8
votes
1answer
92 views

Can all theorems of $\sf ZFC$ about the natural numbers be proven in $\sf ZF$?

I know a proof of Hindman's theorem that uses ultrafilters on the natural numbers, and ultimately, the axiom of choice. But the theorem itself is essentially a combinatorial property of the natural ...
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
35 views

Clarification regarding inner model, standard model, transitive model and Mostowski

After reading books before lectures, here's my thought regarding inner models and so on. Correct me if I am wrong. So there's universe $V$, which we assume to be the true universe. By Gödel's ...
5
votes
2answers
63 views

Removing sets from models of set theory

I have a naive and open-ended question: How can one remove a set from a model of set theory in such a way that the result is again a model of set theory? Directly related: what kinds of sets can ...
2
votes
1answer
49 views

Countable transitive model of ZFC that is well-founded externally

As I am studying set theory, I came to realize that there exists a countable "well-founded" model of ZFC. But I am curious whether countable models can ever be well-founded externally. What would be ...
4
votes
1answer
199 views

Construction of *ZFC

In the following paper, page 11 (Appendix), there is a construction of a model of a theory $^*ZFC$ (see the definitions in the paper included) from a model of $ZFC$. I have been trying really hard to ...
3
votes
2answers
91 views

Non-well-founded models viewing well-founded models as non-well-founded.

I'm currently thinking about how different models of set theory view each other. In particular I'm looking at how well-foundedness behaves between different models. So we have the Axiom of ...
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 ...
4
votes
0answers
46 views

Categorical description of permutation-invariance of models

One of my projects (which I should really leave to professionals, but I'm trying anyway) is trying to find a categorical description of the permutation-invariance of models of NF, if there is such a ...
7
votes
2answers
205 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 ...
5
votes
1answer
224 views

How best to formalize propositions suffering from “size issues”?

Suppose we want to formalize a proposition (say, from category theory or model theory) that has "size" issues. For concreteness, lets take the following statement as a fairly typical example. ...
2
votes
2answers
55 views

every element of $V_{\omega}$ is definable

My attempt by $\in$-induction. I am trying find formula that will work: $N=(V_{\omega},\in)\models rank(\varnothing) =0<\omega$ Assume,given $x\in V_\omega$ that $\forall y\in x$ are definable too ...
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? ...
2
votes
1answer
53 views

Elementary submodels of $(V_{\omega},\in)$ are equal to it

I read that all the ESMs of $(V_{\omega},\in)$ will be equal to it . But what if the ESM's universe is a finite set?
3
votes
1answer
77 views

Elementary Submodels in Set Theory

I was reading the following summary on elementary submodels: http://boolesrings.org/mpawliuk/2012/01/26/a-practical-guide-to-using-countable-elementary-submodels/ Say $M\prec N$. The link above ...
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?
2
votes
1answer
57 views

Possible Turing degrees of countable models of ZFC

Let $M$ be a countable model in a signature $\Sigma$. We assume $\Sigma$ is finite, and (for convenience) has no function or constant symbols. Without loss of generality, we can assume that $M$'s ...
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 ...
0
votes
1answer
96 views

A Question Regarding Uncountable Standard Models of ZFC Where CH is False

Let M be an uncountable standard model of ZFC, let $\frak c$ be the cardinality of the continuum, and let (just for the sake of argument) $\mathfrak c=\aleph_2$. If one assumes M has 'all' the ...
6
votes
1answer
179 views

Using the Reflection Theorem

I've been reading about the Reflection Theorems in Kunen's 2011 Set Theory book. The idea that $ZFC \not \vdash \exists \gamma [V_\gamma \models ZFC]$, but $ZFC \vdash \exists \gamma [V_\gamma \models ...
1
vote
0answers
65 views

Name of, and (if I'm lucky) references on, a particular property of an interpretation

So here I am studying the Ackerman interpretation (via Kaye-Wong) to try and suss what the fragment of arithmetic associated with KF (Mac Lane minus Foundation and Infinity, with separation restricted ...
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", ...
2
votes
0answers
104 views

Defining “structured sets”

In his Notes on Set Theory (p. 44) Moschovakis defines: A structured set is a pair $U = (A,S)$ where $A$ is a set, the space of $U$, and $S$ is an arbitrary object, the frame of $U$. But even ...
7
votes
1answer
97 views

Showing that a statement is absolute.

After reading about various properties of $V_\alpha$ and how it can be used to model various axioms of Set Theory, Kunen mentions that in $ZFC$, one cannot prove that there is an $\alpha$ such that ...
4
votes
2answers
155 views

Using the Downward Lowenheim-Skolem-Tarski Theorem.

I've been reading about the models of Set Theory in Kunen's most recent Set Theory text, and working on exercises since this is my first time working with Model Theory. There is one exercise that I've ...
5
votes
1answer
103 views

A question dealing with conditions for which $V_\alpha$ models $ZFC$

I've been reading through models of Set Theory in Kunen's most recent Set Theory text and practicing exercises. He mentions that $V_\alpha$ can be used to satisfy certain axioms of $ZFC$ when $\alpha$ ...
5
votes
1answer
67 views

Finding a counterexample in model theory

I'm currently reading about models of Set Theory, and I'm working on exercises to better understand the concepts. In Kunen's most recent Set Theory text, he mentions that if we have a transitive model ...
5
votes
2answers
133 views

Is there a Second-Order Axiomatization of ZF(C) which is categorical?

A theory $T$ is called categorical if it only has one model upto isomorphism. (Note: this has nothing to do with category theory.) The Lowenheim-Skolem theorem states that no first-order theory with ...
1
vote
1answer
255 views

Neither Even Nor Odd Natural Numbers

I confused myself and the OP when I tried to answer a recent question. Modular arithmetic (MA) has the same axioms as first order PA except $\forall x(Sx \neq 0)$ is replaced with $\exists x(Sx=0)$. ...
9
votes
2answers
220 views

Can a model of set theory think it is well-founded and in fact not be?

ZF's axiom of regularity implies that no infinite descending sequence of sets $x_1 \ni x_2 \ni x_3 \ni \cdots$ exists. Precisely this theorem asserts the non-existence of a map from $\mathbb{N}$ to ...
3
votes
1answer
115 views

Crashcourse Models in Set Theory

I am currently working through the lecture notes. In the end we had a short introduction to Models in Set Theory, but since it was quite to the end, we did not really go into details. So my ...
4
votes
1answer
91 views

Do we assert the existence of set theory when reasoning about L-structures?

In model theory, if L is a first order language, by the definition of a L-structure $\mathcal{M}$ it is partly given by a non-empty set $M$ called the universe or domain of $\mathcal{M}$. From where ...
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 ...
2
votes
1answer
104 views

Not Skolem's Paradox

Assume we have a countable, non-standard model of Peano Arithmetic (PA) in ZFC. http://en.wikipedia.org/wiki/Non-standard_model_of_arithmetic Let $N^*$ be the universe of this model and let $m \in ...
2
votes
0answers
119 views

Crankery: Is there a perfect inner model of ZFC?

In the book "Aspects of Vagueness", the article "The alternative set theory and its approach to Cantor's set theory" by A. Sochor proposes the following definition: We will say that a set universe ...
4
votes
1answer
114 views

Definiteness of omega

A homework(ish) problem from models of set theory: Define $\varphi(x) :\leftrightarrow Lim(x) \land \forall y\in x \, (Lim(y)\rightarrow y=0)$ where $Lim(x)$ means that $x$ is a limit ordinal. ...
1
vote
1answer
106 views

Standard models being non-standard?

If there is a ''set'' W in V which is a standard model of ZF, and the ordinal κ is the set of ordinals which occur in W, then Lκ is the L of W. If there is a set which is a standard model of ...
3
votes
1answer
155 views

Question on the non-existence of a satisfaction formula in $\mathbb{L}$

I know this may sound trivial, but I'm having trouble figuring out what the problem is. Let $\mathbb{L}$ be the class of constructible sets. We know that $(\mathbb{L}_{\omega + \omega}, \in)$ is a ...
2
votes
0answers
112 views

Formalizing model-theoretical large cardinals in a formal system for ZFC

I work with the Metamath formal system, which is expressive enough to define ZFC and prove some nontrivial stuff, but my current goal is to define large cardinals, and I'm hitting a wall somewhere ...
5
votes
1answer
158 views

In ZF, does there exist an ordinal of provably uncountable cofinality?

Question is in the title. In ZFC, one can prove that $\aleph_{\alpha+1}$ is regular, so there is a large source of cardinals with uncountable cofinality, but in ZF, it is consistent that ${\rm ...
5
votes
1answer
172 views

Questions about the concept of Structure, Model and Formal Language

When we start to define mathematical logic (specifically, propositional, first order, and second order logic) we start defining the concept of a language. At the begining this is done in a purely ...
3
votes
1answer
124 views

Model theory/stability theory

I was flipping through Baldwin's Stability Theory book and saw an example that has me confused... The example is a 1st-order theory $T$ of refining equivalence relations $E_i(x, y), i< ω$, where ...
6
votes
2answers
235 views

Why continuum hypothesis implies the unique hyperreal system, ${}^{\ast}{\Bbb R}$?

On page 33, Robert Goldblatt, Lectures on Hyperreals(1998): Now it has been shown under certain set-theoretic assumption called continuum hypothesis the choice of $\mathcal F$ is irrelevant: All ...
3
votes
1answer
116 views

Model theory question with finiteness

It should be pretty elementary, but I can't really see it.. Cheers to anyone who can help me Let $\Psi$ be a transitive set that is a model of $ZF$. Then, $a$ is finite if and only if $\Psi \models ...
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 ...
3
votes
0answers
113 views

Does ZFC allow self-reference? [closed]

I heard that, "ZFC theory doesn't allow self-reference." But I don't know exactly what it means. As we can see in the proof of Godel's incompleteness theorem, we can use method of "Diagonalization" ...
1
vote
1answer
113 views

understanding provability

I am still confused about provability. . . Let a statement P is, sort-of-says like this. P: ( "X is provable" ∧ "P is provable" ) If ( X is provable ∧ P is provable ) is provable → (P is ...
12
votes
3answers
202 views

Axiom of Choice and Determinacy

In my set theory course we have been talking about the axiom of determinacy. One of the first things we showed was that $AD$ and $AC$ are incompatible. We later showed that $ZF+AD$ implies the ...
8
votes
1answer
145 views

Independence results that cannot be established by forcing.

I read the Wikipedia article on Absoluteness recently and found mention of Shoenfield’s Absoluteness Theorem, which states that if $ \phi $ is any $ \Sigma^{1}_{2} $- or $ \Pi^{1}_{2} $-sentence of ...