3
votes
2answers
40 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 ...
-4
votes
0answers
33 views

What is the pure essence of a definition of semantics? [on hold]

What is the very essence of the definition of semantics (and interpretation, structure, model) for a logician or for an algebraist? We all know the usual definitions. But is it not the essence of ...
1
vote
0answers
49 views

Can one prove existence of incommensurables without the Pythagorean theorem?

Euclid's proof that the side and the diagonal of a square have no common measure, probably going back to Pythagoreans, reduces it to proving the irrationality of $\sqrt{2}$. This reduction uses the ...
2
votes
1answer
45 views

Suppose $R \sim_\omega R'$. Then for every $k$-tuple $a$ in $E$ and every natural number $p$, there is a $k$-tuple $b$ in $E'$ such that $a \sim_p b$

Sorry to bother you guys again with a Poizat question, but I'm struggling a little bit with the material (as it must be obvious) and I want to check if I got the main idea correctly or if I'm totally ...
4
votes
1answer
60 views

Elementary Model Theory

I'm working through section 4.3. on model theory from Dirk van Dalen's Logic and Structure (fifth ed.) and am struggling with van Dalen's sometimes sloppy way of presenting proofs. As usual let a ...
2
votes
1answer
33 views

Equality of sets of local isomorphisms between relations

I'm still working on the first pages of Poizat's A Course in Model Theory. I'll state the basic definitions again, in order to avoid referring back to an early question: Poizat defines an isomorphism ...
2
votes
0answers
47 views

On the back and forth conditions for a set of partial isomorphisms

I've recently begun reading Poizat's A Course in Model Theory and already in the first pages I had some doubts. One odd (not necessarily bad) thing is that he defines notions such as isomorphism only ...
2
votes
1answer
112 views

Keisler Order: Saturated Ultrapowers

Keisler's paper "Ultraproducts which are not Saturated" states the following theorem as a corollary to a (much more) generalized theorem. However, I cannot figure out how to prove it for the specific ...
2
votes
1answer
23 views

Density and Saturated Models.

Consider $(\mathbb{Q}; \leq)$ and let $T$ be the theory of dense linear orderings without endpoints. Let $\mathfrak{A}$ be the $\omega_1$ saturated model of $T$. Note that ...
0
votes
0answers
25 views

When is the higher-order theory of a model categorical?

I'm interested in (classical) type theories $L$ with the following property. For $M$ any model of $L$ (in Set), let $T(M)$ be the type theory of $M$, i.e., the strongest extension of $L$ satisfied by ...
3
votes
0answers
25 views

Let $h: A \to B$ be a weak homomorphism. Is h$[A]$ a substructure of $B$?

A little bit more precise: let $\mathfrak{A}$ and $\mathfrak{B}$ be two structures. Define a weak homomorphism as a function $h: \mathfrak{A} \to \mathfrak{B}$ such that the folowing conditions are ...
1
vote
1answer
44 views

Consistent Set of Sentences is Consistent in Expanded Language

Suppose that we have a set $\Phi$ of sentences over a first-order language $\mathcal{L}$ and that $\Phi$ is consistent. Suppose we have another first-order language $\mathcal{L}'$ such that ...
2
votes
1answer
35 views

Compactness and Arithmetic Confusion

Let $T$ be some theory capable of arithmetic and construct a provability predicate (which we will call $Prb_T$). Let $\mathbb{N} \models T$. Expand our language to include a new constant symbol $c$. ...
4
votes
1answer
75 views

How can I imagine a model of $\mathsf{(ZF-Inf)} + \neg \mathsf{Con(ZF-Inf)}$?

Gödel's second incompleteness theorem states that if $\mathsf{ZF-Inf}$ is consistent, then $\mathsf{ZF-Inf} \nvdash \mathsf{Con(ZF-Inf)}$. Moreover, if $\mathsf{(ZF-Inf)} + \neg \mathsf{Con(ZF-Inf)}$ ...
3
votes
1answer
72 views

Relations between Theories and Categories

I'm just toying around with some thoughts, trying to grock some concepts: It seems that every formal theory induces a locally small category via interpretations: its objects are structures that ...
3
votes
1answer
48 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
33 views

Relative Interpretations alla Kunen

at the moment I try to figure out some details of Kunen's "Relative Interpretation" Definition (within the 2013 Edition of his "Set Theory", p. 99 to 100): Definition If $\Lambda$ is some axioms ...
4
votes
1answer
66 views

Decision and the Uncountable Spectrum

In 2000, Hart, Hrushovski, and Laskowski classified all complete first order theories in a countable language up to their uncountable spectra. However, does this also imply that given a $any$ ...
1
vote
1answer
58 views

Are algebraic structures required to satisfy axioms?

Is it a requirement for algebraic structures, when studying universal algebra, to satisfy axioms? The reason I ask is because algebraic structures are only defined by a underlying set, a signature, ...
1
vote
2answers
71 views

An Uncountable language , A Model of $\mathbb{N}$, A Problem.

Edit 1: I messed up my original question, but Arthur Fischer answered my question anyway. Edit 2: We can actually restrict $L$ to the language in arithmetic with the prdicate $P_{\mathbb{P}}$. ...
5
votes
1answer
70 views

Is infinitary logics $\mathcal{L}_{\infty\omega}$ an abstract logic?

Infinitary logics $\mathcal{L}_{\infty\omega}$ is an extension of first-order logics such that $\bigvee\Phi \in \mathcal{L}_{\infty\omega}$ if $\Phi$ is a set of ...
6
votes
1answer
86 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 ...
1
vote
0answers
45 views

Model-theory : questions regarding back-and-forth sets

See my previous post for the basic definitions from Jouko Väänänen, Models and Games (2011), page 54-on. See page 64 for : Definition 5.14 Suppose $\mathcal A$ and $\mathcal B$ are ...
1
vote
1answer
52 views

Model-theory : questions regarding partial isomorphism

I'm having problems with the first pages of Bruno Poizat, A Course in Model: Theory An Introduction to Contemporary Mathematical Logic (ed or 1985), specifically with local isomorphism and back- and ...
7
votes
1answer
157 views

Gödel's way of teaching non-standard models to Takeuti.

In Memoirs of a Proof Theorist, Gaisi Takeuti relates how Gödel taught him about nonstandard models in an "interesting" way: It went as follows. Let T be a theory with a nonstandard model. By ...
1
vote
1answer
70 views

Show that if $L$ is countable and contains a two-place predicate symbol, there are $2^{2^{\aleph_0}}$ classes of $L$-structures closed under $\equiv$

We say that a class of structures $K$ is closed under elementary equivalence ($\equiv$) if for all $A, B$, if $A \in K$ and $A \equiv B$, then $B \in K$. How to show that if $L$ (as a set of specific ...
0
votes
1answer
52 views

Why are there $\leq$ $2^{\operatorname{card}(\operatorname{Form}(L))}$ elementarily nonequivalent structures for $L$?

Let $L$ be a set of specific symbols and $\operatorname{Form}(S)$ be the set of all first-order formulas over $L$. Why are there $\leq$ $2^{\operatorname{card}(\operatorname{Form}(L))}$ ...
2
votes
1answer
58 views

How to prove that $max(\aleph_{0}, card(X)) = max(\aleph_{0}, card(L(X)))$?

I struggle with the following problem. Let $X$ be a set of elementary sentences and $L(X)$ be the smallest elementary language in which we can express all the sentences from $X$. How to prove that ...
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 ...
4
votes
1answer
52 views

A model which has only one undefinable element over a language with only a finite number of symbols

I try to solve the problem 1.3.14 in Chang and Keisler's Model theory: For each $n\in\omega$, find a model $\mathfrak{A}_n$ for $\mathcal{L}$ a language with only a finite number of symbols, which ...
6
votes
3answers
137 views

Can $(\Bbb N,\leq)$ have an $\aleph_0$-categorical theory (in a larger language)?

One of the nicer consequences of compactness is that there is no statement in first-order logic whose content "$\leq$ is a well-order". So we can show that there are countable structure $(M,\leq)$ ...
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
72 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
105 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 ...
1
vote
2answers
52 views

Prove that ⊨ is not symmetric

Another question from a beginner: We want to prove that ⊨ is not symmetric by finidng concrete formulas φ and ψ for which we can show that φ ⊨ ψ and ψ ⊭ φ. Thank you for your help!
3
votes
1answer
61 views

Is first-order logic complete with respect to countable structures?

Question. Let $\Gamma$ denote a first-order signature, and consider a sentence $\tau$ and a set of sentences $\Sigma$ in the language generated by $\Gamma$. If every countable $\Gamma$-structure that ...
5
votes
1answer
53 views

Nonstandard models of PA with a decidable order relation.

There this exercise in Models of Peano Arithmetic (Kaye 1991, p.157), which asks to define a recursive binary relation on $\mathbb{N}^2$, such that $M \upharpoonright < $ is isomorphic to ...
3
votes
0answers
73 views

Examples of Jónsson Models

Let $T$ be a complete first order theory. Suppose that $M\models T$. Then, $M$ is said to be a Jónsson Model of $T$ if for all $N$, such that $N\prec M$ and $N\models T$, we have $|N|<|M|$ (Note ...
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
78 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 ...
4
votes
1answer
51 views

The class of finite groups (models) and that of countable groups are not elementary classes (a generalized version).

First some definitions: For a set $\Sigma$ of $\mathcal{L}$-sentences, $Mod(\Sigma)$ denotes the class of all models that satisfy $\Sigma$. For a class $\mathcal{M}$ of models, we say it is $EC$ if ...
4
votes
0answers
66 views

Models of infinite groups and 'Group-like' objects

Let $G$ be an infinite group, and for simplicity, we will assume that $G$ is also countable. Now, with $G$ in mind, we construct a new language $L_G=\{f_{a_i\_},f_{\_a_i}:a_i\in G\}$ where ...
5
votes
2answers
481 views

There is no smallest infinitely large prime

I'm reading Kunen's Foundations of Mathematics and trying to solve Exercise II.16.19, which constructs an elementary extension of the reals as an ordered field, and asks the reader to prove various ...
2
votes
1answer
49 views

How can we get rid of function symbols in model theory?

I understand the presentation of a language in logic as having relations with arities, functions with arities and constants. I understand that a constant can be thought as a function with arity $0$. ...
1
vote
1answer
71 views

What is the point of (Compactness Theorem in the) Overspill Principle?

I am trying to understand the basics of computation theory. The Overspill Principle (also at google) basically says if you are cool you can do everything Let Г be a sentence of predicate logic ...
1
vote
1answer
93 views

Existence of an automorphism of $\Bbb C$ that fixes a finite set pointwise but does not fix $\Bbb R$ setwise

I would like to construct a (ring-theoretic) automorphism of $\Bbb C$ that fixes a finite set $A$ pointwise but does not fix $\Bbb R$ setwise. Marker's Model Theory, Corollary 1.3.6 does that in this ...
1
vote
1answer
35 views

Partial order version of elementary equivalence

Elementary equivalence is an important concept in mathematical logic. Two models $\mathfrak{M}$ and $\mathfrak{N}$ of the same signature are elementarily equivalent, written $\mathfrak{M} \equiv ...
3
votes
2answers
54 views

Does having a filter which is not maximal implies the negation of Łoś theorem?

If we have a family $(\mathfrak{M}_i)_{i\in I}$ of $L$-structures, and a filter $\mathcal{F}$ over $I$, we can define the reduced product $\prod_{i\in I}\mathfrak{M }_i/\mathcal{F}$. If $\mathcal{F}$ ...
4
votes
1answer
78 views

Showing $2$ is not definable in $(\mathbb{Q},+)$.

As stated, I'm to show that $2$ is not definable in $(\mathbb{Q},+)$. I tried proving it by contradiction by showing that if $2$ were definable, then we could define $\mathbb{N}$ and multiplication ...
3
votes
1answer
74 views

Showing there does not exist a formal proof of a formula $\phi$.

My problem:Suppose $R$ is a binary predicate and use the soundness theorem to show that there does not exist a formal proof of $$ \phi =\forall y\exists xR(x,y)\rightarrow \exists x\forall yR(x,y).$$ ...