I was reading Hodges' Model Theory when I came across this question in the first chapter:(adsbygoogle = window.adsbygoogle || []).push({});

Specify a structure of cardinality w_{2}which has a substructure of cardinality w but no substructure of cardinality w_{1}. (Working in ZFC)

I am assuming w_{2}means 2^{w1}but I'm not sure. I haven't really encountered this cardinality before, and I'm not really sure what would have cardinality greater than an uncountable set or what that means intuitively (if there even is an intuitive explanation, or if it is just a bit of logical symbol pushing).

I know that I could take the reals with the field operations, an ordering symbol and 0,1 as a signature and specify the integers as a substructure of order w, and I could take the complex numbers and specify R as a substructure of order w1 in the same way, but I have no idea what to make of w2 or what might have w2 as its cardinality.

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# Structure of cardinality W2?

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