# Linked Questions

1answer
419 views

### ZFC and irrational numbers [duplicate]

I understand how integers and rationals are expressed/derived in ZFC. But what about the irrational numbers? Can they also be expressed? If not, are there other axiomatic set theories able to express ...
2answers
123 views

### Integers, rationals and reals as sets? [duplicate]

Natural numbers can be represented as pure sets by defining them to contain every number that is smaller than them. Arithmetic can be performed on them using the Peano axioms. Are there any similar ...
5answers
11k views

### Completion of rational numbers via Cauchy sequences

Can anyone recommend a good self-contained reference for completion of rationals to get reals using Cauchy sequences?
8answers
8k views

### Why does the Dedekind Cut work well enough to define the Reals?

I am a seventeen year old high school student and I was studying some Real Analysis on my own. In the process, I encountered the Dedekind Cut being used to construct the Reals. I just can't get the ...
3answers
16k views

### How can an ordered pair be expressed as a set?

My book says \begin{equation} (a,b)=\{\{a\},\{a,b\}\} \end{equation} I have been staring at this for a bit and it doesn't make sense to me. I have read several others posts on this, but none made ...
1answer
3k views

### True Definition of the Real Numbers

I've found lots of resources that say this is a real number if it's not rational, but what is a real number, really? I mean what is the definition of a real number? If nothing else, anyone know of a ...
3answers
3k views

### Building the integers from scratch (and multiplying negative numbers)

Now I understand that what I am about to ask may seem like an incredibly simple question, but I like to try and understand math (especially something as fundamental as this) at the deepest level ...
3answers
3k views

### The real numbers and the Von Neumann Universe

So I'm going to prefix this question by saying that I probably don't have a great understanding of what I'm asking. We build the cumulative hierarchy as follows: $V_0=\emptyset$ For every $\alpha$, ...
4answers
928 views

### How to write $\pi$ as a set in ZF?

I know that from ZF we can construct some sets in a beautiful form obtaining the desired properties that we expect to have these sets. In ZF all is a set (including numbers, elements, functions, ...
3answers
1k views

1answer
165 views

### Can we prove that set of irrational numbers is a set using Zermelo-Fraenkel axioms?

To remove paradoxes of naive set theory, We started with the axioms of Zermelo-Fraenkel and developed a set theory. Where we are building sets starting from a empty set. How to construct set of ...

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