All Questions
Tagged with model-theory elementary-set-theory
62
questions
1
vote
1
answer
160
views
Permutation model in which infinite sets are weakly Dedekind-infinite but not Dedekind-infinite
I’m trying to create a permutation model $N$ in which every infinite set is weakly Dedekind-infinite (i.e. for every infinite set $A\in N$ there is a surjective map $f:A\rightarrow\omega$ in $N$), but ...
- 312
3
votes
2
answers
222
views
How to create infinitely many disjoint sets from infinitely many sets
Suppose we have a countably infinite set $X$ and we have (countably) infinitely many subsets $A_1,A_2,\cdots\subseteq X$ which are non-empty and distinct (i.e. for any $i\neq j$ either $A_i\setminus ...
- 312
0
votes
2
answers
155
views
Can every set in the Von-Neumann universe be obtained from finite ZFC steps [closed]
For me, ZFC feels like saying either something is a set, or from a set, we know another thing is a set. On the other hand, something like doing the powerset operations for $\omega$ a number of times ...
- 1,722
-3
votes
1
answer
159
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order property vs. antisymmetric property
The definition of order property is well known:for a first-order theory $T$ the order property means that for some first-order formula $\phi(\bar{x},\bar{y})$ linearly orders in $M$ some infinite $\...
- 3,978
1
vote
1
answer
91
views
Resources for "Bell Machover"
Recently, I've been reading through A Course In Mathematical Logic by John Bell and Moshé Machover. However, it's not always the easiest book to understand. What might be some good supplements to have ...
- 183
1
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1
answer
192
views
Are there simple examples of two distinct transitive sets that are elementary embeddable in each other? [closed]
This question had be edited altogether in a more concise manner.
Suppose $M$ and $N$ are transitive sets such that there exist elementary embeddings $f:(M,\in)\to(N,\in)$ and $g:(N,\in)\to (M,\in)$. ...
- 4,631
5
votes
1
answer
56
views
Hyper-extensions of Hom space
We fix an ultrafilter $\mathcal{F}$ of $\mathbb{N}$ which contains the cofinite filter.
Let $A,B$ be sets and ${}^{*}A,{}^{*}B$ their hyper-extensions. Then is
$$
{\rm Hom}({}^{*}A,{}^{*}B)
$$
equal ...
- 161
1
vote
1
answer
60
views
elementary embeddings $j$ in set theory with $V$ and $M$
I'm confused by a variety of elementary non-trivial elementary embedings $j$ we might have.
There are 9 "syntactical" possiblities;Here $M$ is a transitive model. I'll name them with a wish ...
- 3,978
2
votes
1
answer
140
views
Cardinality of integer parts of real closed fields
Every real closed field $R$ has in integer part $I$. That is, $I$ is a discrete ordered subring of $R$ such that for each $x \in R$ there is $z \in I$ such that $z \leq x < z + 1$.
If $R$ is ...
- 133
2
votes
0
answers
53
views
Projecting Skolem's Paradox Upwards
My understanding of the resolution of Skolem's Paradox is that although in a countable model of ZFC there does not exist a bijection between a countable set and its powerset, we can still construct a ...
- 855
1
vote
0
answers
125
views
Universal quantifier over an uncountable set
To prove that a segment has the same number of points with half a segment one might say that one can find a bijective function mapping every point from the segment to the half segment. Let' say:
$$\...
- 19
3
votes
1
answer
226
views
Are there any interesting non standard models of $Q$?
I've encountered a few of the models of Robinson arithmetic here a quick list:
$\mathbb{N}\cup {\infty}$ (used to show Robinson arithmetic has a non standard model)
$\mathbb{N}\cup \{a,b\}$ where the ...
- 1,926
2
votes
1
answer
231
views
Cardinality of the ranks of the constructible universe
First time I encounter the Constructible universe $L$ and the definition given in Jech is the following:
$L_0=\emptyset$,
$L_{\alpha+1}=\operatorname{def}(L_\alpha)$,
$L_\lambda=\bigcup_{\alpha<\...
- 1,926
1
vote
2
answers
209
views
Infinity in ZFC vs infinity in the metatheory
I was reading about different ways to formalize the notion of infinity in ZFC. The classic axiom is of course to have the existence of an inductive set, but you can also assert the existence of a ...
- 166
5
votes
1
answer
529
views
In what sense is logical entailment a set-theoretic relation?
In set theory a relation is a subset of a Cartesian product. I suppose that in logics this product is the Cartesian square of the powerset of all propositions(?).
Semantic/model-theoretic entailment ...
- 536