1
vote
1answer
61 views

set theory, Incompleteness and axiomatic systems

Is the number of theorems that can be proved (decidable) within a certain set of axioms (for instance ZFC) is finite or infinite ? in other words, are we going to fully exhaust that set of axioms ...
0
votes
0answers
32 views

Quotient respect to an equivalence relation.

I have the natural numbers $\mathbb{N}$, in ZF. I want to construct the integers $\mathbb{Z}$ taking the quotient respect to usual the equivalence relation $R$ on $\mathbb{N}\times\mathbb{N}$ that is ...
0
votes
1answer
57 views

axiom of regularity and empty set

So we have axiom of regularity which says that any set (let's call it $A$) has a subset $B$ such that $A\cap B = \emptyset$. But thinking about such sets as $C = \{1, 2\}$ and $D = \{3, 4\}$ we still ...
1
vote
1answer
43 views

Axiom of extension

I am learning Set Theory from the book Naive Set Theory by Halmos as part of my course. The first chapter is on the Axiom of Extension. I understand what it is but what I don't understand is why it ...
1
vote
1answer
29 views

Ill-Founded Sets in ZFC

Let $\mathbb{N}$ be the set of all finite ordinals defined as the intersection of all ordinals including the empty set and closed under successor. Consider the following set: $S_0 = \mathbb{N}$ ...
3
votes
3answers
323 views

Axiom of Pairing

Axiom of Pairing states that if $a,b$ are sets, $\exists$ a set $A$ such that $A=\{{a,b\}}$. My question is that why we can't use Axiom of specification to define $A=\{x|x=a \vee x=b\}$?
2
votes
0answers
63 views

Simple Proofs in ZFC Set Theory

So I'll keep this real short and simple. In this document on page 23 there is a list of axioms. On page 24 there is a list of theorems that come from said axioms. I can prove them all except 3 and 5. ...
2
votes
2answers
66 views

Axiom of unrestricted comprehension

I'm doing some research on naive set theory and was a little confused over the statement of the axiom of unrestricted comprehension, $\exists$B$\forall$x(x$\in$B$\iff$$\phi$(x)). I am curious as to ...
1
vote
3answers
102 views

Question regarding axiom of unions

By axiom of union for any set A there is a set B such that x belongs to B if and only if x belongs to some z which belongs to A. According to this everything is a set.My question is what would union ...
2
votes
4answers
77 views

Are different constructions of an algebraic structure always isomorphic?

Any two complete ordered fields are isomorphic (as proved, e.g., in Spivak's Calculus; see also this question). While I understand this proof, I cannot yet appreciate why it is necessary. Given any ...
0
votes
0answers
23 views

How to translate these defitions of bipartiteness to each other?

Let $(V,E)$ be a bipartite graph. I'm trying to capture this property and I've come up with two definitions and am surprised/confused that one uses a negation and the other doesn't - still I can't ...
4
votes
1answer
124 views

Can one define $\langle x,y\rangle$ in $P(C)$?

I study at course Foundations of Mathematics the below definitions and lemma: $\langle x,y\rangle:=\{\{x\},\{x,y\}\}$ (from Kuratowski 1921) $\langle x,y\rangle:=\{\{\{x\},\varnothing\}\{\{y\}\}\}$ ...
1
vote
2answers
116 views

Axiom of Extensionality - Why not called equality?

Axiom of Extensionality, I understand that if two sets have exactly the same members they are equal. However why it is called Extensionality? Why not equality?
5
votes
6answers
258 views

What is the right interpretation of the axiom of extensionality

A set $a$ can be called extensional if it has the following propery: $$\forall b\left[\forall x\left[x\in b\iff x\in a\right]\Rightarrow a=b\right]$$ Based on this the axiom of extensionality can be ...
1
vote
1answer
52 views

Proving via axioms, that for given set $A$, $P(P(A))$ exists

The question itself: For a given set A, prove P(P(A)) exists. You may only use the axiom of pairing, axiom of union and axiom of empty set. This is how I solved it: Let A be the given set. ...
1
vote
1answer
38 views

Axiom of regularity and ordinal ranks

I am trying to prove that the following two statements are equivalent: Axiom of regularity $\forall x \exists \alpha (\alpha $ is an ordinal and $ x \in V_\alpha)$ I believe I understand how to ...
2
votes
2answers
74 views

ZF Set Theory Axiom of Infinity

Could someone please state and explain the axiom of infinity in ZF set theory? This isn't homework, it's just something that has interested me for awhile.
1
vote
1answer
73 views

Doubts on the axiom of union?

I'm reading Guerino Mazzolla's Comprehensive Mathematics for Computer Scientists 1: Axiom 3 (Axiom of Union) If $a$ is a set, then there is a set $$\{x| \text{ there exists an element } b \in ...
1
vote
2answers
52 views

Showing a class is a set.

The problem is to determine that a given class is or is not a set only using the basic axioms of modern set theory and some examples of classes that are or are not sets. The class in question is ...
3
votes
2answers
284 views

A set is not an element of itself.

I know that in modern set theory that for a given set $A$, $A \notin A$, specifically by the axiom of regularity. However, I'm not permitted to use this axiom in my proof. What I am permitted to use ...
4
votes
2answers
84 views

Is it necessary to use the axiom of Regularity to prove the successor function being injective?

Basically the problem is that given an inductive set $X$ we can define the successor function on $X$ such that $S:X\longrightarrow X$ and for all $x\in X$, $S(x)=x\cup \{x\}$. So, one of Peano axioms ...
2
votes
1answer
89 views

weaker version of some axioms

I'm complementing my knowledge of set theory using other sources in addition to the book that normally I use. Working with the Jech's book (great book, by the way), in one exercise of the first ...
1
vote
1answer
92 views

Apparent equivalent notations for the axiom of infinity

I'm just begining to build the systems of numbers based on the axioms of set theory ($\mathsf{ZF}$). Accordingly the axiom of infinity is no more than assuming the existence of $\mathbb{N}$ (of course ...
7
votes
3answers
227 views

Axiom of extensionality in ZF - pointless?

In every textbook on ZF set theory I come across the Axiom of extensionality, which basically says that if two sets have the same elements, then those two sets are equal: $$\forall x \forall y ...
0
votes
2answers
133 views

relationship between Zorn's lemma and Axiom of Completeness

For me , they look like they are 'similar' to each other , just that one is used in set and another one is used in numbers. Can anyone tell me is there any relationship between Zorn's Lemma and Axiom ...
9
votes
3answers
338 views

How does the axiom of regularity forbid self containing sets?

The axiom of regularity basically says that a set must be disjoint from at least one element. I have heard this disproves self containing sets. I see how it could prevent $A=\{A\}$, but it would seem ...
1
vote
1answer
58 views

In class universe, on what basis can a conclusion that every element of a set is set be made?

Is every element in a set set? In a set model it is obvious true. However in the class universe it is another story since it need to be shown not a proper class. Let $A$ be a set, $a \in A$. ...
1
vote
2answers
201 views

The real numbers and the axiom of foundation

I am having a bit of confusion about the real numbers and ZF set theory (I asked a question about it a few days ago). I am a bit unsure as to why the real numbers can be in any model of ZF as they ...
1
vote
3answers
45 views

Is this definition true about comparisons of sets?

I'm trying to find the error in a proof that yields a contradictory result, and I'm beginning to think that one of the definitions I start with is incorrect or self-contradictory. Is the following ...
0
votes
2answers
504 views

Axioms of set theory and logic

Zermelo–Fraenkel set theory is the most common foundation of mathematics with eight axioms and axiom of choice (ZFC): http://plato.stanford.edu/entries/set-theory/ZF.html But one can see that the ...
-3
votes
4answers
250 views

What are some primary mathematical utilities of the axiom schema of separation?

I read a discussion concerning the axiom schema of specification, which I yet take as saying that for every set and a class-defining condition, those elements of the set satisfying this condition ...
12
votes
6answers
657 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 ...
2
votes
2answers
522 views

what is the relationship between ZFC and first-order logic?

In Wikipedia, it says that ZFC is a one-sorted theory in first-order logic. However, I was not really able to comprehend the later parts that seem to elaborate on that point. Can anyone explain the ...
4
votes
1answer
186 views

How is the Cantor's paradox resolved in the ZFC system?

How is the Cantor's paradox resolved in the ZFC system? Thanks.
4
votes
1answer
464 views

Need help understanding Axiom of Extensionality

I'm attempting to learn Set Theory and I'm currently working through Halmos' Naive Set Theory. I will say that I completely understand the essence of the Axiom of Extensionality. However, where I'm ...
5
votes
3answers
623 views

How do the separation axioms follow from the replacement axioms?

It has come to my attention that the pairing axiom and the separation axiom schema are rarely listed since they follows from the replacement axioms. I see how this works for the pairing axiom, since ...
3
votes
1answer
277 views

Using the Subset Axioms to prove the existence of a set

I'm going through Enderton's Elements of Set Theory, as I heard it is a gentle introduction to set theory. I'm a little confused on how the subset axioms are used. In the text, the axiom is given as: ...