5
votes
0answers
50 views

Indiscernible subsequences

Work in a saturated model $\cal U$ of sufficiently large cardinality. Are there assumptions on $\kappa<|\cal U|$ that guarantee that any sequence $\langle a_i:i<\kappa\rangle$ has an ...
-1
votes
0answers
104 views

Is the axiom of choice constructive in the constructible universe?

Even ZF has some non-constructive elements mostly due to contradiction proofs. For example one may be able to construct a sequence of objects some of which have a given property without being able to ...
4
votes
2answers
79 views

Is it meaningless to say $M\prec N$ for two proper class models?

Kunen in page 88 of his "Set Theory" book says: ... For a specific given $\varphi$, the notion $M\prec_{\varphi}N$ (i.e. $\forall \overline{a}\in M~~~M\models \varphi ...
6
votes
4answers
360 views

Are there counter intuitive interpretations of ZF or ZFC?

Are there interpretations of ZF or ZFC that are non standard in the sense that $\epsilon$ is interpreted in a counter intuitive way that intuitively has nothing to do with "belongs to" or "is part of" ...
3
votes
1answer
102 views

Why is the powerset axiom more acceptable than the axiom of choice?

The key step in Zermelo's proof of the well ordering theorem is to use $\text{AC}$ to simultaneously choose the next elelment for all possible partial chains in prospective well orderings, but that ...
3
votes
2answers
69 views

Clarifications on proof of compactness theorem

I've been reading through the following proof of compactness theorem: http://www.princeton.edu/~hhalvors/teaching/phi312_s2013/compactness.pdf One thing that struck me is that this proof seems to ...
2
votes
2answers
89 views

Uncountable models for integers

Part of Asaf Karagila's brilliant answer to one of my other questions puzzles me a lot. Namely, I find it hard to understand how there can be a model for ZFC with uncountably many integers. My ...
2
votes
1answer
91 views

Maximal model for $\Bbb R$?

I have not dealt professionally with set theory, so excuse me if my way of formulating this question does not completely follow standard terminology. Actually, my question is about whether or not the ...
3
votes
1answer
49 views

A question about second-order logic and inaccessible cardinals.

Let $\kappa$ denote an inaccessible cardinal, and suppose $T \in V_\kappa$ is a second-order theory. Now consider some mathematical structure $X \in V_\kappa$. Then I think it is clear that $X \models ...
1
vote
1answer
27 views

Is “constructible from” a transitive relation?

In Jech's Set Theory, exercise 13.27, it is hinted that $X \in L[Y]$ and $Y \in L[X]$ together imply $L[X]=L[Y]$. I tried to prove this fact without success, although I suspect the proof is simple. ...
6
votes
1answer
87 views

Construction of Ultrafilters

I've been doing a lot of work with ultrapowers and saturation recently. In particular, I am reading chapter 6 of Chang and Keisler as well as Keisler's paper on "Ultraproducts which are not ...
2
votes
1answer
53 views

Why is $\alpha \mapsto L_{\alpha}[A]$ $\Delta_{1}$?

On page 187 of Jech's Set Theory, there is a proof sketch of the fact that $\alpha \mapsto L_{\alpha}$ is $\Delta_{1}$. As far as I can tell, Jech's argument only shows that this operation is ...
0
votes
0answers
38 views

Sets Constructible Relative To A Unary Predicate

The class $L$ of constructible sets is defined by recursion using the operation def$(M)=\{x \subset M: x$ is definable over $(M, \in) \}$. By adding a unary predicate, $P$, to our language, we can ...
4
votes
1answer
75 views

Model of complete extension of Zermelo set theory

Chang and Keisler's Model theory gives the following exercise problem: Prove that there is a complete extension $T$ of Zermelo set theory which has arbitrary large natural models. (A model ...
2
votes
1answer
107 views

Cardinalities of Collections of Models

Let $T$ be a complete theory in a countable language (with only infinite models). Recall the spectrum function: $I(\aleph_\alpha,T)=$ the number of non-isomorphic models of $T$ of cardinality ...
1
vote
1answer
80 views

On the number of countable models of complete theories of models of ZFC [duplicate]

Fix the language of set theory $\mathcal{L}=\{\in\}$. Let $\langle M,\in\rangle$ be a set or proper class model of ZFC (e.g. $M$ could be $L$, $HOD$, $V_{\kappa}$ for some inaccessible cardinal ...
3
votes
0answers
85 views

Countable transitive $T \vDash ZFC-P$ with $\approx$ not absolute: why do we need $H(\aleph_3)$?

I am trying to solve Exercise IV.3.31 from Kunen's Foundations of Mathematics. I think I have a solution but I am confused by one of the hints. By request, here is the text of the exercise. ...
2
votes
1answer
75 views

Absolute confusion! (A question about absolute *sentences*)

I'm seriously confused about absoluteness. A formula in the language of a theory $T$ is absolute for $T$ structures if its truth value is the same in all standard transitive models of $T$ (this may ...
2
votes
0answers
106 views

How large or small can the gap in Shelah's main gap theorem be, up to consistency?

Shelah's main gap theorem in model theory says: For each first order complete theory $T$ in a countable language if $I(T,\kappa)$ denotes the number of its models of size $\kappa$ then one of ...
0
votes
0answers
51 views

Show that $(V_\omega,\in) \models \mathrm{Power\ set}$

Let "$\mathrm{Power\ set}$" denote the Power set axiom. I'd like if some of you could tell me if the solution of the following exercise is correct. I want to prove that $(V_\omega,\in)\models ...
0
votes
0answers
48 views

Absolute formula

I'm trying to study the model of Set Theory and I have a problem. If I know that a formula is absolute for a class, could I infer that the formula is true in my class? Namely, I'm trying to prove in ...
7
votes
1answer
58 views

Every elementary submodel of $H(\aleph_1)$ is transitive

I am trying to solve the first part of Exercise II.17.30 in Kunen's Foundations of Mathematics which asks: Prove that every $A \preccurlyeq H(\aleph_1)$ is transitive. Here we are working over ...
4
votes
1answer
79 views

Ultraproducts of models of ZFC

Let $D$ be a non-principal ultrafilter over $\mathbb{N}$. Let $A_i$ for $i\in\mathbb{N}$ be (countable) models of ZFC such that $A_i\models \mathfrak{c}=\aleph_i$. Then, what is the size of the ...
3
votes
2answers
113 views

Universe cardinals and models for ZFC

I'm reading through Joel David Hamkins' set theory lecture notes. On page 14, on the subject of inaccessible cardinals and submodels of ZFC in $V$, he defines a universe cardinal to be a cardinal ...
6
votes
1answer
123 views

Is it possible to prove the axiom of infinity from the real number axioms?

What I mean by the title is whether, given that there is a class $\Bbb R$ and operations $+,\cdot,<$ that satisfy the ordered field axioms and the least upper bound axiom, you can prove the ...
1
vote
3answers
144 views

ZFC Axioms to be extended?

Sorry if this is going to be a really loaded question. I was told several times that for virtually all theorems/corollaries/propositions of mathematics (except those cases not compatible with ZFC ...
8
votes
1answer
137 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 ...
4
votes
3answers
113 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
43 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
70 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
76 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
223 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
128 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
195 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
48 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
257 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
236 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
62 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
137 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
89 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
55 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
110 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
89 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
66 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
129 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
102 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
186 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
66 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
143 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
131 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 ...