34. Transfinite Induction. Well-ordering property. Ordinals.
Let P(α) be a property defined for all ordinals α. Suppose that whenever P(β) is true for all β < α, then P(α) is also true. Then transfinite induction tells us that P is true for all ordinals.
Usually the proof is broken down into three cases:
Zero case: Prove that P(0) is true.
Successor case: Prove that for any successor ordinal α+1, P(α+1) follows from P(α) (and, if necessary, P(β) for all β < α).
Limit case: Prove that for any limit ordinal λ, P(λ) follows from [P(β) for all β < λ].
well-ordering property: Every nonempty subset of the set of positive integers has a least element.
2)The well-ordering property states, that every nonempty subset of the Natural Numbers has a least member.
∀S ≠ ∅ ⊆ N ∃x ∈ S ∀z ∈ S (x ≤ z)
An ordinal is a set of all previous ordinals.
35. Definition of an ordinal. Examples. Limit and non-limit ordinals. Properties of ordinal. Difinition of w0 and w1.
An ordinal is a set of all previous ordinals.
Ex: ᴓ, {ᴓ}, {ᴓ,{ᴓ}}, {ᴓ{ᴓ},{ᴓ{ᴓ}}}
n+1=nU{n}
n+1-is
a non-limit ordinal.
– is a
union of all finite ordinals, w0 not=
for any ordinal .
w0-is limit-ordinal.
w0+1=w0U{w0} is non-limit. w0+2,… w0+n
w0+w0=w0*2 is limit.
w0*3, w0*n,…w0*w0 are limit ordinal.
w1 is the unions of all countable ordinals.