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I am reading D. J. H. Garling: "A Course in Mathematical Analysis: Volume I Foundations and Elementary Real Analysis ... ...
I am currently focused on Garling's Section 1.7 The Foundation Axiom and the Axiom of Infinity ... ...
I need some help with Theorem 1.7.4 ... and in particular with the notion of a successor set $$Z^+$$ ... ... ... ... the relevant text from Garling is as follows:
View attachment 6153
In the above text we read the following:
" ... ... Suppose that $$S$$ is a successor set. Let
$$Z^+ = \cap \{ B \in P(S) : B \text{ is a successor set } \}$$ ... "
Note also that Garling defines a successor set as follows:
" ... ... A set $$A$$ is called a successor set if $$\emptyset \in A$$ and if $$a^+ \in A$$ whenever $$a \in A$$ ... ... "and
Garling defines $$a^+$$ as follows:
" ... ... If $$a$$ is a set, we define $$a^+$$ to be the set $$a \cup \{ a \}$$ ... ... "
Now my problem is that I do not understand the definition/construction of $$Z^+$$ ... ... in each example I construct I seem to get $$Z^+ = \emptyset$$ ... ... and this cannot be right ...
For example ...
Suppose that
$$S = \{ \emptyset , a , a \cup \{ a \} , \{ a , b \}, \{ a , b \} \cup \{ \{ a , b \} \} \}$$$$Z^+ = \cap B_i$$ where $$B_i \in P(S)$$ and each $$B_i$$ is a successor set ...... ... then we find ... ...$$B_1 = \{ \emptyset, a , a \cup \{ a \} \}$$
$$B_2 = \{ \emptyset, \{ a , b \} , \{ a , b \} \cup \{ \{ a , b \} \} \}$$
$$B_3 = S$$ ... so ... ...$$B_1, B_2, B_3$$ seem to me to be the only subsets of $$P(S)$$ that are successor sets and we find that ...$$\cup B_i = \emptyset $$BUT ... surely this cannot be right ...!Can someone clarify this issue and show me how Z^+ is meant to be constructed ...
Hope someone can help ...
Peter====================================================
In order to enable readers to get a better understanding of Garling's notation and approach I am providing the first two pages of Section 1.7 ... as follows:View attachment 6154
View attachment 6155
I am currently focused on Garling's Section 1.7 The Foundation Axiom and the Axiom of Infinity ... ...
I need some help with Theorem 1.7.4 ... and in particular with the notion of a successor set $$Z^+$$ ... ... ... ... the relevant text from Garling is as follows:
View attachment 6153
In the above text we read the following:
" ... ... Suppose that $$S$$ is a successor set. Let
$$Z^+ = \cap \{ B \in P(S) : B \text{ is a successor set } \}$$ ... "
Note also that Garling defines a successor set as follows:
" ... ... A set $$A$$ is called a successor set if $$\emptyset \in A$$ and if $$a^+ \in A$$ whenever $$a \in A$$ ... ... "and
Garling defines $$a^+$$ as follows:
" ... ... If $$a$$ is a set, we define $$a^+$$ to be the set $$a \cup \{ a \}$$ ... ... "
Now my problem is that I do not understand the definition/construction of $$Z^+$$ ... ... in each example I construct I seem to get $$Z^+ = \emptyset$$ ... ... and this cannot be right ...
For example ...
Suppose that
$$S = \{ \emptyset , a , a \cup \{ a \} , \{ a , b \}, \{ a , b \} \cup \{ \{ a , b \} \} \}$$$$Z^+ = \cap B_i$$ where $$B_i \in P(S)$$ and each $$B_i$$ is a successor set ...... ... then we find ... ...$$B_1 = \{ \emptyset, a , a \cup \{ a \} \}$$
$$B_2 = \{ \emptyset, \{ a , b \} , \{ a , b \} \cup \{ \{ a , b \} \} \}$$
$$B_3 = S$$ ... so ... ...$$B_1, B_2, B_3$$ seem to me to be the only subsets of $$P(S)$$ that are successor sets and we find that ...$$\cup B_i = \emptyset $$BUT ... surely this cannot be right ...!Can someone clarify this issue and show me how Z^+ is meant to be constructed ...
Hope someone can help ...
Peter====================================================
In order to enable readers to get a better understanding of Garling's notation and approach I am providing the first two pages of Section 1.7 ... as follows:View attachment 6154
View attachment 6155