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Results tagged with set-theory
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user 109573
forcing, large cardinals, descriptive set theory, infinite combinatorics, cardinal characteristics, forcing axioms, ultrapowers, measures, reflection, pcf theory, models of set theory, axioms of set theory, independence, axiom of choice, continuum hypothesis, determinacy, Borel equivalence relations, Boolean-valued models, embeddings, orders, relations, transfinite recursion, set theory as a foundation of mathematics, the philosophy of set theory.
9
votes
1
answer
361
views
Definitional complexity of truth in $L$ without $0^{\#}$
Cross posted from MSE at commenter's suggestion: https://math.stackexchange.com/questions/2269319/definability-of-truth-in-l-without-0
I'm interested in the relationship between the existence of $0^{ …
- 4,316
11
votes
Accepted
Does ZF minus infinity imply collection?
ZF - Inf does imply Collection. Fix a set $X$ and a property $P$ (which can be formalized in terms of a formula and a parameter). Since we have separation, we may assume for all $x \in X,$ there is $y …
- 4,316
9
votes
Accepted
Example of a $\Pi^2_2$ sentence?
The Suslin hypothesis is $\Pi^2_2,$ and $T = ZFC + GCH + LC$ (LC an arbitrary large cardinal axiom) does not prove it to be equivalent to any $\Sigma^2_2$ sentence. Suppose toward contradiction $T$ pr …
- 4,316
7
votes
1
answer
359
views
Absoluteness, reflection to ctms, and choice in outer models
Last night I was thinking about some related statements which follow from ZF+DC, but it actually seems they only need DC to hold in some outer model of the universe. In particular, let $M \models ZF.$ …
- 4,316
7
votes
2
answers
392
views
Definability of isomorphisms between class well-orderings
Inspired by this question, I've been trying to figure out for myself the basic properties of definable class well-orderings in transitive models $M$ of ZFC: What is $\omega_1^{CK}(\mathsf{Ord})$?
The …
- 4,316
9
votes
0
answers
257
views
How many iterations of inner models/generic extensions are sufficient?
Let $M=M_0$ be a ctm of ZF.
If $(M_0, \ldots, M_n)$ is a sequence of ctms of ZF, where for all $i,$ either $M_{i+1}$ is an inner model of $M_i$ or a generic extension of $M_i,$ then call $M_n$ an $n …
- 4,316
9
votes
Accepted
Can one exhibit an explicit Kuratowski infinite set without invoking Replacement?
The answer is no. Let ZC' be ZFC without replacement and infinity and with the assertion there is a Kuratowski infinite set. We will construct a model $M$ of ZC' such that only hereditarily finite ele …
- 4,316
4
votes
Consistency of embedding cardinals in linear orderings
Partial answer for the newly formulated axiom, which I'll just call Ax.
First, Ax does imply GCH. Assume Ax and suppose GCH first fails at $\kappa.$ Let
$L=(2^{\kappa} \setminus \{\alpha \mapsto 0, \ …
- 4,316
3
votes
If $X$ a subset of ordinals, is $M[X]$ a generic extension?
It is not true in general that $M[X]$ is a generic extension of $M,$ even in the case $X \subset \omega.$ For example, such an $X$ could code a well-ordering of $\omega$ of type $o(M),$ in which case …
- 4,316
8
votes
Accepted
Products of Cohen forcings
The answer is yes.
Lemma: There is a function $F: 2^{\lambda} \rightarrow 2^{\lambda}$ such that for every $S \in [\lambda]^{\lambda}$ and function $g: \lambda \setminus S \rightarrow 2,$ $F$ restric …
- 4,316
7
votes
Accepted
Ideal-like filter on a ring not generated by ring ideals
This does not hold for all commutative rings, but it does hold for Noetherian rings and for valuation rings (assuming the convention that filters don't contain $\emptyset,$ or else $\mathcal{P}(R)$ is …
- 4,316
15
votes
Accepted
Is Global Choice conservative over Zermelo with Choice?
Edit: Here's my preprint filling in the details of the construction in this answer: https://arxiv.org/pdf/2312.11902.pdf
Global Choice is not conservative over ZC. We'll build a model of ZC which sati …
- 4,316
11
votes
1
answer
367
views
Is $V=L$ equivalent to there being a $\Sigma_1$ well-ordering of the universe?
Working in ZF, it's well-known that for any $n \ge 2,$ the claim that there is a $\Sigma_n$ well-ordering of the universe is equivalent to the axiom $V=HOD.$ It seems natural to believe there should b …
- 4,316
12
votes
1
answer
318
views
Which $L$-like principles are known to be relatively consistent with large cardinals?
For which of the standard large cardinal axioms $\varphi_{LC}$ and which $L$-like principles $\psi$ (e.g. GCH, $\mathrm{V}=\mathrm{HOD},$ the ground axiom, and various diamond and square principles) i …
- 4,316
13
votes
0
answers
460
views
Does Foundation increase the strength of second-order logic?
Thinking about the recent threads on structural consequences of the Axiom of Foundation (AF) over ZF-AF, I've been trying to find some conservativity result which explains why AF doesn't seem to have …
- 4,316