4
votes
1answer
218 views

How can I prove that there is no set containing itself without using axiom of foundation?

I've already found some similar questions in here (and other sites), but in most of the case, the use of axiom of foundation is required to complete the proof. Is there any way to prove $\not\exists ...
3
votes
6answers
603 views

Question about the Continuum Hypothesis

The Continuum Hypothesis hypothesises There is no set whose cardinality is strictly between that of the integers and the real numbers. Clearly this is either true or false - there either exists ...
3
votes
1answer
63 views

Size of topological space depending on the size of local basis. (With elementary submodels)

Recall that the character of a topological space $\chi(X)$ is the minimum cardinal $\kappa$ such that every point in $X$ has a local basis of size $\kappa$. I need to prove that if $X$ is compact ...
1
vote
1answer
64 views

Can set-theoretic forcing exist without law of excluded middle?

Of course the law of excluded middle is accepted by almost every mathematician except a few constructivists, but then I was wondering if set-theoretic forcing can exist without law of excluded middle. ...
1
vote
2answers
117 views

Existence of an axiom question in relation to $\mathsf{Infinity}$

Original Post This may be a stupid question, but does there axist an axiom $\phi$ that is independent of $\mathsf{ZFC}$, and not equivalent to the axiom of $\mathsf{Infinity}$, such that ...
2
votes
1answer
85 views

Well-ordering the reals: finding a certain model of $\mathsf{ZFC}$

How would one go about constructing a model $\mathfrak{M}$ of $\mathsf{ZFC}$ such that under $\mathfrak{M}$, no formula defining a well-ordering of $\mathbb{R}$ exists? I am certain such models are ...
0
votes
0answers
43 views

The lexicographic order [duplicate]

If it is given ordinals $\alpha$ and $\beta$, the lexicographic order on $\alpha \times \beta$,$\leq_{lex}$ is given by: $(\gamma_0,\delta_0)<_lex(\gamma_1,\delta_1)$ if and only if either ...
2
votes
2answers
106 views

$\mathsf{ZF}$ is not finitely axiomatizable

As we know a first order theory $T$ is finitely axiomatizable if there is a finite set $F\subseteq T$ of axioms such that $F\vdash \sigma$ for every $\sigma \in T$. How we can prove if $\mathsf{ZF}$ ...
3
votes
3answers
97 views

How are models constructed?

As I understand, a model $\langle S,\sim\rangle$ of $\mathsf{ZFC}$ is a set $S$ coupled with some binary operation $\sim$, in which the axioms of $\mathsf{ZFC}$ hold (please correct me if I am wrong). ...
1
vote
1answer
67 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 ...
1
vote
0answers
35 views

How is it possible that the well-ordering theorem is strictly stronger than the axiom of choice in second-order logic? [duplicate]

If I am not wrong, the well-ordering theorem is strictly stronger than the axiom of choice in second-order logic. I am not sure to understand how this is possible. The reason is that second order ...
8
votes
6answers
347 views

Countable axiom of choice: why you can't prove it from just ZF

This is a follow-up question to the discussion about the finite axiom of choice here. Suppose we have a countable collection of non-empty sets $\{A_1, A_2, A_3,\cdots\}$ Reasoning as indicated in ...
4
votes
1answer
126 views

What is a non-constructible real?

I am not sure to fully understand the concept (I read many of the wikipedia definitions for many of these issues but I am still confused). For instance, is a surreal number an example of ...
-2
votes
2answers
106 views

Unbounded Finity?

Consider the successor of the largest finite ordinal that will ever be considered alone. But then it wasn't the largest finite ordinal that will ever be considered alone. How do we get around this ...
2
votes
0answers
37 views

Transfert principle of a conservative extension of ZFC

In the following paper, there is a theory called $^*ZFC$ in the language $(^*,\in)$. The *-map is (more or less) defined on the Von Neuman hierarchy $S$ and verifies the following axiom schemata true ...
5
votes
2answers
349 views

Is maths = set theory + logic? [closed]

It seems to me that most branches of mathematics are just an application of set theory and the rules of logic. You just definite particular types of sets and study their properties. For example, in ...
10
votes
2answers
167 views

Founding Arithmetic on geometry

In the past I found some fleeting references that some (Frege in his later years being one of them) tried to found arithmetic not on set-theory and logic but on geometry and logic. Unfortunedly Frege ...
2
votes
1answer
44 views

Are there ordinals other than the set of natural numbers which satisfy this property?

Let $\alpha$ be an ordinal. We say that $\alpha$ is good iff for every $\beta\in \alpha$, there exists $\gamma\in \alpha$ such that $|\scr{P}(\beta)|\leq |\gamma|$. Question: Is the set of natural ...
3
votes
1answer
88 views

A Question Regarding Forcing in Gödel's Constructible Universe in Infinitary Logics

In his answer to the MathOverflow question Gödel's Constructible Universe in Infinitary Logics, Prof. Hamkins gives a very interesting answer and proof to user46667's second question: (2) What is ...
3
votes
1answer
130 views

Are there classes with different sizes?

Are there classes with different sizes ? I will put a precise statement of my question below: Are there two well formed formulas $P,Q$ each with one free variable such that there is no well formed ...
6
votes
0answers
111 views

Skolem Hulls in $H_{\omega_2}$

Consider a model of the form $\mathfrak{A} = (H_{\omega_2}, \epsilon, \prec, f_0, f_1, ...)$, some expansion of $H_{\omega_2}$ in a countable language, with $\prec$ giving a well-order. Does there ...
4
votes
2answers
159 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 ...
0
votes
1answer
48 views

On Counted Languages

In my recent question on Godel Completeness I mentioned that there was a related question I wanted to ask, but would keep separate. I have been recently studying "non-well ordered sets" and Chapter 7 ...
7
votes
2answers
187 views

A truth definition, wrong, but where

Consider the theory ZFC of language with connectives $\neg$, $\rightarrow$, predicative symbol $\in$ and quantifier $\forall$. Assume that we are working in ZFC. Let $\mathcal{V}$ be the class of ...
0
votes
1answer
49 views

Consistency result vs. True in every model of Axiom X

Suppose a forcing extension of ZFC has been found which satisfies statement $A$. For example, say the extension is formed by Cohen or Laver forcing, so that the model satisfies $\neg$CH. At this ...
6
votes
1answer
178 views

Problem 24 from Chapter 1 of Kunen's Set Theory: An Introduction to Independence Proofs

Just want to make sure I'm tracking Kunen here, and hopefully the proof I have is correct. Comments / Suggestions welcome. Thanks! Problem 24. Let T be any consistent set of axioms extending ZF. ...
7
votes
5answers
250 views

Statement that is provable in $ZFC+CH$ yet unprovable in $ZFC+\lnot CH$

My understanding of logic is really basic, and I ask this question out of curiosity. Is there an explicit example of a statement whose proof uses the continuum hypothesis and is unprovable in $ZFC + ...
2
votes
2answers
117 views

Natural numbers in Set Theory

We seem to accept the fact $(\omega,+,\times,<,0,1)^{V}$, where $V:=x=x$ is the set theoretic universe, properly reflects what is intuitively understood to be the set of natural numbers, i.e. we ...
2
votes
2answers
53 views

How come that two inductive subsets can be different

In Enderton's "Mathematical Introduction To Logic". Author says that if we have two operations $f(x,y)$ and $g(x)$ and two sets $B$ and $U$ such that $B \subseteq U$. We say that $S \subseteq U$ is ...
7
votes
2answers
206 views

Number of Non-isomorphic models of Set Theory

Assume that the meta theory allows for model theoretic techniques and handling infinite sets etc (The meta theory itself is, informally, "strong as ZFC"). Also assume that I'm studying ZFC inside this ...
4
votes
1answer
113 views

Gödel Incompletness theorem

I am not familiar with model theory. As a matter of fact, I only had my first Logic and Set theory courses last semester. But still, there is a question that bothers me, and It could be nice if ...
5
votes
3answers
138 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 ...
2
votes
2answers
95 views

Does iterating any definable in ZFC functional relation give a $\mathbb N$-indexed family of sets?

Consider an extension of ZFC (the first-order theory with signature $\in$ and the usual axioms) by a constant $x$, and a binary relation $N$ that is functional, i.e., we have the additional axiom ...
1
vote
1answer
132 views

Questions on a sentence in ZFC asserting that ZFC has a model

Question 1. Let $\operatorname{con}(\mathsf{ZFC})$ be a sentence in $\mathsf{ZFC}$ asserting that $\mathsf{ZFC}$ has a model. Let $S$ be the theory $\mathsf{ZFC}+\operatorname{con}(\mathsf{ZFC})$. ...
1
vote
1answer
86 views

Embedding models of ZF into another model

I had some ideas regarding models of ZF. My ideas (phrased as questions) are: Given two models of ZF, what are the condition for a model containing both models (in the sense of embedding) to exist? ...
6
votes
1answer
108 views

Do second-order categoricity proofs require a background concept of set?

In his article "The Set-Theoretic Multiverse", Joel David Hamkins (as part of his reply to Donald Martin's argument that the set-theoretic universe is unique, found in "Multiple Universes of Sets and ...
1
vote
0answers
62 views

Are there intensional classes independent of the set universe?

The hereditarily finite sets can be regarded as purely extensional sets. Furthermore, they are quite independent of the underlying set universe (at least if we look at them from an extensional point ...
2
votes
1answer
85 views

Some theorems of model theory

Do theorems like "omitting types theorem", "Extended completeness theorem" etc.. hold inside arbitrary countable transitive models of ZFC?
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 ...
2
votes
1answer
111 views

Set of formulas in Model Theory

I'm reading the book Model Theory by Chang and Keisler and there is one thing that always bugs me. Very frequently we have something like $\Sigma(x)$ representing the set of all formulas in a language ...
6
votes
2answers
79 views

Generic filters in the ground model (forcing)

In his book Set Theory (second edition), page 203, Thomas Jech writes Lemma 14.4 Let $\mathfrak{M}$ be a countable model of ZFC and $P$ be a partially ordered set. If $p\in P$, there exists a ...
4
votes
1answer
110 views

Jech's proof of Silver's Theorem on SCH

Jech's textbook proves Silver's Theorem–that if the Singular Cardinals Hypothesis holds for all singular cardinals of countable cofinality, then it holds for all singular cardinals–by breaking up the ...
4
votes
0answers
144 views

Foundational theories, their uses, interactions and comparisons?

Until now, I heard that there are some theories for building mathematical objects (or at least it is what it seems to my poor knowledge). Some of these are: Set theory; Logic; Category theory; Type ...
3
votes
1answer
79 views

Existence of inaccessible cardinals implies the consistency of ZFC

I wonder if any can sketch for me in very broad lines the proof of the fact that the existence of inaccessible cardinals implies the consistency of ZFC? I don´t know much about set theory, but I find ...
3
votes
2answers
125 views

Can the collection of definable sets be a set?

Trevor pointed out, that there are models of $\mathsf{ZFC}$ in which every set is definable. In this case the collection of definable sets of the model is equal to the universal class of the ...
3
votes
2answers
70 views

A Question Regarding Reverse Mathematics and V=L

Can all of "ordinary mathematics" ("ordinary mathematics" as understood by practitioners of reverse mathematics) be formulated in ZF[0]+V=L (ZF[0]is simply ZF without the Power Set Axiom)? It is my ...
2
votes
1answer
59 views

A representation of well-orderings?

Is there a well-ordering $P$ of the set of real numbers $\mathbb{R}$ such that there is NO function $f: \mathbb{R}->\mathbb{R}$ satisfying the property: for all $x,y \in \mathbb{R}$, $xPy$ iff ...
3
votes
1answer
96 views

$L-$rank (Kunen exercise)

I'm stuck with exercise 5 at page 180, where it asks me to compute explicitly the $L-$rank $\rho(\bigcup x)$ of $\bigcup x$ in terms of $\rho(x)$. You can obviously define $\bigcup x$ from the ...
0
votes
2answers
194 views

A (too?) simple argument for the undefinability of definable sets

Preliminaries (see e.g. Jech, Set Theory, p. 5): To every formula $\varphi(x)$ of ZF set theory corresponds a class $C = \lbrace x : \varphi(x)\rbrace$, but only to some formulas corresponds a set. ...
6
votes
1answer
128 views

Question on existential sentences

A sentence is called existential if it is of the form $\exists x_1 \ldots \exists x_n \ \phi(x_1, \ldots, x_n)$, where $\phi$ is quantifier free. We know that (see Chang-Keisler "Model Theory", ...