# Tagged Questions

233 views

### Upper bound on cardinality of a field

Is there an upper bound on the cardinality of a field? The "biggest" fields I know are the field of real numbers, or the field of complex numbers. Is there a field with cardinality greater than ...
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### 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 ...
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### 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 ...
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### 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.
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### 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)? ...
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### 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. ...
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### 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, ...
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### 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 ...
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### 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 $a_1, \ldots, a_n \in M$, ...
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### 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.
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### 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, ...
### 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 $xRz$, ...