# Tagged Questions

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### Question about the Continuum Hypothesis

The Continuum Hypothesis hypothesises There is no set whose cardinality is strictly between that of the integers and the real numbers. Clearly this is either true or false - there either exists ...
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### Size of topological space depending on the size of local basis. (With elementary submodels)

Recall that the character of a topological space $\chi(X)$ is the minimum cardinal $\kappa$ such that every point in $X$ has a local basis of size $\kappa$. I need to prove that if $X$ is compact ...
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### Can set-theoretic forcing exist without law of excluded middle?

Of course the law of excluded middle is accepted by almost every mathematician except a few constructivists, but then I was wondering if set-theoretic forcing can exist without law of excluded middle. ...
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### Existence of an axiom question in relation to $\mathsf{Infinity}$

Original Post This may be a stupid question, but does there axist an axiom $\phi$ that is independent of $\mathsf{ZFC}$, and not equivalent to the axiom of $\mathsf{Infinity}$, such that ...
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### Well-ordering the reals: finding a certain model of $\mathsf{ZFC}$

How would one go about constructing a model $\mathfrak{M}$ of $\mathsf{ZFC}$ such that under $\mathfrak{M}$, no formula defining a well-ordering of $\mathbb{R}$ exists? I am certain such models are ...
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### The lexicographic order [duplicate]

If it is given ordinals $\alpha$ and $\beta$, the lexicographic order on $\alpha \times \beta$,$\leq_{lex}$ is given by: $(\gamma_0,\delta_0)<_lex(\gamma_1,\delta_1)$ if and only if either ...
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### $\mathsf{ZF}$ is not finitely axiomatizable

As we know a first order theory $T$ is finitely axiomatizable if there is a finite set $F\subseteq T$ of axioms such that $F\vdash \sigma$ for every $\sigma \in T$. How we can prove if $\mathsf{ZF}$ ...
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### How are models constructed?

As I understand, a model $\langle S,\sim\rangle$ of $\mathsf{ZFC}$ is a set $S$ coupled with some binary operation $\sim$, in which the axioms of $\mathsf{ZFC}$ hold (please correct me if I am wrong). ...
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### How to extract the sets “produced” by a axiomatic set theory, into some newly introduced collection?

Say I know well how to reason in a set theory, which for the sake of this question I'll call $\bf{ST}$, say one of those, it can by $\bf ZFC$. It's principally written down via first order logic and ...
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### How is it possible that the well-ordering theorem is strictly stronger than the axiom of choice in second-order logic? [duplicate]

If I am not wrong, the well-ordering theorem is strictly stronger than the axiom of choice in second-order logic. I am not sure to understand how this is possible. The reason is that second order ...
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### Countable axiom of choice: why you can't prove it from just ZF

This is a follow-up question to the discussion about the finite axiom of choice here. Suppose we have a countable collection of non-empty sets $\{A_1, A_2, A_3,\cdots\}$ Reasoning as indicated in ...
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### What is a non-constructible real?

I am not sure to fully understand the concept (I read many of the wikipedia definitions for many of these issues but I am still confused). For instance, is a surreal number an example of ...
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### Unbounded Finity?

Consider the successor of the largest finite ordinal that will ever be considered alone. But then it wasn't the largest finite ordinal that will ever be considered alone. How do we get around this ...
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### Transfert principle of a conservative extension of ZFC

In the following paper, there is a theory called $^*ZFC$ in the language $(^*,\in)$. The *-map is (more or less) defined on the Von Neuman hierarchy $S$ and verifies the following axiom schemata true ...
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### Is maths = set theory + logic? [closed]

It seems to me that most branches of mathematics are just an application of set theory and the rules of logic. You just definite particular types of sets and study their properties. For example, in ...
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### Founding Arithmetic on geometry

In the past I found some fleeting references that some (Frege in his later years being one of them) tried to found arithmetic not on set-theory and logic but on geometry and logic. Unfortunedly Frege ...
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### Are there ordinals other than the set of natural numbers which satisfy this property?

Let $\alpha$ be an ordinal. We say that $\alpha$ is good iff for every $\beta\in \alpha$, there exists $\gamma\in \alpha$ such that $|\scr{P}(\beta)|\leq |\gamma|$. Question: Is the set of natural ...
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### A Question Regarding Forcing in Gödel's Constructible Universe in Infinitary Logics

In his answer to the MathOverflow question Gödel's Constructible Universe in Infinitary Logics, Prof. Hamkins gives a very interesting answer and proof to user46667's second question: (2) What is ...
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### Are there classes with different sizes?

Are there classes with different sizes ? I will put a precise statement of my question below: Are there two well formed formulas $P,Q$ each with one free variable such that there is no well formed ...
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### Skolem Hulls in $H_{\omega_2}$

Consider a model of the form $\mathfrak{A} = (H_{\omega_2}, \epsilon, \prec, f_0, f_1, ...)$, some expansion of $H_{\omega_2}$ in a countable language, with $\prec$ giving a well-order. Does there ...
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### What is the relation between axiomatic set theory and logical quantifiers?

On the one hand, the logical predicates $\forall$ and $\exists$ are defined using the concept of a Domain of Discourse, which itself is defined as a set (at least according to wikipedia). On the ...
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### On Counted Languages

In my recent question on Godel Completeness I mentioned that there was a related question I wanted to ask, but would keep separate. I have been recently studying "non-well ordered sets" and Chapter 7 ...
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### A truth definition, wrong, but where

Consider the theory ZFC of language with connectives $\neg$, $\rightarrow$, predicative symbol $\in$ and quantifier $\forall$. Assume that we are working in ZFC. Let $\mathcal{V}$ be the class of ...
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### Consistency result vs. True in every model of Axiom X

Suppose a forcing extension of ZFC has been found which satisfies statement $A$. For example, say the extension is formed by Cohen or Laver forcing, so that the model satisfies $\neg$CH. At this ...
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### Problem 24 from Chapter 1 of Kunen's Set Theory: An Introduction to Independence Proofs

Just want to make sure I'm tracking Kunen here, and hopefully the proof I have is correct. Comments / Suggestions welcome. Thanks! Problem 24. Let T be any consistent set of axioms extending ZF. ...