2
votes
1answer
55 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 ...
1
vote
1answer
157 views

Recommendation on a rigorous and deep introductory logic textbook

In this post, I don't mean any word by its somewhat "mathematical or logical" meaning but just "literally". It's been three years since I started "formal" mathematics, and now I'm familiar with set ...
0
votes
0answers
17 views

Nonaxiomatizability one-dimensional vector spaces [duplicate]

Prove, class of all one-dimensional vector spaces over R isn't axiomatizable in the signature σ=< +, α, 0>, α* is single function vector multiplication by a scalar α of R.
2
votes
1answer
227 views

Logic: cardinality of the set of formulas

How can you proof that $||L||=|L|$ if $L$ is infinite (where $||L||$ stands for the cardinality of the set of all $L$-formulas and $|L|$ the number of all constants, function and relation symbols)? ...
7
votes
3answers
173 views

Is there any countable ordinal number which has a member undefinable?

Let $Q\colon=\{\alpha \in Ord \mid \exists \beta \in \alpha(\beta \text{ is undefinable in } \langle\alpha,<\rangle )\}$ denote the set of all ordinal numbers which has an undefinable member. ...
3
votes
1answer
202 views

What's correspondence between the model theoric and the set theoric kernel of homomorphism?

A kernel of a mapping $h$ from $\mathfrak{A}$ to $\mathfrak{B}$, generally, is an equivalence relation $\{(a,a') \in \mathfrak{A} \times \mathfrak{A} \mid h(a)=h(a')\}$. However, in model theory, ...
3
votes
1answer
83 views

construction set of natural number logic

I identify the natural number $0$ with the empty set $\emptyset$, $1$ with $S(0)$, $2$ with $S(1)$, etc, etc. The axiom of infinity says $\exists x (\emptyset\in x\wedge \forall z\in x\space ...
6
votes
2answers
190 views

The Axiom of Choice and definability

I've seen a lot of relations between the notion of the existence of a definable set with a given property and the necessity of AC is proving that there is a set with the property. For example: Under ...
1
vote
1answer
52 views

Order isomorphism help

Let $A=B=(0,1]$ and let $\{a_i\}_{i\in\mathbb{N}}\subset A$ and $\{b_i\}_{i\in\mathbb{N}}\subset B$ are be two sequences and let $a_i\le a_j$ iff $b_i\le b_j$. is there any order preserving embedding ...
2
votes
1answer
59 views

Showing a model does not have a particular substructure and understanding satisfaction relations.

Out of Winfried Just and Martin Weese's Set theory book: Show that the model $\mathfrak B=\langle \Bbb Z, +, \le, 0 \rangle$ does not have any substructure whose universe is $\{-1,0,1\}$. In a ...
3
votes
1answer
100 views

Question about cumulative hierarchy

In the following let $\mathbf{V} = \bigcup_{\alpha \in \mathbf{ON}} V_\alpha$ denote the cumulative hierarchy. Let $\{\varphi_0, \dots, \varphi_n, \dots \}$ denote a list of all $ZF$ axioms. I am ...
5
votes
1answer
107 views

Proving that $\langle \omega, \in \rangle $ models empty set and extensionality

Let (A1) be the axiom of extensionality: $\forall x,y ( x = y \longleftrightarrow \forall z \in x \leftrightarrow z \in y))$ and let (A2) be the empty set axiom $\exists x \forall y (y \notin x)$. ...
1
vote
1answer
43 views

Model of a language $L$ vs. model of a theory $T$ in $L$

I am reading Just/Weese and they seem to use "model of a language $L$", for example, p. 90: and, more disturbingly, p. 91: Isn't this a "typo" (or perhaps sloppy writing)? If $L$ is any language ...
3
votes
2answers
69 views

Does $E$ in a model $\langle M, E\rangle$ of ZFC have to be wellfounded?

Consider the following exercise from Just/Weese: My first reaction was "Of course $E$ has to be strictly wellfounded otherwise it wouldn't model $\in$" but apparently I am missing something since I ...
2
votes
2answers
67 views

Confused about models of ZFC and passage of book

I have a question about a passage of Just/Weese. In the following, let $\mathcal M = \langle M, E \rangle$ be a model of ZFC. Here is the first half of what I'm about to ask a question: So I asked ...
7
votes
3answers
388 views

Complete first order theory with finite model is categorical

I am trying to prove that if $T$ is a complete first order theory that has a finite model then it has exactly one model up to isomorphism. To this end, I assumed that $T$ is complete with a finite ...
4
votes
2answers
217 views

Proof of compactness theorem

I wanted to prove the compactness theorem, p 79 Just/Weese: The (i) <= (ii) direction is not obvious to me. I thought I could prove it by showing not (i) implies not (ii) as follows: Assume ...
0
votes
3answers
72 views

Understanding models and valuations

I'm thinking about part (a) of the following exercise in Just/Weese page 77: Here is the definition of valuation: For example, say we have a model of the language of group theory, $( \mathbb Z/ 2 ...
2
votes
4answers
271 views

Model of theory of real closed field

I heard somewhere that models of theory of real closed field are isomorphic. However, there is also a statement in Internet which seems to say the opposite. Are the models of theory of reals ...
2
votes
1answer
240 views

Proof of the Löwenheim-Skolem theorem

For each first-order $\sigma \,$-formula $\varphi(y,x_1, \ldots, x_n) \,,$ the axiom of choice implies the existence of a function $f_\varphi: M^n\to M$ such that, for all $a_1, \ldots, a_n \in M$, ...
1
vote
2answers
102 views

Difference between axiomatization and model

As I study through set theory, I find the definition of axiomatization and models somewhat confusing. The question is what is the difference between axiomatization and model? Thanks.
1
vote
1answer
133 views

Lowenheim-Skolem theorem confusion

In Wikipedia ( http://en.wikipedia.org/wiki/L%C3%B6wenheim%E2%80%93Skolem_theorem ), it says: In mathematical logic, the Löwenheim–Skolem theorem, named for Leopold Löwenheim and Thoralf Skolem, ...
2
votes
1answer
127 views

What is a model of a formal system?

Please give the most illuminating example of a model for a formal system, and a simple example of its use. I also wish an example of an interpretation, and what its useful for.
9
votes
2answers
1k views

Showing any countable, dense, linear ordering is isomorphic to a subset of $\mathbb{Q}$

I'm trying to knock out a few of the later exercises from Enderton's Elements of Set Theory. This problem is #17, found on page 227. A partial ordering $R$ is said to be dense iff whenever $xRz$, ...