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I am reading "Introduction to Set Theory" (Third Edition, Revised and Expanded) by Karel Hrbacek and Thomas Jech (H&J) ... ...

I am currently focused on Chapter 1: Sets and, in particular on Section 3: The Axioms where Hrbacek and Jech set up an axiomatic systems (which they do NOT call ZFC ... but it seems to mirror ZFC ... )

I need some help with Example 3.8 (c) which effectively defines the union of two sets ... . In particular I need some help in fully understanding how this definition requires H&J's Axiom of Pair and Axiom of Union in order to justify the definition ...

Example 3.8, the notes following it and the Axiom of Pair and the Axiom of Union read as follows:

https://www.physicsforums.com/attachments/7569

https://www.physicsforums.com/attachments/7570

In some remarks following Example 3.8 (c) we read the following ... ...

" ... ... The Axiom of Pair and the Axiom of Union are necessary to define union of two sets ... ... "

I can see how the Axiom of Union is necessary for the definition of the union of two sets ... BUT ... how exactly is the Union of Pair necessary? Indeed how is it implicated in this definition ... ...

*** NOTE *** The way I see it ... Example 3.8 (c) seems to 'work' perfectly if we assume that \(\displaystyle S = \{ M, N \}\) and 'apply' the Axiom of Union ... so ... nothing else seems to be needed ... except, of course, as Hrbacek and Jech note, the Axiom of Extensionality to guarantee that the union is unique ...

Help will be appreciated ...

Peter

====================================================================================

So that readers of the above post have access to Hrbacek and Jech's axiom system I am providing the relevant text ... as follows:

https://www.physicsforums.com/attachments/7571

https://www.physicsforums.com/attachments/7572

https://www.physicsforums.com/attachments/7573

Hope access to the above text helps ...

Peter

I am currently focused on Chapter 1: Sets and, in particular on Section 3: The Axioms where Hrbacek and Jech set up an axiomatic systems (which they do NOT call ZFC ... but it seems to mirror ZFC ... )

I need some help with Example 3.8 (c) which effectively defines the union of two sets ... . In particular I need some help in fully understanding how this definition requires H&J's Axiom of Pair and Axiom of Union in order to justify the definition ...

Example 3.8, the notes following it and the Axiom of Pair and the Axiom of Union read as follows:

https://www.physicsforums.com/attachments/7569

https://www.physicsforums.com/attachments/7570

In some remarks following Example 3.8 (c) we read the following ... ...

" ... ... The Axiom of Pair and the Axiom of Union are necessary to define union of two sets ... ... "

I can see how the Axiom of Union is necessary for the definition of the union of two sets ... BUT ... how exactly is the Union of Pair necessary? Indeed how is it implicated in this definition ... ...

*** NOTE *** The way I see it ... Example 3.8 (c) seems to 'work' perfectly if we assume that \(\displaystyle S = \{ M, N \}\) and 'apply' the Axiom of Union ... so ... nothing else seems to be needed ... except, of course, as Hrbacek and Jech note, the Axiom of Extensionality to guarantee that the union is unique ...

Help will be appreciated ...

Peter

====================================================================================

So that readers of the above post have access to Hrbacek and Jech's axiom system I am providing the relevant text ... as follows:

https://www.physicsforums.com/attachments/7571

https://www.physicsforums.com/attachments/7572

https://www.physicsforums.com/attachments/7573

Hope access to the above text helps ...

Peter

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