Axioms of Set Theory .... and the Union of Two Sets ....

[Math Amateur]

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/7570In 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 $$S = \{ M, N \}$$ 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/7573Hope access to the above text helps ...

Peter

 

[castor28]

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 ...

Hi Peter,

The way I see it, the axiom of union allows you to define the union of a set of sets.

To define $A\cup B$ using that axiom, you need a set whose elements are $A$ and $B$, and this requires the axiom of pairs.

 

[Math Amateur]

Hi Peter,

The way I see it, the axiom of union allows you to define the union of a set of sets.

To define $A\cup B$ using that axiom, you need a set whose elements are $A$ and $B$, and this requires the axiom of pairs.

Yes ... what you suggest seems correct to me ...

Thanks for the help ... appreciate your assistance ...

Peter