I'm teaching myself bits and pieces of forcing at the moment, for the purposes of translating them into sheaf-theoretic versions. I'm trying to write down what I feel is a cleaner description of the Easton product of forcing posets, by which I mean a global description rather than one in terms of elements like
$$
\left|{(\kappa,α,β) ∈ dom(p) : \kappa\leq \lambda}\right| \lt \lambda \qquad (1)
$$
for $p$ in the Easton product $\prod^E P(\kappa)$, which is not very useful at the level of the category of sets if it is not the category of ZF(C)-sets.


I'm confident I understand what this condition does when you do the forcing, namely I think it puts a bound on how many sets are added to any given $\lambda$, else you might get silly things like a proper class of sets added to some 
set.


However, at the risk of embarrassing myself, and in the interest of educating others, here is my best guess for what this condition translates to. Consider, as Jech does (Set theory, 3rd edition), first the Easton product over some set $A$ of regular cardinals. For $\kappa\in A$, let $P(\kappa)$ be the set of functions $p:D(p)\to 2$ where $D(p)\subset \kappa\times B(\kappa)$ has cardinality less than $\kappa$. Here $B(\kappa)$ is some cardinal, not necessarily
given by an Easton function on regular cardinals, and forcing using the 'usual order'
on this poset will add $B(\kappa)$ subsets to each $\kappa\in A$. There is a distinguished
element $\top$ of each $P(\kappa)$, namely the unique function $\emptyset \to 2$.


Let $P=\prod P(\kappa)$ be the product over $\kappa\in A$. The Easton product is the subset 
consisting of those $p\in P$ such that the support condition (1) holds. So what
does this mean? An element $p\in P$ is a collection of functions $p\_\kappa$, one for each $\kappa\in A$. Then let $supp(p) \subset A$ be the set of those $\kappa\in A$ such that $p\_\kappa\not=\top$.
Analogously, define $supp\_\lambda(p) \subset A$ to be the set of those $\kappa\leq \lambda$ such that $p\_\kappa\not=\top$. This last definition is where I am the most unsure, as the definition in Jech really involves $supp(p)\cap \lambda$, which doesn't make sense from a
structural set theory point of view unless $\lambda$ is viewed as a subset of $A$ (even though it makes perfect sense in a material set theory such as ZFC).


Now assuming the definition of $supp\_\lambda(p)$ is correct, the support condition on $p$ as first stated in Jech is that
$$
\forall \text{ regular } \lambda, \left|supp\_\lambda(p)\right| \lt \lambda.\qquad (2)
$$
Apparently it's enough to enforce (2) whenever $\lambda$ is weakly inaccessible, (though Friedman just says 'inaccessible'), and that's fine by me. However, this doesn't look the same as the condition (1), which has me slightly worried. The condition (1) imposes a condition on the size of the domain of $p$ as well as (2), and in fact implies that the domain of $p\_\kappa$ is smaller than $\kappa$ for all $\kappa$.


So where has my reasoning gone wrong? At this point I really just want to understand the set underlying the Easton product, rather than any intricacies of class forcing.



I'll admit it: I don't understand the definition of the Easton product.
Here is a general way to think about product forcing, which may be
helpful. One has forcing notions $\mathbb{Q}\_\gamma$ for each
$\gamma$ in a class $D$ of ordinals. (In your example, $D$ is the
class of regular cardinals, and
$\mathbb{Q}\_\gamma=\text{Add}(\gamma,E(\gamma))$ for the Easton
function $E$.) One wants to define the notion of product forcing
$\Pi\_\gamma\mathbb{Q}\_\gamma$.


The idea is that fundamentally different product forcing notions
are obtained by allowing different kinds of support in the
product. This is exactly the same issue as the difference between
direct sums and direct products in algebra. For each ideal $I$ on
$D$, one may use that ideal as collection of allowed supports, so that the product $\Pi\_\gamma\mathbb{Q}\_\gamma$ with supports
in $I$ consists of the functions $p:\gamma\mapsto
p(\gamma)\in\mathbb{Q}\_\gamma$, such that $\{\gamma\mid
p(\gamma)\neq\top\}\in I$. That is, we insist that the support of
a condition $p$ is in the ideal. It is very convenient for the collection of allowed supports to be an ideal, since we often want to manipulate conditions, either by weakening them on each coordinate or by combining two conditions that are compatible on each coordinate, and this leads us to want an ideal, which is closed under subsets and finite unions. Several natural cases arise:


* If we use the ideal of all finite sets, then this product is
called the finite-support product, and this corresponds to taking
what in set theory is called the direct limit at each limit stage,
and this can be characterized in terms of a nice universal
property of how the product relates to the factors
$\mathbb{Q}\_\gamma$. This is like the direct sum.
* If we use the ideal of all countable sets, the product is called
a countable-support product.
* If we use the (improper) ideal of *all* subsets of $D$, then the
product is called the full-support product, and this corresponds
to using inverse limits at every stage. This also has a nice
characterization by a universal property, like the direct product.
* The Easton support ideal consists of sets $A$, which are bounded
below every inaccessible cardinal. This ideal corresponds to
taking direct limits at every inaccessible cardinal stage, and
inverse limits at all other limit stages.


This ideal perspective makes sense whether one speaks of product
forcing or iterated forcing. Although Easton used his support with
product forcing, there are now numerous applications of Easton
forcing with iterated forcing.


Each of these ideals and others finds numerous applications in
forcing. For example, the finite support iteration of ccc forcing
is necessarily ccc. The countable support iteration of proper
forcing remains proper.


Note that the Easton ideal is exactly the same as the improper
ideal of all sets, if there should happen to be no inaccessible
cardinals. Easton realized, however, that he could prove his
theorem on the continuum function, without splitting into cases as
to the existence of inaccessible cardinals, by using this hybrid
support. Easton support is important because with it, one is often
able to show that cardinals are preserved, when they would not be
preserved with a full-support iteration. The insertion of direct
limits at inaccessible stages enables one to prove better chain
conditions, which are the means by which one shows that cardinals
are not inadvertently collapsed by the forcing.


Lastly, let me say that the particular confusion about
inaccessible versus regular cardinals in the Jech presentation is
resolved when one realizes that he is forcing only at *cardinal*
stages. And so if $\lambda$ is a successor cardinal
$\lambda=\delta^+$, then there at most $\delta$ many cardinals
below $\lambda$, and so the support will have size less than
$\lambda$, for free. So the only operative restriction there
occurs at inaccessible cardinals. (My belief is that this
presentation in the book could be improved to explain this better,
since numerous students have had exactly this confusion.)


