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Topic: Wellorder

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	[No title] (Site not responding. Last check: 2007-10-12)
		We give a wellorder of all finite trees such that equality in the ordering is the usual equality between trees.
		This wellorder gives a notation system for all ordinals less than the small Veblen ordinal and all common ordinals from proof theory - as \epsilon_0, \Gamma_0,...
		In our wellordering the ordinals less than \epsilon_0 are the trees with at most binary branching.
	www.ii.uib.no /forskningsgrupper/algo/seminar/data/jervell.txt (186 words)

	[No title] (Site not responding. Last check: 2007-10-12)
		Suppose X cross X for all infinite X. For definiteness, we'll show that you can wellorder R, the set of all reals; you can substitute any other set for R. Let Theta be the Hartog number of R; that is, Theta is the least ordinal such that there is no surjection from R to Theta.
		Now we're going to let X be the disjoint union of R and Theta; that is, we assume we're coding reals and ordinals in such a way that no real is an ordinal, and then we just take X = (R union Theta).
		You can apply this argument to any other infinite set (though you'll sometimes have to take care of the technical matter of the disjoint union, which was automatic for the reals).
	www.math.niu.edu /~rusin/papers/known-math/99/infin_arith (1448 words)

	Transfinite induction - Wikipedia, the free encyclopedia
		However it is very often the case that proofs or constructions using the technique do use AC.
		For example, consider the following construction of the Vitali set: First, wellorder the reals, say into a sequence
		The above argument uses AC in a blatant way at the very beginning, by wellordering the reals.
	en.wikipedia.org /wiki/Transfinite_induction (562 words)

	AUTOMATA, WORDS AND LOGIC - TALKS
		However, an infinite number of iterations might be needed.
		To each linear order L, there is a hierarchy of semi-models s(k,L), each of which is a finite ordered set whose domain has k nested subsets (one within the other).
		To keep the talk interesting and brief, we will describe these results by comparing the differences between the k-quantifier theories of Wellorder and of Linear Order.
	www.math.helsinki.fi /logic/FMTF/meetings/AWL2005/talks.html (1462 words)

	Bernays Paul - new and used books
		North-Holland, Amsterdam Classic comprehensive text presents a formal system of axiomatic set theory, including the original Zermelo-Fraenkel form of set-theoretic axiomatics; frame of logic and class theory; transfinite recursion; power, order, wellorder; completing axioms; analysis, cardinal arithmetic.
		This book features a historical introduction by Abraham A. Fraenkel.
		Chapters include: The Frame of Logic and Class Theory; The Start of General Set Theory; Ordinals, Natural Numbers, Finite Sets; Transfinite Recursion; Power, Order, Wellorder; The Completing Axioms; Analysis, Cardinal Arithmetic, Abstract Theories; and Further Strengthening of the Axiom System.
	www.isbn.pl /A-BERNAYS-Paul (396 words)

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