Implementation of mathematics in set theory: Difference between revisions

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starting section on ordinals
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and write <math>W_1 \sim W_2</math> just in case there is a bijection f from the field of
<math>W_1</math> to the field of <math>W_2</math> such that <math>x W_1 y \leftrightarrow f(x)W_2f(y)</math> for all x and y.
 
Similarity is shown to be an equivalence relation in much the same way that equinumerousness
was shown to be an equivalence relation above.
 
In [[NFU]] we define the <strong>order type</strong> of a well-ordering W as the set of
all well-orderings which are similar to W. We define the set of ordinal numbers as the
set of all order types of well-orderings.
 
In [[ZFC]] we cannot do this, because the equivalence classes are too large. It would
be formally possible to use Scott's trick to define the ordinals in essentially this way
nonetheless, but instead we use a device of [[von Neumann]].
 
For any partial order <math>\leq</math>, the corresponding <strong>strict partial order</strong>
<math><</math> is defined as <math>\{(x,y) \mid x \leq y \wedge x \neq y\}</math>. Strict
linear orders and strict well-orderings are defined similarly.
 
A set A is said to be <strong>transitive</strong> if <math>\bigcup A \subseteq A</math>:
each element of an element of A is also an element of A. A <strong>(von Neumann) ordinal</strong>
is a transitive set on which membership is a strict well-ordering.
 
In [[ZFC]], the order type of a well-ordering W is then defined as the unique von Neumann ordinal
which is equinumerous with the field of W and membership on which is isomorphic to the strict
well-ordering associated with W. (the equinumerousness condition distinguishes between well-orderings with fields of size 0 and 1, whose associated strict well-orderings are indistinguishable).