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Results tagged with lo.logic
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user 7206
first-order and higher-order logic, model theory, set theory, proof theory, computability theory, formal languages, definability, interplay of syntax and semantics, constructive logic, intuitionism, philosophical logic, modal logic, completeness, Gödel incompleteness, decidability, undecidability, theories of truth, truth revision, consistency.
3
votes
1
answer
156
views
Rigid structure which is generically homogeneous
Is it possible to have a structure $T$ in some language which is rigid in $V$, but in a cardinal-preserving extension $T$ is homogeneous (in a suitable sense of the word)?
If this is not possible, is …
- 39.8k
6
votes
1
answer
238
views
Definability of defining classes
Suppose that $M$ is a transitive class, denote by $\mathrm{HOD}(M)$ the class of all those sets which are hereditarily definable from ordinals and parameters in $M$.
Some trivial examples include $\m …
- 39.8k
8
votes
1
answer
436
views
Uncountable models of Kelley-Morse set theory with only a countable number of sets
The Kelley-Morse set theory can be thought as the "full-secondorderification of $\sf ZFC$", where we switch from sets to classes and allow the comprehension schema to include quantifiers on class vari …
- 39.8k
12
votes
1
answer
1k
views
Can there be only one (uncountable transitive model of ZFC)?
It is an immediate consequence of Cohen's forcing that if there is one countable transitive model of $\sf ZFC$, then there are many of them. Even if all of these models are of the same height, there a …
- 39.8k
10
votes
1
answer
671
views
Is Collection really stronger than Replacement?
The two powerhouse schemata of set theory are Replacement and Collection:
Replacement. For every definable function $f$ and every set $x$, $f"x$ is a set.
Collection. For every definable relation $R$ …
- 39.8k
10
votes
1
answer
2k
views
Generating family for the Lebesgue $\sigma$-algebra
Let $X$ be a set, and $\cal F$ a family of subsets of $X$, let $\Sigma(\cal F)$ denote the smallest $\sigma$-algebra containing $\cal F$. We can also define $\Sigma(\cal F)$ internally using a transfi …
- 39.8k
22
votes
2
answers
2k
views
If ZFC has a transitive model, does it have one of arbitrary size?
It is known that the consistency strength of $\sf ZFC+\rm Con(\sf ZFC)$ is greater than that of $\sf ZFC$ itself, but still weaker than asserting that $\sf ZFC$ has a transitive model. Let us denote t …
- 39.8k
20
votes
2
answers
2k
views
What is $\omega_1^{CK}(\mathsf{Ord})$?
We know that if $\alpha<\omega_1^{CK}$ then there is some recursive $R$ such that $(\omega,R)$ has order type $\alpha$.
Let's consider now the ordinals, $\mathsf{Ord}$ with their natural order. This …
- 39.8k
17
votes
1
answer
1k
views
Is there an $L$ like inner model for $\sf Z$?
Godel proved the consistency of the axiom of choice with the axioms of $\sf ZF$ by showing that given any model of $\sf ZF$, there is a definable class which satisfies $\sf ZFC$.
The proof uses a lot …
- 39.8k
4
votes
1
answer
289
views
Can we always add sets without collapsing cardinals or adding [very] bounded sets?
Given a model of $\sf ZFC$, and an infinite ordinal $\alpha$. Can we prove that there is always a cardinal $\kappa$, and a forcing $\Bbb P$, such that:
$\Bbb P$ does not add sets of rank $\leq\alpha …
- 39.8k
13
votes
1
answer
758
views
Do we know the consistency strength of the Singular Cardinal Hypothesis failing on an uncoun...
Suppose that $\kappa$ is a strong limit cardinal. The singular cardinal hypothesis states $2^\kappa=\kappa^+$. We know that the failure of SCH requires large cardinals, and in fact is equiconsistent w …
- 39.8k
5
votes
0
answers
261
views
Generic properties of dominating/etc. reals with non-Cohen working parts
The first and foremost forcing that adds a real is the Cohen forcing which approximates a new real number by giving finite approximations to its characteristics function.
But very quickly after that, …
- 39.8k
13
votes
2
answers
631
views
Adding a real with infinite conditions
Consider the forcing $\Bbb P$ whose conditions are partial functions $p\colon\omega\to2$ with $\operatorname{dom}(p)$ a co-infinite subset of $\omega$, ordered by reverse inclusion.
Does $\Bbb P$ col …
- 39.8k
5
votes
1
answer
282
views
Broken families
Assume $\sf GCH$.
Let $\kappa$ be a regular cardinal, we say that $\{A_\alpha\mid\alpha<\kappa^+\}\subseteq\mathcal P(\kappa)$ is an almost disjoint family, if whenever $\alpha\neq\beta$, $A_\alpha\c …
- 39.8k
3
votes
1
answer
137
views
Preserving distributivity with finite support products
We work in $\sf ZFC+GCH$. Let $D$ be a class of uncountable regular cardinals, and for every $\alpha$ let $\Bbb Q_\alpha$ be either trivial if $\alpha\notin D$, or forcing with these two properties:
…
- 39.8k