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**The smallest natural number that cannot be described in sixteen words without use of self reference**.

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- First of all, it is
**Feferman**-Schütte (after Solomon Feferman).

Next, there are larger ordinal notation systems "from below", notably Schütte's Brackets (or*Klammersymbole*) which go up to the so-called large Veblen ordinal. These are impredicative in the technical sense, but only barely so, and hence they (and the systems whose strength they measure) are sometimes called*metapredicative*(but we don't have a technical definition of "metapredicative" yet!).

Ordinal notation systems using collapsing, e.g., for the Howard ordinal, are on the face of it "gappy", since they employ symbols that can be thought of as uncountable ordinals. After the fact, though, we can just think of Ω in the notation system for the Howard ordinal as denoting the Howard ordinal itself (essentially taking the Mostowski collapse of the image of the notations in the ordinals). Then the system is ungappy, but doesn't embed nicely in larger systems using larger cardinals (or their recursive analogs).Apr 25, 2012 - Thanks, +Ulrik Buchholtz - I'm trying to learn this stuff, going rather slowly because it's just a hobby so far, though it would be fun to think of a new idea someday... I don't even understand the technical definition of 'impredicative' yet! There's not a
*This Week's Finds in Infinity*, is there?Apr 25, 2012 - I'm going to try to understand Schütte's
*Klammersymbole*. Are they discussed in his book or only in his "Kennzeichnung von Ordinalzahlen durch rekursiv definierte Funktionen"? This stuff is hard enough already without the language barrier!Apr 25, 2012 - +John Baez , I'm afraid there's no
*This Week's Finds in Infinity*yet; perhaps someone should create it!

Technically (following the analysis of Feferman and Schütte), a predicative ordinal is the ordertype of a relation that can be defined and*proved to be wellfounded*using only previously secured sets (say, in the ramified hierarchy). Feferman has an article discussing the various possible meanings of "predicative", and why the above definition is reasonable (http://math.stanford.edu/~feferman/papers/predicativity.pdf).

Schütte's Klammersymbole are a way of extending the Veblen functions of finite arity to infinite arity. I think Veblen already discussed the idea in his work, but Schütte worked out a way to make a recursive notation system from it. I don't remember if it's discussed in his book (but I guess it is). A shorter treatment is in Crossley and Kister's*Natural Well Orderings*(http://dx.doi.org/10.1007/BF02017491), which also treats of many other systems and the history of their discovery. If that's too short, try this article by Schütte's student, Hilbert Levitz: http://dx.doi.org/10.5169/seals-37163Apr 26, 2012 - Great, thanks! I'll read those.Apr 26, 2012
- +NotReal Naame - The use of English, is, of course, just a trick for starting a conversation. If we go for a formalization using Turing machines or something like that, as +Sridhar Ramesh and you suggest, it seems maybe my idea will morph into the Kritchman-Raz proof of Goedel's 2nd incompleteness theorem, as +Terence Tao pointed out. A summary of that proof is here, by the way:

http://johncarlosbaez.wordpress.com/2011/10/06/chaitins-theorem-and-the-surprise-examination-paradox/

In my dreams, I'd like to formalize the notion of 'avoiding self-reference' well enough to get that to play a real role in the formalization of what I wrote. Logicians try to do this using the idea of 'predicative arithmetic'. But I don't understand that well enough to proceed, so I just thought I'd throw the idea out for discussion. I got the idea from this sentence:

"Specifically, we expect that any proof within predicative mathematics of the existence of TR(2,4) will have incomprehensibly many symbols; e.g., more than A(5,5) symbols."

at the end of here:

http://www.math.osu.edu/~friedman.8/pdf/EnormousInt112201.pdfApr 26, 2012