Linked Questions

158
votes
8answers
17k views

How do we prove that something is unprovable?

I have read somewhere there are some theorems that are shown to be "unprovable". It was a while ago and I don't remember the details, and I suspect that this question might be the result of a total ...
79
votes
10answers
13k views

Is mathematics just a bunch of nested empty sets?

When in high school I used to see mathematical objects as ideal objects whose existence is independent of us. But when I learned set theory, I discovered that all mathematical objects I was studying ...
57
votes
10answers
6k views

Is it an abuse of language to say “*the* integers,” “*the* rational numbers,” or “*the* real numbers,” etc.?

I'm finding that the more math I learn, the more concepts I thought were well-defined seem to be intuitive and naive. Here I'm asking about whether it's an abuse of language to refer to "the integers,"...
49
votes
6answers
5k views

Are there any objects which aren't sets?

What is an example of a mathematical object which isn't a set? The only object which is composed of zero objects is the empty set, which is a set by the ZFC axioms. Therefore all such objects are ...
65
votes
8answers
7k views

Does mathematics become circular at the bottom? What is at the bottom of mathematics? [duplicate]

I am trying to understand what mathematics is really built up of. I thought mathematical logic was the foundation of everything. But from reading a book in mathematical logic, they use "="(equals-sign)...
23
votes
3answers
3k views

Why can't Russell's Paradox be solved with references to sets instead of containment?

My background is in computer science, and I'm keeping the Java implementation in my mind as a model. Included in the Java language is the notion of sets. Now I understand that this is different from ...
37
votes
6answers
3k views

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 Zermelo–Fraenkel set theory on first order logic, does that mean first order logic is not ...
15
votes
5answers
3k views

How can we know we're not accidentally talking about non-standard integers?

This question is mostly from pure curiosity. We know that any formal system cannot completely pin down the natural numbers. So regardless of whether we're reasoning in PA or ZFC or something else, ...
5
votes
5answers
733 views

Set theoretic concepts in first order logic

I have been reading introductory texts on first order logic (for example, Leary&Kristiansen). All of them used concepts that I have heard in set theory courses - ordered pairs, functions, ...
8
votes
3answers
1k views

Consistency of ZFC and proof by contradiction

I will start off by saying that I am an elementary student of mathematics and do not possess the deep and rigorous knowledge of most members of this site. Nonetheless, whilst learning how to do a ...
6
votes
4answers
754 views

Decidability and “truth value”

One can read in the Wikipedia page for "Gödel's incompleteness theorems": Undecidability of a statement in a particular deductive system does not, in and of itself, address the question of ...
18
votes
1answer
898 views

Computability viewpoint of Godel/Rosser's incompleteness theorem

How would the Godel/Rosser incompleteness theorems look like from a computability viewpoint? Often people present the incompleteness theorems as concerning arithmetic, but some people such as Scott ...
10
votes
2answers
1k views

Why do we find Gödel's Incompleteness Theorem surprising?

Gödel's First and Second Incompleteness Theorems are well-known, and usually taught by most colleges in undergrad logic courses. In my logic course I'm taking, we went over the proof of Gödel's ...
2
votes
4answers
2k views

How could a statement be true without proof?

Godel`s incompleteness theorem states that there may exist true statements which have no proofs in a formal system of particular axioms. Here I have two questions; 1) How can we say that a statement ...
2
votes
2answers
461 views

About the existence of the diagonal set of Cantor

The classic proof of the Cantor set start with the assumption that the set $$B=\{x\in A:x\notin f(x)\}$$ exists, where $f: A\to\mathcal P(A)$ is a bijective function. I understand the proof but I ...

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