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1
Beginning Number Theory
Well-Ordering. and. Induction. Theorem 1.8 Consider the usual ordering on N,
namely, if a,b € TV, then a < b if and only if b - a > 0. This yields 1 < 2 < 3 < etc.
This ordering has the following properties: If a,b, c are distinct natural numbers,
then ...
2
Discrete Mathematics with Applications
The well-ordering principle for the integers looks very different from both the
ordinary and the strong principles of mathematical induction, but it can be shown
that all three principles are equivalent. That is, if any one of the three is true, then
so ...
Every countable set has a well-ordering. Having proved that R is uncountable,
Cantor soon asked whether it too may be well-ordered. Initially Cantor felt that
such an ordering must be possible, and went on to suggest that every set has a ...
If a collection 6 of well ordered sets is a chain with respect to continuation, and if
U is the union of the sets of 6, then there is a unique well ordering of U such that
U is a continuation of each set (distinct from U itself) in the collection 6. Roughly ...
5
Introduction to Model Theory
Consequently, if (X, <) is narrow, but not itself a set, then X is the only segment of
(X, <) that is not a set. In particular, X cannot be a section, which means that (X, <)
has no greatest element. A well-ordering is, by definition, a narrow linear ...
‹0,1, 2,...,...,–3,–2,–1› is not a well-ordering: it does not single out any whole
number as the first to succeed all the natural numbers. On the other hand the
standard imposition of order on ‹0, 1,2,...› is a well-ordering. So too the non-
standard ...
CHAPTER 3 WELL-ORDERINGS AND ORDINALS §1. WELL-ORDERINGS What
is the efiect of this condition? If A is a non-empty well-ordering then A must have
a least element no because A itself is a non-empty subordering ofA. If {ao} does ...
Joseph G. Rosenstein, 1982
One reason, why Gentzen's result was so much appreciated, when it was
presented, was that he reduced the consistency of Peano Arithmetic to the
principle of well-ordering up to eo, which we intuitively believe to be correct. The
argument ...
S. Barry Cooper, John K. Truss, 1999
9
Proofs of the Cantor-Bernstein Theorem: A Mathematical Excursion
19.7 CBT from the Well-Ordering Theorem Poincaré, determined to bar any
possibility to prove CBT except by assuming complete induction, added the
following (Poincaré 1906b p 315): It is possible to deduce Bernstein's theorem
from the ...
10
The Structure of the Real Line
Consider a strict well-ordering W of [0, 1], i.e., the set [0, 1] is well ordered by the
relation (x, y) G W. We denote the diagonal by D = {{x,x} : x G [0, 1]}. Then the sets
W, D and W^1 form a partition of [0,1] x [0,1]. The descriptive complexity of the ...
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Axiom Of Choice - The Programmer's Guide
... there is an order relation that can be applied to the elements so that every subset has a least element under the order. This is Zermelo's well-ordering theorem. «iProgrammer, 七月 13»
Tech grad competes on reality show
I consider it an honor, but as a mathematician I have to reject the notion that the set of nerds has a naturally-induced well-ordering, so I consider any sort of ... «Technique, 一月 13»
