stylistic:tone
The countable chain condition

An (strong) antichain A {\displaystyle A} of P {\displaystyle \mathbb {P} } is a subset such that if p, q ∈ A {\displaystyle p,q\in A} and p ≠ q {\displaystyle p\neq q}, then p {\displaystyle p} and q {\displaystyle q} are incompatible (written p ⊥ q {\displaystyle p\perp q}), meaning there is no r {\displaystyle r} in P {\displaystyle \mathbb {P} } such that r ≤ p {\displaystyle r\leq p} and r ≤ q {\displaystyle r\leq q}. In the example on Borel sets, incompatibility means that p ∩ q {\displaystyle p\cap q} has zero measure. In the example on finite partial functions, incompatibility means that p ∪ q {\displaystyle p\cup q} is not a function, in other words, p {\displaystyle p} and q {\displaystyle q} assign different values to some domain input.

P {\displaystyle \mathbb {P} } satisfies the countable chain condition (c.c.c.) if and only if every antichain in P {\displaystyle \mathbb {P} } is countable. (The name, which is obviously inappropriate, is a holdover from older terminology. Some mathematicians write "c.a.c." for "countable antichain condition".)

It is easy to see that Bor (I) {\displaystyle \operatorname {Bor} (I)} satisfies the c.c.c. because the measures add up to at most 1 {\displaystyle 1}. Also, Fin (E, 2) {\displaystyle \operatorname {Fin} (E,2)} satisfies the c.c.c., but the proof is more difficult.

Given an uncountable subfamily W ⊆ Fin (E, 2) {\displaystyle W\subseteq \operatorname {Fin} (E,2)}, shrink W {\displaystyle W} to an uncountable subfamily W 0 {\displaystyle W_{0}} of sets of size n {\displaystyle n}, for some n < ω {\displaystyle n<\omega }. If p (e 1) = b 1 {\displaystyle p(e_{1})=b_{1}} for uncountably many p ∈ W 0 {\displaystyle p\in W_{0}}, shrink this to an uncountable subfamily W 1 {\displaystyle W_{1}} and repeat, getting a finite set { (e 1, b 1), …, (e k, b k) } {\displaystyle \{(e_{1},b_{1}),\ldots,(e_{k},b_{k})\}} and an uncountable family W k {\displaystyle W_{k}} of incompatible conditions of size n − k {\displaystyle n-k} such that every e {\displaystyle e} is in Dom (p) {\displaystyle \operatorname {Dom} (p)} for at most countable many p ∈ W k {\displaystyle p\in W_{k}}. Now, pick an arbitrary p ∈ W k {\displaystyle p\in W_{k}}, and pick from W k {\displaystyle W_{k}} any q {\displaystyle q} that is not one of the countably many members that have a domain member in common with p {\displaystyle p}. Then p ∪ { (e 1, b 1), …, (e k, b k) } {\displaystyle p\cup \{(e_{1},b_{1}),\ldots,(e_{k},b_{k})\}} and q ∪ { (e 1, b 1), …, (e k, b k) } {\displaystyle q\cup \{(e_{1},b_{1}),\ldots,(e_{k},b_{k})\}} are compatible, so W {\displaystyle W} is not an antichain. In other words, Fin (E, 2) {\displaystyle \operatorname {Fin} (E,2)} -antichains are countable.

The importance of antichains in forcing is that for most purposes, dense sets and maximal antichains are equivalent. A maximal antichain A {\displaystyle A} is one that cannot be extended to a larger antichain. This means that every element p ∈ P {\displaystyle p\in \mathbb {P} } is compatible with some member of A {\displaystyle A}. The existence of a maximal antichain follows from Zorn's Lemma. Given a maximal antichain A {\displaystyle A}, let

Then D {\displaystyle D} is dense, and G ∩ D ≠ ∅ {\displaystyle G\cap D\neq \varnothing } if and only if G ∩ A ≠ ∅ {\displaystyle G\cap A\neq \varnothing }. Conversely, given a dense set D {\displaystyle D}, Zorn's Lemma shows that there exists a maximal antichain A ⊆ D {\displaystyle A\subseteq D}, and then G ∩ D ≠ ∅ {\displaystyle G\cap D\neq \varnothing } if and only if G ∩ A ≠ ∅ {\displaystyle G\cap A\neq \varnothing }.

Assume that P {\displaystyle \mathbb {P} } satisfies the c.c.c. Given x, y ∈ V {\displaystyle x,y\in V}, with f: x → y {\displaystyle f:x\to y} a function in V {\displaystyle V}, one can approximate f {\displaystyle f} inside V {\displaystyle V} as follows. Let u {\displaystyle u} be a name for f {\displaystyle f} (by the definition of V {\displaystyle V}) and let p {\displaystyle p} be a condition that forces u {\displaystyle u} to be a function from x {\displaystyle x} to y {\displaystyle y}. Define a function F: x → P (y) {\displaystyle F:x\to {\mathcal {P}}(y)}, by

By the definability of forcing, this definition makes sense within V {\displaystyle V}. By the coherence of forcing, a different b {\displaystyle b} come from an incompatible p {\displaystyle p}. By c.c.c., F (a) {\displaystyle F(a)} is countable.

In summary, f {\displaystyle f} is unknown in V {\displaystyle V} as it depends on G {\displaystyle G}, but it is not wildly unknown for a c.c.c.-forcing. One can identify a countable set of guesses for what the value of f {\displaystyle f} is at any input, independent of G {\displaystyle G}.

This has the following very important consequence. If in V {\displaystyle V}, f: α → β {\displaystyle f:\alpha \to \beta } is a surjection from one infinite ordinal onto another, then there is a surjection g: ω × α → β {\displaystyle g:\omega \times \alpha \to \beta } in V {\displaystyle V}, and consequently, a surjection h: α → β {\displaystyle h:\alpha \to \beta } in V {\displaystyle V}. In particular, cardinals cannot collapse. The conclusion is that 2 ℵ 0 ≥ ℵ 2 {\displaystyle 2^{\aleph _{0}}\geq \aleph _{2}} in V {\displaystyle V}.