All Questions
Tagged with inner-model-theory inner-models
10 questions
12
votes
1
answer
532
views
Why do we need the comparison lemma?
An inner model is a standard transitive (proper class) structure which satisfies all the axioms of ZFC and contains all the ordinals. The simplest and most well-known inner model is Gödel’s $L$, which ...
- 704
11
votes
1
answer
429
views
Coding the universe into a real over better core models
One of the most incredible results in modern set theory, due to Jensen, is that given any model of $\sf ZFC$, there is a class forcing which adds a real number $r$ and in the extension $V=L[r]$. ...
- 39.7k
10
votes
0
answers
288
views
How wealthy are canonical inner models?
One of the way a person shows their wealth is by having many diamonds. The same can be said about models of $\sf ZFC$. We can add generic diamond sequences, while preserving the old ones, so in some ...
- 39.7k
8
votes
1
answer
339
views
Inner model theory without choice
How much of the inner model project can be constructed without assuming the axiom of choice? I.e. which large cardinals provably have canonical inner models not assuming choice?
- 419
7
votes
1
answer
344
views
Characterizing L(R) Cardinals in HOD
We're working in L(R) under AD.
We know that
$\omega_1$ is the least measurable in HOD, $\Theta$ is the least woodin, $\delta^2_1$ is the least strong to the woodin, etc.
My question is about ...
- 603
4
votes
0
answers
270
views
What does $L(A,\mathbb{R})$ mean?
I many papers by Woodin, and on some answers here on MathOverflow (like the first answer of this question), I see the expression "$L(A,\mathbb{R})$" being used, but I have never seen it defined. I ...
- 615
3
votes
2
answers
249
views
Existence of inner models of $\mathrm{ZFC} \ +$ forcing axioms, under incompatible assumptions
I am curious about the existence of inner models of $\mathrm{ZFC}$ in conjunction with forcing axioms, under assumptions inconsistent with such theories. For example:
can we prove under any extension ...
- 1,328
3
votes
1
answer
241
views
Do all limit $\alpha \in \omega_1^L$ satisfy $L_\alpha \models V=HC$?
In Gaps in the constructible universe, Marek and Srebrny, 1973 a gap ordinal and the start of a gap are defined as follows
$\alpha$ is a gap ordinal iff $(L_{\alpha+1}-L_\alpha)\bigcap \mathcal{P}(\...
- 421
3
votes
0
answers
152
views
Why are the sharps of sets of big ordinals not in $\mathcal{P}(\omega)$?
In his talk A Condensed History of Condensation, Welch presents the following recursive sharp function, that is total when all sharps exist:
\begin{align*}
\# \colon ON &\to \mathcal{P}(ON) \\
\...
- 421
2
votes
1
answer
255
views
Why can't $L_\beta$ contain a real coding a well-ordering of order-type $\beta$, when $\beta$ is a gap ordinal?
In Gaps in the constructible universe, Marek and Srebrny, 1973 a gap ordinal is defined as follows
$\alpha$ is a gap ordinal iff $(L_{\alpha+1}-L_\alpha)\cap \mathcal{P}(\omega) = \emptyset$
Their ...
- 421