Linked Questions

-1
votes
1answer
117 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 ...
0
votes
2answers
56 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 ...
35
votes
5answers
5k 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?
25
votes
8answers
2k 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 ...
11
votes
3answers
2k 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 is not making since to me. I have read several others posts on this, but none made ...
7
votes
3answers
683 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 ...
10
votes
2answers
424 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 ...
4
votes
3answers
651 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 $\alpha$, ...
9
votes
3answers
298 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, ...
4
votes
3answers
205 views

Power set difference on the same set.

I've been arguing about the following expression: Given the following set $ S := \{1,2,3,4,5\}$ evaluate the expression: $$ \wp S - S = $$ I think that the result is $$\wp S - S = \wp S $$ Because ...
2
votes
4answers
281 views

Arithmetic on real numbers

So far, I have studied elementary set theory and I have some questions. I know how to add or multiply natural numbers and ordinals, but how do I subtract or divide or root or log? Is there any ...
2
votes
2answers
327 views

What is the difference between $\omega$ and $\mathbb{N}$?

What is the difference between $\omega$ and $\mathbb{N}$? I know that $\omega$ is the "natural ordering" of $\mathbb{N}$. And I know that $\mathbb{N}$ is the set of natural numbers (order doesn't ...
2
votes
1answer
245 views

How to prove $2+2=4$ using axioms of real number system?

How to prove $2+2=4$ using axioms of real number system? How do you make sense of the axioms for real number system when you cannot define the operations. You don't give an algorithm to calculate the ...
2
votes
2answers
116 views

Treatise on foundational mathematics

I read the construction of real numbers by Dedekind cuts today in a book(it was quite incomplete).I was wondering if someone could please point me to a source or a book containing the construction of ...