Products and Easton’s Theorem
1. Product Forcing
Let P “ xP, ďP y, Q “ xQ, ďQ y be partial orders. We define their product by
P ˆ Q “ txp, qy : p P P ^ q P Qu. This is ordered by pp1 , q 1 q ďPˆQ pp, qq iff p1 ďP p
and q 1 ďQ q (note: we will frequently use pp, qq instead of the more formal xp, qy
when details of the pair coding are irrelevant).
For example, the forcing for adding two real, FNpω ˆ 2, 2q is isomorphic to the
product FNpω, 2q ˆ FNpω, 2q (which in this case is isomorphic to FNpω, 2q itself).
If G Ď P and H Ď Q are filters, then G ˆ H Ď P ˆ Q is also easily a filter.
Conversely, if F Ď P ˆ Q is a filter, let G “ tp P P : Dq P Q pp, qq P F u and likewise
H “ tq P Q : Dp P P pp, qq P F u. Easily G and H are filters. If pp, qq P F then
by definition p P G and q P H, so F Ď G ˆ H. For the other direction, suppose
p P G and q P H. then pp, q 1 q P F and pp1 , qq P F for some p1 , q 1 . Let pr, sq P F
with r ď p, p1 , s ď q, q 1 (note: I’ve switched to the other definition of filter now).
Since pp, qq ď pr, sq P F , pp, qq P F . Thus, filters F in P ˆ Q are precisely of the
form F “ G ˆ H where F , G are filters in P , Q respectively. The relation between
generics for P ˆ Q and generics for P, Q is clarified in the following lemma.
Lemma 1.1. A filter F “ G ˆ H is M generic for P ˆ Q iff G is M generic for
P and H is M rGs generic for Q.
Remark 1.2. Of course, the situation is symmetrical with respect to P and Q, so
we equally well say iff H is M generic for Q and G is M rHs generic for P .
Proof. First suppose G ˆ H is M generic for P ˆ Q. Let D Ď P , D P M , be dense.
Then Q ˆ Q is dense in P ˆ Q, so let pp, qq P pG ˆ Hq X pD ˆ Qq. Thus, p P G X D.
This shows G is M generic for P . Let now E Ď Q, E P M rGs, be dense in Q. Let
E “ τG , where τ P M P . Let p0 P P X G with p , pτ is dense in Q̌q. Let
D “ tpp, qq P P ˆ Q : pp K p0 q _ pp ď p0 ^ pp , q̌ P τ qqu.
D is easily dense in P ˆ Q [Let pr, sq P P ˆ Q. If r K p0 , then pr, sq P D. Otherwise,
let pr1 , sq ď pr, sq with r1 ď p0 . Since r1 , pτ is dense q, r1 , Dq ď š pq P τ q. Then
there is a p ď r1 and a q ď s with p , pq̌ P τ q. Thus, pp, qq ď pr, sq and pp, qq P D.]
Let pp, qq P pG ˆ Hq X D. We must have p ď p0 , and so p , pq̌ P τ q. Since p P G,
q P E, so q P H X E. Thus, H is M rGs generic for Q.
Conversely, suppose G is M generic for P and H is M rGs generic for Q. Let
D Ď P ˆ Q be dense. Let E “ tq P Q : Dp P G pp, qq P Du. Clearly E P M rGs. It is
enough to show that E is dense in Q for then we would have q P H XE. By definition
of E there would then be a p P G with pp, qq P D. Hence, pp, qq P pG ˆ Hq X D. To
see E is dense, let s P Q. Let A “ tp P P : Dq pq ď s ^ pp, qq P Du. Clearly A is
dense in P , and A P M . So, let p P G X A. Let q ď s with pp, qq P D. Then q P E
and q ď s.
Thus, if P, Q are partial orders in M , forcing with the product PˆQ is equivalent
to doing a two-step forcing where we first force over M with P to get M rGs, and
then force over M rGs with Q to get M rGsrHs. Note that M rGsrHs “ M rG ˆ Hs,
as G, H are definable from G ˆ H and conversely.
The following technical lemma combines lemmas ?? and ??.
1
2
Lemma 1.3. Let κ be a cardinal and P be κ` -c.c. and Q be ď κ closed in a transitive
model M of ZFC. Let G ˆ H be M generic for P ˆ Q. Then any f : κ Ñ M in
M rGsrHs lies in M rGs.
Proof. Let f : κ Ñ M lie in M rGsrHs. Let τ P M PˆQ with f “ τGˆH . For each
α ă κ let Dα “ tq P Q : @p P P Dp1 ď p Dx P M ppp1 , qq , τ pα̌q “ x̌qu. We claim
that Dα is dense in Q. To see this, let q P Q. Let p0 P P , q0 ď q, and x0 P M with
pp0 , q0 q , τ p0̌q “ x̌. We construct pp0 , q0 q ď ¨ ¨ ¨ ď ppβ , qβ q ď as follows. Assume
ppγ , qγ q is defined for γ ă β, and β ă κ` . If tpγ uγăβ is a maximal antichain on P ,
then we let stop the construction and let q extend all of the qγ , which we can do as
Q is ď κ closed. Otherwise, let pβ be incompatible with all of the pγ , γ ă β. Let qβ
extend all of the qγ for γ ă β and such that for some xβ P M , ppβ , qβ q , τ pβ̌q “ xˇβ .
As P is κ` -c.c. this construction cannot go on κ` times. Thus, for some β ă κ` ,
tpγ uγăβ is a maximal antichain of P . The corresponding q (which extends all of
the qγ ) lies in Dα .
Ş
Using again that Q is ď κ closed, we get that D “ αăκ Dα is dense in Q. Fix
q P H X D. Working in M we may define dense sets Dα , α ă κ, such that for
all α and all p P Dα , there is an x P M such that pp, qq , τ pα̌q “ x̌. In M rGs
we may then compute f , namely f pαq “ x iff there is a p P Dα X G such that
pp, qq , τ pα̌q “ x̌ (we are using the replacement and comprehension axioms in M
to get that f is a set in M ).
As a warm-up for Easton’s theorem, let us give another, perhaps more direct,
proof of lemma ??. So, let M satisfy GCH and κ1 ă ¨ ¨ ¨ ă κn be regular, and
λ1 ď ¨ ¨ ¨ ď λn with cofpλi q ą κi . Let Pi “ FNpλi , 2, κi q be the partial order
for adding λi many subsets of κi . Let P “ P1 ˆ ¨ ¨ ¨ ˆ Pn be the product. Let
G “ G0 ˆ¨ ¨ ¨ˆGn be M generic for P. First we show that P preserves all cardinalities
and cofinalities. Let δ be a regular cardinal of M , but suppose f : ρ Ñ δ is cofinal
where ρ ă δ and f P M rGs. Let P´ “ P1 ˆ ¨ ¨ ¨ ˆ Pi where i is maximal so that
κi ď ρ. Let P` “ Pi`1 ˆ ¨ ¨ ¨ ˆ Pn . Clearly P` is ď ρ closed in M . Also, P´ is
ăρ `
`
p2
ř q “ ρ -c.c. in M (note that P can be viewed as a subset of FNpλi , 2, κi q since
´
jďi pκj ˆ λj q – λi ; use then lemma ??). From lemma 1.3 we have f P M rP s.
From lemma ?? we then have that there is an F : ρ Ñ Ppδq, F P M with |F pαq| ď ρ
for all α ă δ. This is a contradiction as δ is regular in M . Som P preserves all
cofinalities and hence cardinalities (which the same argument also shows directly;
thus the preservation of cardinals only requires M to satisfy ZF).
Clearly p2κi q ě λi qM rPs . To get the upper bound for p2κi qM rPs , write P “
´
P ˆ P` where P´ “ P1 ˆ ¨ ¨ ¨ ˆ Pi and P` “ Pi`1 ˆ ¨ ¨ ¨ ˆ Pn . From lemma 1.3
´
we have Ppκi qM rPs “ Ppκi qM rP s . Since P´ is κ`
i -c.c. in M , there are at most
pλκi i qκi “ λκi i “ λi many nice names for a subset of κi in M (these computations
are done in M ; the last equality uses cofpλi q ą κi and the GCH in M ). Thus,
p2κi ď λi qM rPs . We have thus shown that p2κi “ λi qM rPs for all i ď n.
2. Remarks on Class Forcing
In most applications we will be in the situation where P P M , that is, P is a set
in M (this is what we have been considering up to this point). For some purposes,
including Easton’s theorem, we would like to generalize this to allow P being a class
in M . Note that we are still assuming that M is a transitive set in V , thus there
is no problem in quantifying over the classes of M (as statements in V ). For this
3
section, when we say a class of M , we mean a formula with set parameters from
M.
Let M be a set which is a transitive model of ZF (or ZFC). Let P “ xP, ďy
where P, ďĎ M are classes of M (i.e., definable in M from parameters in M ), and
such that P is a partial order. Note that M also satisfies that P is a partial order in
the sense that, for example, p@x, y, z rppx, yq P ďq ^ ppy, zq P ďq Ñ ppx, zq PďqsqM .
If D Ď P is a class of M , we say D is dense just as before; if @p P P Dq P Dpq ď pq.
For a given D, this is expressible in M . We say G Ď P is M generic for P exactly
as before; if G X D ‰ H for all dense classes D Ď P ofŤM .
We define M P essentially as before. Thus, M P “ αPONM MαP , where we take
unions at limit ordinals and
P
Mα`1
“ tτ P M X Vα`1 : pτ is a relation q ^ dompτ q Ď MαP ^ ranpτ q Ď P u.
Again, the transfinite recursion theorem shows that M P is a well-defined class of
M . Given a filter G Ď P , we define the evaluation map τ Ñ τG exactly as before,
and again define M rGs “ tτG : τ P M P u.
For p P P , φpx1 , . . . , xn q a formula, and τ1 , . . . , τn P M P , we define the forcing
relation p , φpτ1 , . . . , τn q exactly as before. We again have that for all formuals
φpx1 , . . . , xn q that tpp, τ1 , . . . , τn q : p , φpx1 , . . . , xn qu is a class of M .
Finally, the forcing theorem goes through as before. For example, consider the
atomic case φ “ pτ1 P τ2 q. Suppose p P P and p , φ. Then the class
D “ tq P P : Dxσ, ry P τ2 pq ď r ^ r , pτ1 « σqqu
is dense below p. D is now a class, not a set in M , but G being generic still implies
G X D ‰ H. If q P G X D, let xσ, ry P τ2 be such that q ď r ^ r , pτ1 « σq.
So, σG P pτ2 qG and by induction pτ1 qG “ σG . Hence, pτ1 qG P pτ2 qG . The other
direction is also as before.
Consider now which axioms of ZF hold in M rGs. Certainly founadation, extensionality, pairing and union hold in M rGs (and again only require G to be a filter).
We run into problems, though, when we try to show power set, replacement, and
comprehension. The proofs of these axioms given previously for set forcing use the
fact that P is a set in M . For example the proof of power set required us to consider
tσ : dompσq Ď dompτ q ^ ranpσq Ď P u, where τ P M P is fixed. If P is not a set in
M , this will clearly be a proper class of M as well. Similarly, the previous proof
of replacement required the apllication of replacement in M to the set dompτ q ˆ P
for some τ P M P ; here again this will be a proper class.
For general class forcing, the power set, replacement, and comprehension axioms
may fail in M rGs. For example, power set will fail if we add ON many reals, or
if we collapse ON to ω. Thus, some restiction on the forcing is necessary. The
following gives a sufficient condition.
Theorem 2.1. Let P be a class partial order of M , where M is a transitive model
of ZF (or ZFC). Suppose that for arbitrarily large cardinals κ of M that we can
write P “ P´ ˆ P` where P´ is κ` -c.c. and P` is ď κ closed. Let G be M generic
for P. Then M rGs satisfies ZF (or ZFC).
Proof. We show comprehension, power set, and replacement in M rGs.
To show comprehension, fix a1 “ pτ1 qG , . . . ,, an “ pτn qG , A “ τG , and a formula
φpx1 , . . . , xn , y, zq. We must show that tz P A : φM rGs pa1 , . . . , an , A, zqu exists as a
set in M rGs. Let κ ą |τ | be such that P “ P´ ˆ P` with P´ κ` c.c., and P` ď κ
4
closed. As in the proof of lemma 1.3, let Q Ď P ` be those q P P ` such that for
all xπ, py P τ , there is a dense below p set of conditions p1 P P ´ such that pp1 , qq
decides φpτ1 , . . . , τn , τ, πq. More precisely,
Q “tq P P ` : @π P dompτ q @r P P ´ Ds P P ´ rpps, qq , φpτ1 , . . . , τn , τ, πqq_
pps, qq ,
φpτ1 , . . . , τn , τ, πqqs.
As in the proof of lemma 1.3, it follows from the ď κ closure of P` that Q is dense
in P ` . Let pp0 , q0 q P G with q0 P Q. For π P dompτ q let
Dπ “ tp P P ´ : ppp, q0 q , φpτ1 , . . . , τn , τ, πqq _ ppp, q0 q ,
φpτ1 , . . . , τn , τ, πqqu.
Thus, Dπ is dense in P ´ . Moreover, as in lemma 1.3 we may assume each Dπ is
an antichain which is a set of size ď κ. Let
σ “ txπ, pp, q0 qy : pπ P dompτ qq ^ pp P Dπ q ^ ppp, q0 q , φpτ1 , . . . , τn , τ, πqqu.
Note that σ is a valid name, that is, σ is a set in M .
If x P σG , then x “ πG where pp, q0 q P G, and pp, q0 q , φpτ1 , . . . , τn , τ, πq. Thus,
φM rGs pa1 , . . . , an , A, xq. Suppose next that x P M rGs and φM rGs pa1 , . . . , an , A, xq.
Then x “ πG where π P dompτ q. Let pp1 , q1 q P G with p1 P Dπ and pp1 , q1 q ,
φpτ1 , . . . , τn , τ, πq. Since p1 P Dπ , either pp1 , q0 q , φpτ1 , . . . , τn , τ, πq or pp1 , q0 q ,
φpτ1 , . . . , τn , τ, πq. The latter case is impossible as pp1 , q0 q, pp1 , q1 q are compatible.
This shows xπ, pp1 , q0 qy P σ, and hence x “ πG P σG (note that pp1 , q0 q P G, as G is
a filter).
Consider next power set. Let x “ τG P M rGs, let κ ą |τ | and again write
P “ P´ ˆ P` as before. Let ρ “ txσ, 1y : dompσq Ď dompτ q ^ ranpσq Ď P´ u. We
show that Ppxq Ď ρG . Fix y Ď x, say y “ µG . Arguing as in the previous case, we
get a pp0 , q0 q P G such that for all π P dompτ q, the set Dπ Ď P ´ is dense, where
now
Dπ “ tp P P ´ : ppp, q0 q , π P µq _ ppp, q0 q ,
pπ P µqqu.
Let
σ “ txπ, pp, 1qy : pπ P dompτ qq ^ pp P Dπ q ^ ppp, q0 q , π P µqu.
Clearly σG Ď µG (note that if pp, 1q P G, then pp, q0 q P G, since pp0 , q0 q P G). The
other direction, µG Ď σG follows now exactly as in the previous case.
Consider replacement. The proof is again similar to the previous cases. Let
A “ τG , a1 “ pτ1 qG , . . . , an “ pτn qG , and φpx1 , . . . , xn , y, z, wq be a formula.
Assume that
@y P A Dz P M rGs φM rGs pa1 , . . . , an , A, y, zq.
Fix κ ą |τ |, and again write P “ P´ ˆ P` . As in the previous cases (using the fact
that P` is ď κ closed and P´ is κ` -c.c.) we get a pp0 , q0 q P G such that for each
π P dompτ q the set Dπ Ď P´ is a dense set (which we may assume has size ď κ),
where
Dπ “ tp P P ´ : Dz P M ppp, q0 q , φpτ1 , . . . , τn , τ, π, žqu.
Using replacement in M , let S be a set in M such that for all π P dompτ q and all
p P Dπ , there is a z P S such that pp, q0 q , φpτ1 , . . . , τn , τ, π, žq. Let
σ “ txρ, 1y : ρ P Su.
To see this works, let y “ πG P A “ τG , where π P dompτ q. Let pp1 , q1 q P G with
p1 P Dπ . By definition of Dπ , let z P M be such that pp1 , q0 q , φpτ1 , . . . , τn , τ, π, žq.
5
From the definition of S it follows that pp1 , q0 q , φpτ1 , . . . , τn , τ, π, zˇ1 q for some
z 1 P S. Hence, φM rGs pa1 , . . . , an , A, y, z 1 q holds for some z 1 P σG . This verifies
replacement.
If M satisfies AC, then so does M rGs by exactly the same argument as for set
forcing, since if A “ τG , then τ is a set in M , and so there is map f P M from an
ordinal α onto τ . Although G itself is no longer in M rGs (as with set forcing), we
nevertheless still get a map from α onto τG definable from f and G X Vβ for some
β, which is a set in M rGs. Thus, in M rGs we still define an F from α onto τG . If we assume the factoring hypothesis of theorem 2.1 holds for all regular cardinals, then the class forcing P also preserves all cardinals and cofinalities.
Theorem 2.2. Let P be a class partial order of M , where M is a transitive model of
ZFC. Suppose that for all regular cardinals κ of M that we can write P “ P´ ˆ P`
where P´ is κ` -c.c. and P` is ď κ closed. Then all cardinals and cofinalites are
preserved in forcing with P.
Proof. Let δ be a regular cardinal of M , and suppose ρ “ pcofpδqqM rGs ă δ. We
again use the argument of lemma 1.3. Write P “ P´ ˆ P` where P´ is ρ` -c.c. and
P` is ď ρ closed. Let f : ρ Ñ δ be onto, f P M rGs. Fronm lemma 1.3, f P M rG´ s,
where G “ G´ ˆ G` . Since P´ is ρ` c.c., there is an F P M , F : ρ Ñ δ with
ranpf q Ď ranpF q. This contradicts δ being regular in M .
We are now ready to give Easton’s theorem.
Definition 2.3. An Easton function is a class function F of M with domain a
class of regular cardinals of M and range in cardinals of M satisfying:
(1) If λ1 ă λ2 are in dompF q, then F pλ1 q ď F pλ2 q.
(2) @λ P dompF q pcofpF pλq ą λq.
If F is an Easton function for M , we define the Easton forcing PF as follows.
Condition p P PF are functions with domain dompF q, and for λ P dompF q, ppλq P
FNpF pλq, 2, λq. Further, we require p to satisfy the Easton condition: for all regular
κ of M , tλ ă κ : ppλq ‰ 1u has size ă κ.
Thus, the forcing is just the product of the forcings to make 2λ at least F pλq,
except we add the Easton condition which restricts the size of the domains of the
conditions. Note that the Easton condition is only non-trivial when κ is a weakly
inaccessible cardinal.
Theorem 2.4 (Easton). Let M be a transitive model of ZFC`GCH, and F a class
of M which is an Easton function. Assume G is M generic for the Easton forcing
PF . Then M rGs satisfies ZFC, all cardinalities and cofinalities are preserved from
M to M rGs, and for all regular cardinals λ of M rGs we have p2λ “ F pλqqM rGs .
Proof. Fix a regular cardinal λ of M (equivalently, of M rGs). Write PF “ Pďλ ˆ
Pąλ where Pďλ consists of those p P PF with domppq Ď λ ` 1, and Pąλ those p
with domppq Ď CARD ´ pλ ` 1q. Clearly Pąλ is ď λ closed. We show that Pďλ
is λ` c.c. For every regular κ ď λ, FNpF pκq, 2, κq is p2ăκ q` “ κ` c.c. in M , since
M satisfies the GCH. Suppose tpα uαăλ` were an antichain of size λ` in Pďλ . Let
dα “ domppα q. Since |dα | ă λ (by the Easton condition if λ is limit, otherwise
trivially), there are only λăλ “ λ many choices for dα . So, we may assume that all
of the pα have the same domain d. By regularity of λ, each pα may be viewed as a
6
function from d ˆ F pλq Ñ t0, 1u with domain of size ă λ. Since FNpF pλq, 2, λq is
p2ăλ q` “ λ` c.c. in M , this is a contradiction. Thus, Pďλ is λ` c.c.
From lemmas 2.1, 2.2 we know that M rGs satisfies ZFC and all cardinals and
cofinalities are preserved from M to M rGs. We clearly have for all regular cardinals
of M rGs that p2λ ě F pλqqM rGs . To see the other direction, fix a regular cardinal λ
of M (equivalently, of M rGs) and consider PF “ Pďλ ˆ Pąλ as above.
Every subset of λ in M rGs is in M rGďλ s, where G “ Gďλ ˆGąλ , from lemma 1.3.
ďλ
Since Pďλ is λ` c.c., p2λ qM rG s ď p|Pďλ |λ¨λ qM . Also, |Pďλ | ď F pλqăλ 2ăλ “ F pλq
since cofpF pλqq ą λ and the GCH in M (these computations are done in M ). Thus,
ďλ
p2λ qM rG s ď F pλqλ “ F pλq.
Finally, we use class forcing to get a model of GCH.
Theorem 2.5. Let M be a transitive model of ZFC. Then there is a class partial
order P of M such that if G is M -generic for P then M rGs satisfies ZFC ` GCH.
Proof. Let M be a transitive model of ZFC. Let α Ñ iα be the beth function
of M (for this proof, iα always denotes piα qM ). For each ordinal α of M , let
` M
M
“ FNpi`
Pα “ collpi`
α , iα`1 , iα q . Note that Pα is iα closed and
α , iα`1 q
iα `
iα
`
iα iα
iα
piα`1 q “ iα`1 c.c. (iα`1 “ p2 q “ 2 “ iα`1 ).
Let P be the Easton product of the Pα . That is, P consists of functions p with
domain a subset of ordinals, ppαq P Pα for all α P domppq, and p satisfies the Easton
condition: for all inaccessible λ, tα ă λ : ppαq ‰ 1u has size ă λ. For α P ONM ,
let Păα denote those p P P with domppq Ď α. Likewise, Pěα denotes those p with
domppq X α “ H. Clearly P “ Păα ˆ Pěα (at least, up to isomorphism).
First we show that M rGs satisfies ZFC. For α a successor ordinal of M , consider
P “ Păα ˆ Pěα . Easily Pěα is ď iα closed. Any p P Păα can be viewed as a
`
`
`
function from i`
α´1 to iα of size ď iα´1 . Since FNpiα´1 , iα , iα´1 q is iα c.c.,
ăα
`
it follows that P
is iα c.c. From lemma 2.1 it now follows that M rGs satisfies
ZFC.
M rGs
. First we show that
Clearly p|iα`1 | ď piα q` qM rGs . Thus, |iα |M rGs ď ℵα
M rGs
`
M rGs
, and for this it suffices to show that iα is still a cardinal of
|iα |
“ ℵα
M rGs. First assume that α is a successor and write P “ Păα ˆ Pěα as above.
`
`
Thus, Pěα is ď iα closed and Păα is i`
α c.c. If ρ ă iα and f : ρ Ñ iα , then
ăα
ăα
ěα
from lemma 1.3 it follows that f P M rG s, where G “ G ˆ G . Since Păα
`
is i`
α c.c., this gives an onto F : iα Ñ iα in M , a contradiction. Suppose next
that α is limit. We consider cases as to whether iα is regular (i.e., inaccessible).
Suppose first that iα is regular. Again write P “ Păα ˆ Pěα . Pěα is ď iα closed.
From the Easton condition, any p P Păα has domain bounded in α. This gives that
ăα
Păα is i`
of size i`
α c.c., since if there were an antichain in P
α we could assume
the domains of the conditions were constant, and a simple computation would then
show there are ă iα many conditions in the antichain. If f : iα Ñ i`
α were onto
and in M rGs, lemma 1.3 would give a function F : iα Ñ i`
α in M which was also
onto, a contradiction. Suppose then that iα is singular, say ρ “ cofpiα q ă iα .
Let tαi uiăρ be increasing cofinal in iα with iα0 ą ρ. Let D be those p P P such
that there is a sequence tAγi u for i ă ρ, γ ă iαi , each Ai a maximal antichain of
Păαi , such that for all i ă ρ, γ ă iαi , and all q P Aγi there is an ordinal β such
that pq, pěαi q , τ pγ̌q “ β̌. Iterating the argument of lemma 1.3 ρ times shows that
D is dense. Fix p P P X D, and let Aγi be the corresponding antichains. From the
7
Ť
Aγi we may construct in M a set of size | i,γ Aγi | which contains the range of f .
Thus in M we have a set of size iα which contains i`
α , a contradiction.
M rGs
We now know that iα has cardinality ℵα
in M rGs. To show the GCH in
M rGs it is thus enough to show that there are at most iα`1 many subsets of ipαq
in M rGs. Suppose first that α is a successor. Write P “ Păα ˆ Pěα . Every subset
of iα in M rGs lies in M rGăα s, so it is enough to count these. In this case Păα
iα
has size iα , so there are at most iα
“ iα`1 many nice names for subsets of iα .
Suppose next that iα is innaccessible. Again write P “ Păα ˆ Pěα . Every subset
of iα in M rGs again lies in M rGăα s. From the Easton condition, Păα has size iα
iα
in M . So again there are at most iα
“ iα`1 many nice names for subsets of iα .
Finally, suppose α is limit and iα is singular, say ρ “ cofpiα q ă iα . Let β ă α
be a successor with iβ ą ρ. Proceeding inductively, we may assume the GCH
holds in M rGs below iα . Thus it suffices to show that ppiα qρ ď iα`1 qM rGs . Write
P “ Păβ ˆ Pěβ . Every f P piα qρ X M rGs lies in M rGăβ s. Since |Păρ | ď iβ , there
iα
α ρ
are at most pii
“ iα`1 many nice names for functions f P piα qρ .
β q ď2