# Tagged Questions

This tag is for set theory topics typically studied at the advanced undergraduate or graduate level. These include cofinality, axioms of ZFC, axiom of choice, forcing, set-theoretic independence, large cardinals, models of set theory, ultrafilters, ultrapowers, constructible universe, inner model ...

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### What will happen when Ax.Inf is replaced to its negation in ZFC? [duplicate]

Possible Duplicate: What are the consequences if Axiom of Infinity is negated? In ZFC, if we replace Ax.Inf to such a statement that every set is finite, then does this theory satisfiable? ...
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### Ultrapowers by extenders of potential premice

I have a problem with an argument in Fine structure and iteration trees by Mitchell and Steel. Let $E$ be a $(\kappa, \lambda)$-extender. Let $\dot E^{\mathcal{M}}$ the a unary predicate with is ...
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### No $\Delta-$System on a subset of a singular cardinal.

I've been making my way through the new Kunen and I've come across an exercise that I can't work out. The question is this: Let $\kappa$ be a singular cardinal. Show that there is a collection $A$ ...
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### Other ways of proving that the set of all countable ordinals is uncountable

I know that the standard way of proving that the set of all countable ordinals is uncountable is by stating that if the set is countable, then it incurs Burali-Forti paradox. Is there other ways of ...
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### In NFU, is there a bijection between the set of all sets and the set of all one-element sets?

In the set theory NFU (described by M. Randall Holmes in "Elementary Set Theory with a Universal Set"), it is possible to define the set of all sets, and the set of all one-element sets. An object is ...
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### How to associate non-natural numbers with set features/relations?

Pythagoras said "everything is number". Therefore, it might seem that sets (everything) can be described/identified using numbers. But does it really make sense to associate non-natural numbers with ...
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### Existence of function (AC)

I want to make such a construction: Let $X$ be an infinite set. Put: $Y_0 = X \\ y_0 \in Y_0$ $Y_1 = Y_0 - \{y_0\} \\ y_1 \in Y_1$ ... $Y_n = Y_{n-1} - \{y_{n-1}\} \\ y_n \in Y_n$ ... for every ...
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### Importance of Kripke–Platek set theory

What would be importance of Kripke-Platek set theory? I know that Saul Kripke is one of the most important philosophers in the world, but curious in what place he is in set theory.
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### How large is the infinity of real numbers [closed]

Umm ... Can someone disprove my proof that there are aleph-1 number of real numbers? Even comments to make my proof more rigorous are welcome. https://www.dropbox.com/sh/1fz28jlwrprh4jv/rhA7Ad7OtX
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### ZFC set theory,first order theory [duplicate]

Possible Duplicate: What is the difference between Gödel's Completeness and Incompleteness Theorems? what is the relationship between ZFC and first-order logic? I am a bit ...
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### which of the following sets are countable

are the following sets countable? set of all sequences of non-negative integers. The set of all sequences of non-negative integers with only a finite number of non zero terms could any one tell ...
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### Ultrafilters in $\omega$

Let $U$ be a non-principal ultrafilter in $\beta \mathbb{N}$. Can it have a countable character as a point in this topological space? Is there decreasing chain of clopen subsets of $\beta\mathbb{N}$ ...
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### Is this category essentially small?

Let $\mathcal C$ be the category of finite dimensional $\mathbb C$-vector spaces $(V, \phi_V)$ where $\phi_V \colon V \to V$ is a linear map. A morphism $f \colon (V , \phi_V) \to (W , \phi_W)$ in ...
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### Axiom of Regularity

I am having difficulty understanding http://en.wikipedia.org/wiki/Axiom_of_regularity Every non-empty set A contains an element B which is disjoint from A So from my understanding if I have a ...
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### Definition for Empty Set in Logic

$\exists X (\forall Y (\neg(Y \in X)))$ is whats given in my lecture, but I was wondering, is it the same as $\exists X (\forall Y (Y \notin X))$