4
votes
1answer
52 views

Equivalence of Axiom of Regularity

So Axiom of regularity states: every non-empty set A contains an element that is disjoint from A I'm wondering if this is equivalent as any set is not a member of itself? If so, how do we prove it? ...
1
vote
1answer
66 views

How to extract the sets “produced” by a axiomatic set theory, into some newly introduced collection?

Say I know well how to reason in a set theory, which for the sake of this question I'll call $\bf{ST}$, say one of those, it can by $\bf ZFC$. It's principally written down via first order logic and ...
2
votes
1answer
71 views

What is the weakest notion of “set” that we need, so that we can say the Yoneda lemma implies something about sets?

We set up a set theory axiomatically by fixing certain statements about "$\in$". There are many different set theories. The Yoneda lemma (the theorem about functors to Set) is a early but central ...
1
vote
1answer
65 views

Axiom of Infinity, existence of empty set circular?

I am reading Jech's "Set Theory". First, he states that the existence of the empty set follows from the axiom of infinity. The empty set is defined, using the separation schema as $$\emptyset = \{ u ...
4
votes
2answers
158 views

What is the relation between axiomatic set theory and logical quantifiers?

On the one hand, the logical predicates $\forall$ and $\exists$ are defined using the concept of a Domain of Discourse, which itself is defined as a set (at least according to wikipedia). On the ...
3
votes
1answer
50 views

When do surjections split in ZF? Two surjections imply bijection?

We have that the Axiom of Choice is equivalent to the principle that every surjection has a right inverse. However, without the Axiom of Choice we can determine for some $X$ that $X\succeq Y\implies ...
0
votes
2answers
81 views

How do the ZFC axioms produce the ideas of order?

The notion of order (and cardinality, for that matter) seems so basic to me that I can't imagine how it could be derived from anything. In an answer to a previous question I learned that all the ...
5
votes
3answers
136 views

Axiom schema of specification (formula arguments)

Some sources define the formula like this: $$ \forall w_1,\ldots,w_n \, \forall A \, \exists B \, \forall x \, ( x \in B \Leftrightarrow [ x \in A \wedge \varphi(x, w_1, \ldots, w_n , A) ] ) $$ Why ...
1
vote
1answer
58 views

Multiple context available for the AC?

I am surprised at the language used in connection with the axiom of choice. From the answer to a question a made (which turned out to be duplicate) about involvement of AC in Wiles’ proof of Fermat’s ...
3
votes
2answers
107 views

axioms of equality

Every text I've come across uses the Axiom of Extensionality to describe the fact that sets are equal iff they contain each other's elements. $$(\forall x)(\forall y)\bigl((\forall a)(a\in x \iff a\in ...
13
votes
1answer
383 views

Can we prove the existence of $A\cup B$ without the union axiom?

If $A$ and $B$ are sets, then $A\cup B$ is defined as follows: $$A\cup B:=\{x: x\in A \, \, \,\text{ or } \, \, \, x\in B\}.$$ In ZFC, $A\cup B$ exists because $\bigcup\{A,B\}$ exists by union axiom, ...
3
votes
1answer
97 views

Deriving the Axiom of Infinity

I'm currently reading about recursion, working on various exercises since I haven't formally studied the material before. The main book that I'm following is Kunen's Set Theory book There's an ...
3
votes
2answers
111 views

What is meant by one axiom being “weaker” than another?

If we have two axioms $A$ and $B$, what exactly is meant by axiom $A$ being weaker than axiom $B$? This question is a follow-up to A weaker Axiom of Infinity?
4
votes
1answer
158 views

A weaker Axiom of Infinity?

As I understand, the Axiom of Infinity in ZFC theory gives us an infinite set from which the natural numbers can be extracted. (See "Axiom of Infinity" at ...
0
votes
4answers
83 views

Replacing Axiom of Extensionality with a logical formalism

Is it possible to replace the Axiom of Extensionality with a formalism from logic, namely the following one: $\forall a \forall b (a=b\Leftrightarrow \forall P (P (a)\Leftrightarrow P (b)))$ ($ P $ is ...
2
votes
3answers
119 views

If we abandon the axiom of regularity, can the cumulative hierarchy just become a definition?

In ZFC, the axiom of regularity is used to prove that every set is an element of some stage of the cumulative hierarchy. The index of the least such stage is, by definition, the rank of that set. Now ...
2
votes
1answer
62 views

Can I express (only) the syntactical formulation of a set theory within another.

Motivated by this question on the SE philosophy board "Is there a one true set theory", I want to ask for a clarification: I've seen e.g. ZF being expressed in Grothendieck set theory. If I say ...
6
votes
0answers
136 views

What does category theory say about chains of set inclusions $\dots\in c\in b\in a$?

I'm currently trying to understand to what extend a set theory like ZFC can be mirrored within category theory, i.e. topos theory. What appears as an obstacle to me is the axiom of regularity, which ...
11
votes
2answers
470 views

Which axioms of ZFC or PA are known to not be derivable from the others?

Which, if any, axioms of ZFC are known to not be derivable from the other axioms? Which, if any, axioms of PA are known to not be derivable from the other axioms?
6
votes
1answer
127 views

An Exercise in Kunen (A Model for Foundation, Pairing,…)

This is exercise I.4.18 in Kunen's Set Theory. Derive $\forall y (y \notin y)$ from the Axioms of Comprehension and Foundation. Don't use the Pairing or Extensionality Axioms. Then find a 2 element ...
2
votes
1answer
198 views

What's the differences between naive and axiomatic set theory?

I'm at a loss studying math.Recently I decided to begin with set theory as it seems the most fundamental for math.I found the book Naive Set Theory by Halmos,and began to read it because it's so thin ...
4
votes
1answer
143 views

Understanding the Definition of the Axiom Schema of Specification

Consider the Axiom Schema of Separation: If $P$ is a property (with paramter $p$), then for any $X$ and $p$ there exists a set $Y = \{u \in X : P(u,p)\}$ that contains all those $u \in X$ that ...
2
votes
2answers
54 views

Question Regarding the Replacement Schema

For each formula $\phi(x,y,p)$, we have the following axiom: $\forall x \forall y \forall z (\phi(x,y,p) \wedge \phi(x,z,p) \rightarrow y = z) \rightarrow \forall X \exists Y \forall y (y \in Y ...
12
votes
2answers
137 views

How far is it true that statements dependent on Axiom of Choice are not constructive.

Axiom of Choice is often used in mathematics to construct various objects, such as basis of $\mathbb{R}$ as a vector space over $\mathbb{Q}$, unmeasurable subset of $\mathbb{R}$, or a non-principal ...
3
votes
1answer
82 views

Success of Hilbert's Axioms

We know Euclid's axioms were found to be having many loopholes as in there were still many assumptions which weren't being stated in his system of axioms . Are Hilbert's axioms today completely ...
3
votes
1answer
62 views

ZF: Regularity axiom or axiom schema?

I have seen the axiom system ZF for set theory described including a single axiom of regularity (aka "foundation"), namely $$\forall x\neq\emptyset \, \exists y\in x \ y\cap x = \emptyset$$ and also ...
5
votes
3answers
153 views

Gödel's incompleteness wrt weakend versions of ZFC

Suppose for the sake of argument that we look at ZFC with the axiom of infinity removed. http://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems#First_incompleteness_theorem ...
4
votes
1answer
213 views

What set theory axioms do I need to believe in uncountable ordinals?

Math people: The title is the question. I am convinced that uncountable sets exist, thanks to Cantor's diagonal proof. It is not intuitively clear to me that uncountable ordinals or cardinals should ...
3
votes
1answer
118 views

Axiom of Regularity - Transitive set

I just managed to confuse myself completely while studying for Set Theory. We have the Axiom of regularity: $$\forall S (S\not= \emptyset \rightarrow (\exists x\in S)(S\cap x=\emptyset))$$ Now a set ...
6
votes
3answers
144 views

Question Regarding the Axiom of Extensionality

Jech's text on Set Theory states the following: If X and Y have the same elements, then X = Y : ∀u(u ∈ X ↔ u ∈ Y ) → X = Y. The converse, namely, if X = Y then u ∈ X ↔ u ∈ Y, is an axiom ...
8
votes
1answer
145 views

What is the smallest fragment of ZFC that has the same consistency strength as ZFC?

The question in the title is undoubtedly nonsensical, but I am not sure how to state this question properly. Perhaps some examples will help me explain it. Thanks to Godel and Cohen, we know that ...
1
vote
2answers
65 views

¿Can you help me with axiom of regularity? [duplicate]

I understand that if we define a set such that $A=\{A\}$ we have then a contradiction because of regularity axiom, thus, a set can not be a member of itself. My question is; how can it be so, if I ...
4
votes
2answers
95 views

ZF Extensionality axiom

To familiarize myself with axiomatic set theory, I am reading Kenneth Kunen's The Foundations of Mathematics that presents ZF set theory. I haven't gotten really far since I am stuck at the axiom of ...
3
votes
2answers
113 views

Axiom of infinity exceptions?

It is my understanding that the axiom of infinity, or axioms that depend on the axiom of infinity, are the only axioms that are not representable in finite First-Order Predicate Logic. Is this ...
3
votes
2answers
41 views

Schema of separation and set of all sets

The schema of separation states (this is probably simplified) states that if $P$ is a property and $X$ is a set, then there exists a set $Y = \{ x \in X : P(x)\}$. The notes I'm reading say that from ...
4
votes
3answers
377 views

A basic doubt on axiom of foundation of Zermelo-Fraenkel set theory

I have read in a book that the "axiom of foundation prevents anomalies such as a set being an element of itself". Now, axiom of foundation says that there exist an element in every set which is ...
1
vote
0answers
66 views

Is a “model” only a proper model if everything in it's definition is also explicitly constructed?

Say you have some collection of axioms and you find a set $X$ fulfilling them. And the definition of the set $X$ involves another concept, e.g. the real number, but only in a way which refers to the ...
2
votes
1answer
97 views

axiom of foundation of Zermelo–Fraenkel set theory

I have found two different statements on axiom of foundation of Zermelo–Fraenkel set theory in two different books as: 1) every nonempty set contains an element that is not an element of any other ...
3
votes
0answers
46 views

Have people studied weakened forms of Tarski's axiom?

My intuition about Grothendieck universes says the following. Axiom of empty set $\leftrightarrow$ There exists a Grothendieck universe. Axiom of infinity $\leftrightarrow$ There exist at least two ...
5
votes
3answers
180 views

The axiom of infinity for Zermelo–Fraenkel set theory

The axiom of infinity for Zermelo–Fraenkel set theory is stated as follows in the wikipedia page: Let $S(w)$ abbreviate $ w \cup \{w\} $, where $ w $ is some set (We can see that $\{w\}$ is a ...
2
votes
2answers
126 views

Formulation of the Separation Axiom in Set Theory

I have a question regarding this formulation of the Separation Axiom: I quote from the Book "Handbook of Mathematical Logic", the chapter from which this is taken could also be found here. Now we ...
4
votes
1answer
375 views

Is ZFC without Axiom of Infinity consistent?

The incompleteness theorem states that one cannot prove whether ZF or ZFC is consistent, but what about ZFC withouth Axiom of infinity? (Assuming the empty set exists) Furthermore, let $M$ be a ...
17
votes
5answers
557 views

Why hasn't GCH become a standard axiom of ZFC?

I've never seen a text that includes GCH in the ZFC axioms. I presume this means that GCH has not achieved widespread acceptance. This seems surprising to me, given that: The cardinal numbers ...
2
votes
2answers
115 views

Would it be possible to concoct a “harmful” axiom?

Suppose I run an automated theorem prover. It begins with the axioms of ZFC, and using a random number generator, it proves more theorems, and it runs for two days. At the end of the second day, it ...
1
vote
1answer
108 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 ...
5
votes
2answers
325 views

How does (ZFC-Infinity+“There is no infinite set”) compare with PA?

How does (ZFC-Infinity+"There is no infinite set") compare with (first order) PA? Intuitively, neither theory should be more powerful than the other.
3
votes
2answers
128 views

Why does any transitive model satisfy extensionality?

I see it stated as something very clear, but i can't figure it out. i found a proof in Jech(old version) which goes through the concept of restricted formulas, which i don't quite understand. (i'm not ...
3
votes
1answer
425 views

Explain Zermelo–Fraenkel set theory in layman terms

What does Zermelo–Fraenkel set theory mean? According to Wikipedia, Zermelo–Fraenkel set theory is a set theory that is proposed to overcome issues in naive set theory. I really appreciate if ...
2
votes
2answers
132 views

Forcing Question about a sequence of functions from $\aleph_{0}$ into $2 = \{0, 1\}$

I'm trying to go through Timothy Chow's A Beginner's Guide to Forcing (arxiv pdf). My particular question is about the first paragraph of Section $5$ (on p. $7$ of the above pdf). First, my vague ...
6
votes
3answers
497 views

The existence of the empty set is an axiom of ZFC or not?

I found in the Wolfram MathWorld page of the Axiom of the Empty Set that this is one of the Zermelo-Fraenkel Axioms, however on the page about these ZFC Axioms I read that it is an axiom that can be ...