I am reading D. J. H. Garling: "A Course in Mathematical Analysis: Volume I Foundations and Elementary Real Analysis ... ...(adsbygoogle = window.adsbygoogle || []).push({});

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:

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 of ##Z^+## ... ... in each example I construct I seem to get ##Z^+ = \emptyset## ... ... and this cannot be right ...

For example ...

Suppose that a successor set S is such that:

##S = \{ \emptyset , a , a \cup \{ a \} , \{ a , b \}, \{ a , b \} \cup \{ \{ a , b \} \} \}##

... then we have ...

##Z^+ = \cap B_i ## where ##B_i \in P(S)## and each ##B_i## is a successor set ...

then we have ..

##B_1 = \{ \emptyset, a , a \cup \{ a \} \}##

##B_2 = \{ \emptyset, \{ a , b \} , \{ a , b \} \cup \{ \{ a , b \} \} \}##

and

##B_3 = S##

Indeed,

##B_1, B_2, B_3## seem to me to be the only subsets of ##P(S)## that are successor sets

and

##\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:

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# I Axiom of Infinity & Garling, Th. 1.7.4 & the successor set

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