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The class indexed by real numbers is a set? 
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#1
Mar2912, 07:02 PM

P: 110

Let [itex]\mathcal{S} = \{S_{i}:i \in \mathbb{R} \} [/itex] where [itex]S_{i}[/itex] is a set. Then [itex]\mathcal{S}[/itex] is a set? Or, can this notation make sense in some way?



#2
Mar3012, 10:18 AM

P: 606

I can't see why you think S couldn't be a set, as long as each [itex] S_i[/itex] is...What did you have in mind? DonAntonio 


#3
Mar3012, 10:20 AM

P: 642

A set can be of anything. Even {Lincoln, Charizard, {Fish Fingers, Custard}} is a set. Your set is, therefore, valid. Note that its cardinality is one of the alephs, I don't know which one.



#4
Mar3012, 10:48 AM

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The class indexed by real numbers is a set?



#5
Mar3012, 10:53 AM

P: 642




#6
Mar3012, 11:06 AM

P: 800

[itex]\mathcal{S} = \{S_{\alpha}:\alpha \in \mathbb{R} \} [/itex] which provides readers with an indication that we are indexing over a set other than the natural numbers. 


#7
Mar3012, 08:39 PM

P: 110




#8
Mar3012, 10:42 PM

P: 606

I see. But, as already noted by others, it is possible to index by means of any set, no matter its cardinality. In ZFC we can even use Zermelo's Well Ordering Theorem to well order ℝ and then wellorder the so indexed sets, if so wanted... DonAntonio 


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