A description/proof of that fixed-point in proof of Veblen fixed-point theorem is smallest that satisfies condition
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Starting Context
Target Context
- The reader will have a description and a proof of the proposition that the fixed-point shown in the proof of the Veblen fixed-point theorem is the smallest that satisfies the condition that it is a fixed-point that is equal to or larger than the specified ordinal number.
Orientation
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There is a list of propositions discussed so far in this site.
Main Body
1: Description
While a fixed-point, \(o_1\), is shown as a fixed point that is equal to or larger than any ordinal number, \(o_0\), for any monotone and continuous operation, \(f: O \rightarrow O\), from the all the ordinal numbers collection into the all the ordinal numbers collection in the proof of the Veblen fixed-point theorem, in fact, \(o_1\) is the smallest one that satisfies \(o_0 \in= o, f (o) = o\).
2: Proof
By the proposition that for any monotone operation from the all the ordinal numbers collection into the all the ordinal numbers collection and any argument, the value equals or contains the argument, \(o_0 \in= f (o_0)\).
Let us suppose that \(o_0 = f (o_0)\). Then, \(o_1 = o_0\) is the smallest.
Let us suppose that \(o_0 \in f (o_0)\). For any possible fixed-point, \(o\), such that \(o_0 \in= o\) and \(o = f (o)\), \(o_0 \in o\), because \(o_0\) is not any fixed point. \(f (o_0) \in o\), because if it was \(o_0 \in o \in= f (o_0)\), it would be \(o \in= f (o_0) \in f (o)\), a contradiction against \(o\)'s being a fixed point. Likewise, \(f^n (o_0) \in o\) for any natural number, \(n\), such that \(1 \lt n\) where \(f^n\) means \(f \circ f \circ . . . \circ f\) \(n\) times. So, \(o\) is an upper bound of \(\{f^n (o_0)\vert n \in N\}\) where \(N\) is the natural numbers set. So, \(o\) cannot be smaller than the supremum of \(\{f^n (o_0)\vert n \in N\}\), which is \(o_1\).
