Constructing Structures with Cardinality W2: A Puzzle in Model Theory

[Bourbaki1123]

I was reading Hodges' Model Theory when I came across this question in the first chapter:

Specify a structure of cardinality w₂ which has a substructure of cardinality w but no substructure of cardinality w₁. (Working in ZFC)

I am assuming w₂ 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.

 

[g_edgar]

When he says ``cardinality [itex]\omega_2[/itex]'' he means ``[itex]\aleph_2[/itex]'' which may or may not be [itex]2^{\aleph_1}[/itex]. If he actually says [itex]w_2[/itex], then I don't know what he means.

 

[honestrosewater]

I'm not really sure what would have cardinality greater than an uncountable set

Its power set?

 

[AKG]

[itex]\omega = \aleph _0[/itex] is the smallest infinite cardinal, and it's said to be countably infinite. All larger cardinals are said to be uncountably infinite. [itex]\omega _1 = \aleph _1[/itex] is the next infinite cardinal larger than [itex]\omega[/itex], and so it's the first uncountable cardinal. [itex]\omega _2 = \aleph _2[/itex] is the next infinite cardinal; [itex]\omega < \omega _1 < \omega _2[/itex]. Find a structure M (i.e. a set, together with some function(s) and/or relation(s) on that set) such M has a countable substructure (i.e. there's a countable subset of M closed under those functions/relations) but such that there's no substructure of size [itex]\omega _1[/itex] that's closed under those functions/relations.

Hint: Let M be some well-ordered set of size [itex]\omega _2[/itex]. Let N be the subset of M consisting of the first (w.r.t. the well-ordering) [itex]\omega[/itex] elements of M. Define a bunch of functions on M such that N is closed under those functions (and thus N forms a countable substructure), but such that any subset of M containing any element outside of N has to contain ALL elements of M in order to be closed under all your functions.