108 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 ...
4k 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?
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 ...
2k views

How can an ordered pair be expressed as a set?

My book says $$(a,b)=\{\{a\},\{a,b\}\}$$ 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 ...
629 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 ...
353 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 ...
580 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 $\alpha$, ...
279 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, ...
184 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 ...
265 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 ...
301 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 ...
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 ...