# Tagged Questions

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### Uniqueness in transfinite recursion.

Theorem of transfinite recursion: Given a well-ordered set $A$ let $\varphi(g,y)$ be a ZF formula such that for every $a \in A$ and every function $g$ with domain $I_a$ (where $I_a$ is the initial ...
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### How can I make a set theory arbitrarily weak and finitely axiomatizable?

So the background is I'm trying to construct an automated theorem proving system based on first order resolution to formalize problems in mathematics and number theory. I'm familiar with the notion ...
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### A problem in transitivity of set

Let $D$ be a transitive set with the following property : $$\forall a\in D:\quad a\subseteq B\,\Longrightarrow a\in B$$ Prove that $D\subseteq B$. I think there will be needed axiom of ...
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### Difference between impredicative and predicative version of separation axiom

What is the difference between an impredicative and a predicative version of the separation axiom in ZFC: $$\forall x \exists y \forall z ( z\in y \leftrightarrow (z \in x \wedge \phi (z))$$ What ...
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### How can I define $\mathbb{N}$ if I postulate existence of a Dedekind-infinite set rather than existence of an inductive set?

Suppose in the axioms of $\sf ZF$ we replaced the Axiom of infinity There exists an inductive set. with the Axiom of Dedekind-infinite set There exists a set equipollent with its proper ...
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### Well-Ordering Theorem without Axiom of Regularity

Personally, I am not fond of the Axiom of Regularity. Some alternative models in set theory use the negation of the Axiom of Regularity as an axiom (non well-founded theories). I am curious if the ...
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### Problem understanding the Axiom of Foundation

I am just beginning to learn the ZF axioms of set theory, and I am having trouble understanding the Axiom of Foundation. What exactly does it mean that "every non-empty set x contains a member y such ...
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### Is it possible to prove the axiom of infinity from the real number axioms?

What I mean by the title is whether, given that there is a class $\Bbb R$ and operations $+,\cdot,<$ that satisfy the ordered field axioms and the least upper bound axiom, you can prove the ...
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### Non-constructive axiom of infinity

Looking at the ZFC axioms (in the Wikipedia version), the axiom of infinity stands out because it contains an extremely specific construction. This seems rather unelegant to me, therefore I've thought ...
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### How far is it true that statements dependent on Axiom of Choice are not constructive.

Axiom of Choice is often used in mathematics to construct various objects, such as basis of $\mathbb{R}$ as a vector space over $\mathbb{Q}$, unmeasurable subset of $\mathbb{R}$, or a non-principal ...
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### Success of Hilbert's Axioms

We know Euclid's axioms were found to be having many loopholes as in there were still many assumptions which weren't being stated in his system of axioms . Are Hilbert's axioms today completely ...
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### ZF: Regularity axiom or axiom schema?

I have seen the axiom system ZF for set theory described including a single axiom of regularity (aka "foundation"), namely $$\forall x\neq\emptyset \, \exists y\in x \ y\cap x = \emptyset$$ and also ...
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### Gödel's incompleteness wrt weakend versions of ZFC

Suppose for the sake of argument that we look at ZFC with the axiom of infinity removed. http://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems#First_incompleteness_theorem ...
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### What set theory axioms do I need to believe in uncountable ordinals?

Math people: The title is the question. I am convinced that uncountable sets exist, thanks to Cantor's diagonal proof. It is not intuitively clear to me that uncountable ordinals or cardinals should ...
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### Axiom of Regularity - Transitive set

I just managed to confuse myself completely while studying for Set Theory. We have the Axiom of regularity: $$\forall S (S\not= \emptyset \rightarrow (\exists x\in S)(S\cap x=\emptyset))$$ Now a set ...
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### Question Regarding the Axiom of Extensionality

Jech's text on Set Theory states the following: If X and Y have the same elements, then X = Y : ∀u(u ∈ X ↔ u ∈ Y ) → X = Y. The converse, namely, if X = Y then u ∈ X ↔ u ∈ Y, is an axiom ...
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### What is the smallest fragment of ZFC that has the same consistency strength as ZFC?

The question in the title is undoubtedly nonsensical, but I am not sure how to state this question properly. Perhaps some examples will help me explain it. Thanks to Godel and Cohen, we know that ...
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### ¿Can you help me with axiom of regularity? [duplicate]

I understand that if we define a set such that $A=\{A\}$ we have then a contradiction because of regularity axiom, thus, a set can not be a member of itself. My question is; how can it be so, if I ...
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### ZF Extensionality axiom

To familiarize myself with axiomatic set theory, I am reading Kenneth Kunen's The Foundations of Mathematics that presents ZF set theory. I haven't gotten really far since I am stuck at the axiom of ...
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### Axiom of infinity exceptions?

It is my understanding that the axiom of infinity, or axioms that depend on the axiom of infinity, are the only axioms that are not representable in finite First-Order Predicate Logic. Is this ...
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### Schema of separation and set of all sets

The schema of separation states (this is probably simplified) states that if $P$ is a property and $X$ is a set, then there exists a set $Y = \{ x \in X : P(x)\}$. The notes I'm reading say that from ...
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### A basic doubt on axiom of foundation of Zermelo-Fraenkel set theory

I have read in a book that the "axiom of foundation prevents anomalies such as a set being an element of itself". Now, axiom of foundation says that there exist an element in every set which is ...
Say you have some collection of axioms and you find a set $X$ fulfilling them. And the definition of the set $X$ involves another concept, e.g. the real number, but only in a way which refers to the ...