Linked Questions

7 votes
3 answers
1k views

Is the Axiom of Empty Set a canonical ZFC axiom? [duplicate]

As in title. The Axiom of Empty set is sometimes offered as a fundamental axiom in the ZFC framework (example), but sometimes it is not (example). Would you consider it a tenth canonical ZFC axiom? ...
JohnDoeVsJoeSchmoe's user avatar
66 votes
13 answers
15k views

What is exactly the difference between a definition and an axiom?

I am wondering what the difference between a definition and an axiom. Isn't an axiom something what we define to be true? For example, one of the axioms of Peano Arithmetic states that $\forall n:0\...
wythagoras's user avatar
47 votes
11 answers
10k views

Why are integers subset of reals?

In most programming languages, integer and real (or float, rational, whatever) types are usually disjoint; 2 is not the same as 2.0 (although most languages do an automatic conversion when necessary). ...
Aivar's user avatar
  • 597
11 votes
3 answers
3k views

Why is the Axiom of Infinity necessary?

I am having trouble seeing why the Axiom of Infinity is necessary to construct an infinite set. According to a professor of who's mine teaching a class on "infinity," the Peano axioms are only ...
ata's user avatar
  • 165
5 votes
3 answers
1k views

Axiom of infinity and empty set

The axiom of infinity is formulated as $$\exists S ( \varnothing \in S \wedge (\forall x \in S) x \cup \{x\} \in S)$$ Can someone explain why the use of $\varnothing$ in the axiom of infinity makes ...
user193756's user avatar
7 votes
1 answer
952 views

What makes Tarski Grothendieck set theory non-empty?

I'm fighting with Grothendieck set theory for some time now. This is the framework for the automated proof checking system of Mizar and hence there is a formalized version of the axioms here too, and ...
Nikolaj-K's user avatar
  • 12.3k
2 votes
2 answers
860 views

Why does $\exists x\,\ x = x$?

The Wikipedia article on ZFC insists that the empty set exists since it suffices for any set to exist, since the Axiom of Specification for which we always specify "false" will construct the empty set....
VF1's user avatar
  • 1,993
0 votes
1 answer
509 views

Why is the smallest example of an admissible set hereditarily finite set [closed]

As the title says, why is the smallest example of an admissible set hereditarily finite set?
hwe's user avatar
  • 345
0 votes
1 answer
112 views

Why does Munkres call the empty set "a convention"?

On page 6 of Munkres' book Topology (Second Edition) he says "the empty set is only a convention" in a few different spots. I am wondering why he says this... My understanding is that the ...
Satana's user avatar
  • 1,097
0 votes
1 answer
66 views

What is the full axiom of choice (with "$\in$" as the only atomic formula)?

I've seen the axiom of choice as $\forall X\left[\varnothing \notin X\implies \exists f\colon X\rightarrow \bigcup_{A\in X}A,~~~ \forall A\in X\,(f(A)\in A)\right]$. But isn't there a version which ...
Nathan Kaufmann's user avatar
0 votes
0 answers
103 views

How to prove the existence of the empty set from other axioms in ZFC?

In another post, the top answer says: If there are sets at all, the axiom of subsets tells us that there is an empty set: If $x$ is a set, then $\{ y∈x ∣ y≠y \}$ is a set, and is empty, since there ...
YV1999's user avatar
  • 11