# Tagged Questions

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### Axiom of infinity: What is an inductive set?

This Axiom states that there exists an inductive set. But, what is the definition of an inductive set?
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### set theory, Incompleteness and axiomatic systems

Is the number of theorems that can be proved (decidable) within a certain set of axioms (for instance ZFC) is finite or infinite ? in other words, are we going to fully exhaust that set of axioms ...
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### Quotient respect to an equivalence relation.

I have the natural numbers $\mathbb{N}$, in ZF. I want to construct the integers $\mathbb{Z}$ taking the quotient respect to usual the equivalence relation $R$ on $\mathbb{N}\times\mathbb{N}$ that is ...
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### axiom of regularity and empty set

So we have axiom of regularity which says that any set (let's call it $A$) has a subset $B$ such that $A\cap B = \emptyset$. But thinking about such sets as $C = \{1, 2\}$ and $D = \{3, 4\}$ we still ...
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### Axiom of extension

I am learning Set Theory from the book Naive Set Theory by Halmos as part of my course. The first chapter is on the Axiom of Extension. I understand what it is but what I don't understand is why it ...
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### Ill-Founded Sets in ZFC

Let $\mathbb{N}$ be the set of all finite ordinals defined as the intersection of all ordinals including the empty set and closed under successor. Consider the following set: $S_0 = \mathbb{N}$ ...
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### Axiom of Pairing

Axiom of Pairing states that if $a,b$ are sets, $\exists$ a set $A$ such that $A=\{{a,b\}}$. My question is that why we can't use Axiom of specification to define $A=\{x|x=a \vee x=b\}$?
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### Simple Proofs in ZFC Set Theory

So I'll keep this real short and simple. In this document on page 23 there is a list of axioms. On page 24 there is a list of theorems that come from said axioms. I can prove them all except 3 and 5. ...
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### The real numbers and the axiom of foundation

I am having a bit of confusion about the real numbers and ZF set theory (I asked a question about it a few days ago). I am a bit unsure as to why the real numbers can be in any model of ZF as they ...
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### Is this definition true about comparisons of sets?

I'm trying to find the error in a proof that yields a contradictory result, and I'm beginning to think that one of the definitions I start with is incorrect or self-contradictory. Is the following ...
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### Axioms of set theory and logic

Zermeloâ€“Fraenkel set theory is the most common foundation of mathematics with eight axioms and axiom of choice (ZFC): http://plato.stanford.edu/entries/set-theory/ZF.html But one can see that the ...
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### What are some primary mathematical utilities of the axiom schema of separation?

I read a discussion concerning the axiom schema of specification, which I yet take as saying that for every set and a class-defining condition, those elements of the set satisfying this condition ...
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### When does the set enter set theory?

I wonder about the foundations of set theory and my question can be stated in some related forms: If we base Zermeloâ€“Fraenkel set theory on first order logic, does that mean first order logic is not ...
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### what is the relationship between ZFC and first-order logic?

In Wikipedia, it says that ZFC is a one-sorted theory in first-order logic. However, I was not really able to comprehend the later parts that seem to elaborate on that point. Can anyone explain the ...
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### How is the Cantor's paradox resolved in the ZFC system?

How is the Cantor's paradox resolved in the ZFC system? Thanks.
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### Need help understanding Axiom of Extensionality

I'm attempting to learn Set Theory and I'm currently working through Halmos' Naive Set Theory. I will say that I completely understand the essence of the Axiom of Extensionality. However, where I'm ...