We briefly discussed relative constructibility and compared the models ![L[x]](https://s0.wp.com/latex.php?latex=L%5Bx%5D&bg=ffffff&fg=333333&s=0&c=20201002)
An amusing application of the fact that ![L[x]\models\mathsf{AC}](https://s0.wp.com/latex.php?latex=L%5Bx%5D%5Cmodels%5Cmathsf%7BAC%7D&bg=ffffff&fg=333333&s=0&c=20201002)
![\prec\in L[\prec]](https://s0.wp.com/latex.php?latex=%5Cprec%5Cin+L%5B%5Cprec%5D&bg=ffffff&fg=333333&s=0&c=20201002)
![L[\prec]](https://s0.wp.com/latex.php?latex=L%5B%5Cprec%5D&bg=ffffff&fg=333333&s=0&c=20201002)
Question. Find a `choice-free’ argument for Exercise 3.
(See here.)
The main example we will consider of a model of the form 
We introduced the setting to discuss determinacy, namely infinite 2-person games with perfect information. We proved the Gale-Stewart theorem that open games are determined and discussed Martin’s extension to Borel games. A nice reference for the proof of Martin’s result (using the idea of `unraveling‘, which reduces any Borel game to an open game in a different space) is Kechris’s book on descriptive set theory:
MR1321597 (96e:03057). Kechris, Alexander S. Classical descriptive set theory. Graduate Texts in Mathematics, 156. Springer-Verlag, New York, 1995. xviii+402 pp. ISBN: 0-387-94374-9.
Posted by andrescaicedo 
form a continuous increasing sequence and are transitive. It follows that the reflection theorem holds for the 
. Arguing in 
,” and so is any 
. An argument using the Mostowski collapsing theorem allowed us to prove Gödel’s condensation lemma: If 
for 
. These two facts combine to provide a proof that 
holds in 
implies 
, but they also indicate that showing that 
ought to be more complicated. The reason is that the absoluteness of the construction of 
is a transitive proper class model of 
and in fact 
, i.e., the result of running the construction of 
, then running the construction inside 
, so 
, which would be a contradiction, since we are assuming that (provably in 
but 
is independent of 
is consistent with 
formulas are absolute among transitive models of (enough) set theory, and used this to prove that satisfiability for transitive sets is absolute. More precisely, let 
mean that 
is a transitive set 
codes a formula 
, 
is a tuple 
of elements of 
. Then 
such that 
is absolute for transitive models of 
,” although both are 
and therefore relativize downwards.
involves several bounded quantifiers, ranging over finite powers of 
and each class 
formula that defines satisfiability for 
functions are not. We also verified that being a well-ordering is absolute. The same argument actually shows:
is a well-founded relation on 
is absolute for transitive models of 
.
and 
. By combining with an isomorphism between 
and its order type, we may assume that 
. By restricting attention to the subset 
of 
as well.
can be built by a straightforward recursion: Given 
and 
a sequence of elements of 
which is larger than all the previous 
under both orderings.
. We circumvent the problem by only considering ordinals 
whose cofinality is larger than the cofinality of 
, then 
.
be an increasing sequence of regular cardinals cofinal in 
. Consider the subset 
. It must contain a subset of size 
is bounded in 
. Let 
denote the shortest initial segment of 
, we are left with a set of size 
a set of size 
.
let ![X^{[2]}=\{(n,m)\in X^2:n<m\}](https://s0.wp.com/latex.php?latex=X%5E%7B%5B2%5D%7D%3D%5C%7B%28n%2Cm%29%5Cin+X%5E2%3An%3Cm%5C%7D&bg=ffffff&fg=333333&s=0&c=20201002)
. A function ![f:X^{[2]}\to{\mathbb N}](https://s0.wp.com/latex.php?latex=f%3AX%5E%7B%5B2%5D%7D%5Cto%7B%5Cmathbb+N%7D&bg=ffffff&fg=333333&s=0&c=20201002)
is regressive iff 
for all 
in 
. A set 
is min-homogeneous for 
iff 
whenever 
and 
.
such that if 
and 
to be homogeneous, i.e., in general ![f\upharpoonright H^{[2]}](https://s0.wp.com/latex.php?latex=f%5Cupharpoonright+H%5E%7B%5B2%5D%7D&bg=ffffff&fg=333333&s=0&c=20201002)
will not be constant. For example, consider 
. Notice also that without loss of generality 
, since 
. It is natural to try to establish the rate of growth of the function 
that to each 
variables, Kanamori and McAloon showed:
of two variables, defined as 
. Obviously, 
and 
. The situation for 
is less clear, although it is of exponential growth.
, 
, 
, 
.
, so 
for 
.
.
hold?
; for example, every 
and 
that the power set of any ordinal is well-orderable. We want to conclude that choice holds, i.e., that every set is well-orderable. A natural strategy is to proceed inductively, showing that each 
is well-orderable: Clearly, if the result is true, each 
and therefore 
, so a well-ordering of 
thanks to our initial assumption.
is well-ordered by the well-ordering 
. Given a well-ordering 
. By assumption, 
is well-orderable, and any well-ordering of it induces (via 
) a well-ordering of 
into a well-ordering of 
for 
, iff
, or else
, say, and if 
is the well-ordering of 
, then 
.
, let 
Then the 
-first among these elements would be the 
is such a chain then, since 
is a non-increasing sequence of ordinals, there must be 
such that all 
with 
have the same rank 
would be an infinite 
, not that there is any canonical way of choosing one. In order for the argument above to work, we need not just that each 
is well-orderable, but in fact we need to have selected a sequence 
of well-orderings of the 
, we considered an arbitrary set 
, then a well-ordering of 
large enough so each 
for 
.
. (Notice that this makes sense since, inductively, each 
be a well-ordering of 
so that 
we copy the strategy with which we tried to well-order 
: For 
, set 
iff
be the unique ordinal such that there is an order isomorphism
.
, then 
, so 
and the well-ordering 
. Via 
A kind of library 