# Tagged Questions

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### Manifolds of Non-Standard Dimension

Can there exists (non-trivial) manifolds of non-standard dimensions? Certainly, there do exist manifolds of dimension $n$ for any $n \in \mathbb{N}$ (as well as manifolds of countably many ...
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### How much maths can we do in NF(U)?

I have recently become interested in non-standard set theories, particularly in NF and NFU and have been reading some things here and there. Of course, I don't know much about it and I'm still trying ...
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### Absoluteness of $\forall x (x=x)$

Is there any kind of (set-theoretic) absoluteness result for the formula $\forall x (x=x)$? And what about for $\exists x (x=x)$? I know $x=x$ is absolute given that it's $\Delta_0$. Also, if I'm ...
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### Godel's completeness theorem and formula that states consistency of ZF

Godel's completeness theorem, in original formulation, says that every logically valid statement/formula has finite deduction of a formula. Now then there is Godel's incompleteness theorem. Would this ...
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### Are there axiomatizations of first order logic or set theory defined in first order logic or set theory?

There are several axiomatizations for number theory, group theory, and other theories represented in first order logic. Further, these theories are also representable in set theory such as $\sf ZFC$ ...
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### Modal set-theory

In his “The Potential Hierarchy of Sets”, Review of Symbolic Logic 6:2 (2013), 205-28 Øystein Linnebo has proposed a modal set-theory. I was wondering what kind of utility can such a theory have for a ...
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### Model for replacement

If $R_{k}$ is a model for replacement, is necesarly k a strong limit cardinal? And if k is a regular cardinal, are then equivalent the sentences: k is a strongly inaccesible cardinal if and only if ...
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### About the definible sets $L_\alpha$

Let $\alpha$ be an ordinal number. Is that true that $\alpha$ = $\beth_\alpha$ is equivalent to the statement $|L_\alpha|=|R_\alpha|$, where $L_\alpha$ is the $\alpha$-th stage of the constructible ...
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### Every elementary submodel of $H(\aleph_1)$ is transitive

I am trying to solve the first part of Exercise II.17.30 in Kunen's Foundations of Mathematics which asks: Prove that every $A \preccurlyeq H(\aleph_1)$ is transitive. Here we are working over ...
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### Ultraproducts of models of ZFC

Let $D$ be a non-principal ultrafilter over $\mathbb{N}$. Let $A_i$ for $i\in\mathbb{N}$ be (countable) models of ZFC such that $A_i\models \mathfrak{c}=\aleph_i$. Then, what is the size of the ...
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### Is there a formalisation of set theory where unions can be taken over some classes that are not a priori known to be sets?

Of course, fully unrestricted unions in ZFC will immediately lead to inconsistency, but is there a variation where restrictions on allowed classes are given explicitly at least, in terms of the ...
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### Can existence of aleph one be proved without the power set axiom? [duplicate]

Cantor's construction of $\aleph_1$ is to notice that all ordinals constructed after $\omega$ are countable, take the union of all countable ordinals, and then show that this union can not be ...
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### About proving that the Continuum Hypothesis is independent of ZFC

In Mathematical Logic, we were introduced to the concept of forcing using countable transitive models - ctm - of $\mathsf{ZFC}$. Using two different notions of forcing we were able to build (from the ...
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### Hall's marriage Theorem and Tychonoff Theorem

I was reading this paper. In particular the second point. He proves the Hall's marriage Theorem for infinite family using the Tychonoff theorem on topological product of compact $T_2$ spaces and the ...
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### New Axioms of Infinity

Axiom of Infinity says there is an inductive set (i.e. a set which includes $\emptyset$ and is closed under successor operator). Formally: \$Inf:\exists x~(\emptyset\in x~\wedge~\forall y\in ...
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### Can the consistency of ZF be proved in MK?

When adding choice to Zermelo–Fraenkel (ZF) set theory, one can go further than ZFC, by first adding classes, then adding global choice, and then proving that the resulting von Neumann–Bernays–Gödel ...
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### Asymmetric roles that are symmetric in every instance

This is similar to something else I posted, but this time we'll pretend we've never heard of infinite sets or infinite series. \begin{align} & \sin(\alpha+\beta+\gamma+\delta+\varepsilon+\zeta) ...