# 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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### Better representaion of natural numbers as sets?

Natural numbers can be represented as $0=\emptyset$ $1=\{\emptyset\}$ $2=\{\{\emptyset\}\}$ $...$ or as $0=\emptyset$ $1=\{0\}=0\cup\{0\}$ $2=\{0,1\}=1\cup\{1\}$ $...$ What are the names of ...
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### Prüfer groups are countable

I have read that for any prime number $p$ the Prüfer $p$-group is countable. My question is: where can I find a proof of this fact? Thanks.
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### Is the set $S$ of functions $S = \{ f : \mathbb R \to \mathbb R \}$ from the reals to the reals a totally ordered set?

Is the set $S = \{ f : \mathbb R \to \mathbb R \}$ of functions $f$ from the reals to the reals a totally ordered set ? I think the question is clear. Note that there is a bijection to $g(x)$ gives ...
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### Looking for Cantor's original proof of the Cantor-Bernstein theorem that relies on the axiom of choice?

Even a sketch of it would be good enough. Thanks.
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### What are some good intuitions for understanding Souslin's operation $\mathcal{A}$?

What are some good intuitions for understanding Souslin's operation $\mathcal{A}$? Recall the definition: Let $S = \mathbb{N^{<N}} = \bigcup_{n = 1}^\infty \mathbb{N}^n$ be the set of ...
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### Contour Infinites and Vector Spaces

We usually define Hilbert or finite dimensional vector spaces, and even topologies or differential geometry on $\mathbb{R}^n$ , so I wonder what is the implication of doing that on some extended ...
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### Why is the class of all sets a stage?

I want to prove that the class of all sets $\mathbb{S}=\{x \mid x=x \}$ is a stage (p. 15) (and then that it is a limit thus that it is the successor of another stage). One way to do it is to proof ...
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### Is there a categorification of (infinite) ordinal arithmetic?

Background for the curious reader: An ordinal $\beta$ is a transitive set in the sense that $\alpha\in\beta$ implies $\alpha\subset\beta$. Any ordinal is naturally well-ordered under $\in$ (so any ...
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### Why do we need a pullback for the definition or classification of subobjects?

Regarding the subobject classifier construction, why do we need the pullback? Monos from $U$ to $X$ are called subobjects, but I see that there might be injections which just have elements of the X ...
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### Existence of non-atomic probability measure for given measure zero sets

Let $\Omega$ be a set and $\Sigma$ be a $\sigma$-algebra of subsets of $\Omega$. Let $N$ be a collection of measurable subsets of $\Sigma$. Question: What conditions on $\Sigma$ and $N$ guarantee ...
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### What is the definition of the well-founded part of a model of set theory?

I've been trying to understand John Steel's various notes on inner model theory, but the one thing that trips me up is what he calls the well-founded part of a model of set theory. What exactly is the ...
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### A product of 2 Bourbakian posets, ordered by pointwise ordering.

A poset $P$ is called Bourbakian if every order-preserving map $P\rightarrow P$ has a LEAST fixed point. Let $P, Q$ be Bourbakian. Let $P\times Q$ be ordered pointwise, that is $(a,b)\le (c,d)$ if and ...
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### Does the category framework permit new logics?

It appears to me that a topos permits a broader concept of subsets than the yes/no decission of a characteristic function in a set theory setting. Probably because the subobject classifier doesn't ...
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### Maximal ideal space of $c_{\mathcal{U}}$

Let $\mathcal{U}$ be an filter over $\mathbb{N}$. Define $$c_{\mathcal{U}} = \{{(x_n)\in \ell_\infty\colon \lim_{\mathcal{U}, n}x_n =0\}},$$ which is a C*-algebra. Is there an accessible topological ...
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This a specific question about Ramsey type colorings. The arrow notation: If $\kappa$, $\lambda$, $\mu$ are cardinals and $n<\omega$, then $$\kappa\rightarrow(\lambda)^n_\mu$$ if for any function ...
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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 $\exists X (\forall Y (Y \notin X))$
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