Zfc Definition and 34 Threads

It is well known that a vector space always admits an algebraic (Hamel) basis. This is a theorem that follows from Zorn's lemma based on the Axiom of Choice (AC). Now consider any specific instance of vector space. Since the AC axiom may or may not be included in the underlying set theory...

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...

¿Is there a minimal standard methamatematical model for Zermelo set theory in the sense that the other models contains this model ?

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...

Which one do you prefer? Which do you think is better?

Can someone point me to a first-order axiomatization of ZFC? As I've mostly seen ZFC expressed in higher-order logics.

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. [FONT="Arial Black"]Are any...

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...

what is the proof-theoretic strength (largest ordinal whose existence can be proved) for ZFC set theory?

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[/color] The Skolem pardox shows ZFC is inconsistent Undecidability of ZFC is based on the assumption that it is consistent therefore the presence of the...

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.