In set theory, Zermelo–Fraenkel set theory, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is an axiomatic system that was proposed in the early twentieth century in order to formulate a theory of sets free of paradoxes such as Russell's paradox. Today, Zermelo–Fraenkel set theory, with the historically controversial axiom of choice (AC) included, is the standard form of axiomatic set theory and as such is the most common foundation of mathematics. Zermelo–Fraenkel set theory with the axiom of choice included is abbreviated ZFC, where C stands for "choice", and ZF refers to the axioms of Zermelo–Fraenkel set theory with the axiom of choice excluded.
Zermelo–Fraenkel set theory is intended to formalize a single primitive notion, that of a hereditary well-founded set, so that all entities in the universe of discourse are such sets. Thus the axioms of Zermelo–Fraenkel set theory refer only to pure sets and prevent its models from containing urelements (elements of sets that are not themselves sets). Furthermore, proper classes (collections of mathematical objects defined by a property shared by their members where the collections are too big to be sets) can only be treated indirectly. Specifically, Zermelo–Fraenkel set theory does not allow for the existence of a universal set (a set containing all sets) nor for unrestricted comprehension, thereby avoiding Russell's paradox. Von Neumann–Bernays–Gödel set theory (NBG) is a commonly used conservative extension of Zermelo–Fraenkel set theory that does allow explicit treatment of proper classes.
There are many equivalent formulations of the axioms of Zermelo–Fraenkel set theory. Most of the axioms state the existence of particular sets defined from other sets. For example, the axiom of pairing says that given any two sets
a
{\displaystyle a}
and
b
{\displaystyle b}
there is a new set
{
a
,
b
}
{\displaystyle \{a,b\}}
containing exactly
a
{\displaystyle a}
and
b
{\displaystyle b}
. Other axioms describe properties of set membership. A goal of the axioms is that each axiom should be true if interpreted as a statement about the collection of all sets in the von Neumann universe (also known as the cumulative hierarchy). Formally, ZFC is a one-sorted theory in first-order logic. The signature has equality and a single primitive binary relation, set membership, which is usually denoted
∈
{\displaystyle \in }
. The formula
a
∈
b
{\displaystyle a\in b}
means that the set
a
{\displaystyle a}
is a member of the set
b
{\displaystyle b}
(which is also read, "
a
{\displaystyle a}
is an element of
b
{\displaystyle b}
" or "
a
{\displaystyle a}
is in
b
{\displaystyle b}
").
The metamathematics of Zermelo–Fraenkel set theory has been extensively studied. Landmark results in this area established the logical independence of the axiom of choice from the remaining Zermelo-Fraenkel axioms (see Axiom of choice § Independence) and of the continuum hypothesis from ZFC. The consistency of a theory such as ZFC cannot be proved within the theory itself, as shown by Gödel's second incompleteness theorem.
Let ##(M_i)_{i\in I}## be a multiverse of models of ZFC. By that I mean:
Each ##M_i## is a well-founded model of ZFC.
##(I,\leq_I)## is a partially ordered set, and whenever ##i\leq_I j##, there is an embedding ##\tau^j_i:M_i\rightarrow M_j## such that the image of ##M_i## is a transitive...
I am reading Micheal Searcoid's book: Elements of Abstract Analysis ( Springer Undergraduate Mathematics Series) ...
I am currently focused on Searcoid's treatment of ZFC in Chapter 1: Sets ...
I need help in order to fully understand the Axiom of Power Sets and Definition 1.1.1 ...
The...
The ZFC axioms are statements combining "atomic formulas" such as "p ∈ A" and "A = B", using AND, OR, imply, NOT, for all and exists.
But (it seems to me, at least) there is the implicit assumption that the "atomic formulas", "p ∈ A" and "A = B", are considered to be propositions, i.e. they are...
Homework Statement: See attached image.
Homework Equations: ZFC set theory.
Consider the text in the attached image. What is meant with "We require of an axiom system that it be possible to decide whether or not any given formula is an axiom."? Is consistency synonymous with soundness? Is...
I am reading Micheal Searcoid's book: Elements of Abstract Nalysis ( Springer Undergraduate Mathematics Series) ...
I am currently focussed on Searcoid's treatment of ZFC in Chapter 1: Sets ...
I am trying to attain a full understanding of Searcoid's proof of the Pairing Principle ...
The...
So apparently the proof involves a trick that converts the problem of a general power set ##\mathscr{P}(M)## of some set ##M## which has of course the property of not having pairwise disjoint set-elements to a problem that involves disjoint set-elements. I do not understand why this trick is...
I am reading D. J. H. Garling's book: "A Course in Mathematical Analysis: Volume 1: Foundations and Elementary Real Analysis" ... ...
I am at present focused on Part 1: Prologue: The Foundations of Analysis ... Chapter 1: The Axioms of Set Theory ...
I need help with an aspect of the proof of...
I am reading D. J. H. Garling's book: "A Course in Mathematical Analysis: Volume 1: Foundations and Elementary Real Analysis" ... ...
I am at present focused on Part 1: Prologue: The Foundations of Analysis ... Chapter 1: The Axioms of Set Theory ...
I need help with an aspect of the proof of...
I am reading Micheal Searcoid's book: Elements of Abstract Analysis ( Springer Undergraduate Mathematics Series) ...
I am currently focussed on Searcoid's treatment of ZFC in Chapter 1: Sets ...
I am struggling to attain a full understanding of the Axiom of Foundation which reads as shown...
I am reading Micheal Searcoid's book: Elements of Abstract Analysis ( Springer Undergraduate Mathematics Series) ...
I am currently focussed on Searcoid's treatment of ZFC in Chapter 1: Sets ...
I am struggling to attain a full understanding of the Axiom of Infinity which reads as shown...
I am reading Micheal Searcoid's book: Elements of Abstract Analysis ( Springer Undergraduate Mathematics Series) ...
I am currently focussed on Searcoid's treatment of ZFC in Chapter 1: Sets ...
I am struggling to attain a full understanding of the Axiom of Replacement which reads as shown...
I am reading Micheal Searcoid's book: Elements of Abstract Nalysis ( Springer Undergraduate Mathematics Series) ...
I am currently focussed on Searcoid's treatment of ZFC in Chapter 1: Sets ...
I am struggling to attain a full understanding of the Subset Principle which reads as shown below...
I am looking for a book that starts at the standard ZFC axioms and progresses to the point where some recognisable non-trivial mathematical statement is proved. By recognisable I mean something that you may encounter in school/early university level and is not purely set-theoretical (e.g...
I would appreciate any and all feedback regarding this document currently housed in Google docs. Basically, I generalize induction among natural numbers to an extreme in an environment regarding what I call grammatical systems. Then an induction principle is derived from that which holds in...
Other than ZFC, what other axiomatic systems have been shown to be useful to the real world?
I'm a layman that is reading through the Wikipedia articles and it seems like the axioms are becoming more Philosophy than Math. We can use a different axiomatic system other than ZFC, but are there...
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
http://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems#Second_incompleteness_theorem
We would...
Must all models of ZFC (in a standard formulation) be at least countable?
Why I think this: there are countably many instances of Replacement, and so, if a model is to satisfy Replacement, it must have at least countably many satisfactions of it.
Does my question only apply to first-order...
Is it true that for every standard formulation T of ZFC, T ⊢ the power set of {naturals}?
After all, the empty set axiom and the pairing axiom are in T, and so we get N. Then by the power set axiom we get P(N).
Three facts:
(1)The constructible universe L is the minimal model for ZFC;
(2) L is a model of "there exists an inaccessible cardinal \kappa"; and
(3) if V=L, an inaccessible cardinal \kappa with the membership relation \epsilon is a model of ZFC.
So, what is confusing me is: if...
Does the axiom of choice make the class of all sets bigger or smaller? Does it perhaps bring new sets into the universe and kick others out of it at the same time? (Maybe in a way that ensures that the first question doesn't make sense?)
The AC gives us permission to construct certain sets...
I recently learned predicate calculus from Schaum's Outline of Logic.
in this sort of form:
In addition to refutation trees, however; pfff, refutation trees.
I'm reading "Introduction to Set Theory," Hrbacek, Jech. I'm a little "annoyed" by the informal proofs. Are any books that teach...
Just to show how dumb I am, I have to ask, does this implication follow from the ZFC axioms?
x=y\ \land\ y\in z\rightarrow x\in z
It's obviously true if we use the meaning of the = symbol, i.e. that the formula "x=y" means that x and y are different symbols for the same set, but I thought...
I've been following the first few chapters of Yuri Manin's "A Course in Mathematical Logic for Mathematicians," and as an undergraduate who has only had basic logic and naïve set theory, the way he explained a few of the topics rubbed me the wrong way - specifically, the definitions of the...
It's often said that the axioms of ZFC set theory are the foundation of mathematics, but the same people who say that also use the term "class" a lot. For example, "the class of all ordinals", is apparently too large to qualify as a set. What's bugging me right now is that I read that there's no...
I'm thinking about buying . Both books are getting excellent reviews at Amazon, especially Goldrei.
I would like to learn about the ZFC axioms, cardinals and ordinals, etc. I assume both books will cover those topics. I'm also very curious about something I heard for the first time today...
If you assume that ZFC is consistent, then by the main theorem of model theory ZFC has a model, let the model be countable.
Since ZFC proves: "there is a set consisting of all real numbers" there is a point a belonging to M such that:
M satisfies " a is the set of all real numbers"
But since...
since a lot of talking is going on with sets, will somebody write down the axioms in ZFC theory as a point of reference , when a discussion is opened up.
thanx
The Australian philosopher colin leslie dean argues that
The Skolem paradox destroys the incompleteness of ZFC
Crackpot link removed
The Skolem pardox shows ZFC is inconsistent
Undecidability of ZFC is based on the assumption that it is consistent
therefore
the presence of the Skolem...
How does ZFC manage to block Russell's paradox? I've read through the axioms extensively, and it's not clear how to prove Russell's paradox is impossible.
In particular, I'm talking about Russell's paradox that shows {x| x not in x} is not a well-defined set.