Past exam of the mathematics course of the University of Cambridge / 2013 / iii / Paper 19 / 5 / i / b

Solution

a

, often an ordinal, has the form

σ = {(\overset{x}{ˇ}, q) : x \in a, q \in A_{x}},

where each

A_{x}

is an antichain in

Q

and the entire construction belongs to

M

. Its value consists of the

x

for which

H

meets

A_{x}

. The antichains need not be maximal. Under the countable chain condition they are countable in

M

, which makes nice forcing names useful for counting possible subsets in extensions.

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