# Questions tagged [set-theory]

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.

3,476 questions
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### Injection into a proper class and choice without regularity

In $\sf ZF$, we have that the axiom of choice is equivalent to: For all sets $X$, and for all proper classes $Y$, $X$ inject into $Y$ and For all sets $X$, and for all proper classes $Y$, $Y$ ...
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### Why should we believe in the axiom of regularity?

Today I started reading Maddy's Believing the axioms. As I knew beforehand, it includes some discussion of ZFC axioms. However, I really hoped for a more extensive discussion of axiom of foundation/...
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### Set-theoretical foundations of Mathematics with only bounded quantifiers

It seems that outside of researchers in Mathematical Logic, mathematicians use almost exclusively bounded quantifiers instead of unbounded quantifiers. In fact, I haven't observed any other practice ...
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### Is there a standard meaning for this notation for ordinals in set theory?

In the paper Berkeley Cardinals and the Structure of $L(V_{\delta+1})$", by Raffaella Cutolo, a use is made of the notation $\alpha < < \beta$ for a pair of ordinals $\alpha, \beta$, without ...
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### Solutions to the Continuum Hypothesis

Related MO questions: What is the general opinion on the Generalized Continuum Hypothesis? ; Completion of ZFC ; Complete resolutions of GCH; How far wrong could the Continuum Hypothesis be?; When was ...
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### Can every first order theory have a finitely axiomatizable conservative extension in this sense?

Note: This is an edit of the previous question. By $T_2$ conservatively extends $T_1$ if and only if: There exists a function $F$ such that $T_2$ extends $T_1$ through $F$; and for every function $G$...
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### Very Large Cardinal Axioms and Continuum Hypothesis

Are very large cardinal axioms like $I_0$, $I_1$, $I_2$ consistent with $CH$ and $GCH$?
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### Injectivity implies surjectivity

In some circumstances, an injective (one-to-one) map is automatically surjective (onto). For example, Set theory An injective map between two finite sets with the same cardinality is surjective. ...
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### Radin generics from iterated ultrapowers

Let $u$ be a measure sequence derived from some embedding $j:V\rightarrow M$. I heard that from iterating $j$ is possible to define a generic filter for some Radin forcing $\mathbb{R}_w$. Specifically,...
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### Proofs of the uncountability of the reals.

Recently, I learnt in my analysis class the proof of the uncountability of the reals via the Nested Interval Theorem. At first, I was excited to see a variant proof (as it did not use the diagonal ...
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### Characterizing “bounded” distributivity in terms of dense open sets

Prikry forcing is a well-known example of a forcing which adds an $\omega$-sequence to some cardinal $\kappa$, but does not add bounded subsets to $\kappa$. There are other examples of forcings which ...
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### Proving that family of sets has non-empty intersection

Let's say I have an object which can be viewed as family of sets $\mathfrak{S} \subseteq 2^S$, and I want to prove that its intersection is non-empty. What is known already: $S$ is set of measurable ...
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### Number of ultrafilters in an extender

Assume GCH. Suppose $j: V\to M$ is an elementary embedding such that $\mathrm{crit}(j)=\kappa$, ${}^\kappa M \subset M$, $M\supset V_{\kappa+2}$. We can assume $j$ is defined from some extender of ...
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### Stationarity and Fodor's lemma for a (nice) poset?

The notion of a stationary set is peculiar in that it applies to subsets of certain very particular posets -- ordinals or powersets. At least to a non-set-theorist, the situation seems to beg for the ...
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### Sets = structured sets without structure

Motivation There is presumably no single and widely accepted formal definition of structured sets = sets plus structure based on sets as primitive objects, but several approaches are around. See e.g. ...
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### A consistent way to “decide” independent statements [closed]

Let us fix some mathematical theory T=T(0), such as ZFC. The aim is to develop algorithm A, which takes any statement S independent of T(0) as an input, and outputs axiom a(S) such that T(1)=T(0)+a(S) ...
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### Sunflower / $\Delta$-system lemma in a more general poset?
The sunflower lemma (or $\Delta$-system lemma) may be viewed as a statement about the poset $P_\omega(\omega_1)$, and the generalized sunflower lemma may be viewed as a statement about the poset \$P_\...