seminar "Exploring the Frontiers of Incompleteness", Woodin described his recent thinking on mathematical truth. While making a point about how large infinite sets ("large cardinals") can have meaningful implications, he gave an example which I think might be of interest to this forum:
There is a Godel-like sentence that a has real-life consequence if true, yet our 'evidence' for it, while convincing, depends on the existence of things too large to possibly be meaningful for us. His point was that the phenomena of "large cardinal hypotheses" exists on the finite level as well.
The setting is a special version of set theory, ZFC_0, that has as an axiom "there is a set which contains all sets". These axioms are satisfied by all of the standard finite universes for set theory, called V_n, for any whole number n. The sentence X he contructs "says" (using the magic of Godel numbering) that "I (this sentence) am unprovable by any proof from ZFC_0 less than 10^24 in length AND the set V_N exists", where N is some very big finite whole number. Crucially, Woodin chooses N to be so big actually that there is no reason at all to believe that our universe contains that many things, and certainly so big that we could never hope to represent such an object in a digital non-quantum computer. Say N = 10^10^124.
By Godel's logic, which can be summarized as "Suppose this sentence is false. Then by what is says, it's provable, and therefore true", X is true, and we cannot find a proof of length 10^24 or less.
The amazing conclusion: that we are persuaded by Woodin's proof that our real-world computers will never verify such a proof of X (with current technology, we could in principle check if a given "proof" of length less than 10^24 were valid), yet how can we believe this and not believe that V_N also has a physical meaning, despite the lack of evidence?
Last edited by Peter Barendse on 09-26-2011 at 05:00 PM
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