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Results tagged with lo.logic
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user 23835
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.
5
votes
Accepted
Does PA prove a sentence asserting that all of I-sigma(n) theories are consistent?
No. Given any $\varphi\in \mathcal{L}_{PA}$, if $PA\vdash \varphi$ then there exists $n$ such that $I\Sigma_n\vdash \varphi$ (by finitarity of proofs). So $\forall n con(I\Sigma_n)$ implies for any $n …
- 3,038
6
votes
2
answers
321
views
A question regarding strong cardinals and measure sequence
Let $E$ be a $(\kappa, \lambda)$-extender such that $j: V\to M\simeq Ult(V,E)$ is the corresponding elementary embedding with critical point $\kappa$, $M\supset V_{\kappa+2}$, $M^\kappa\subset M$. Let …
- 3,038
1
vote
A question regarding strong cardinals and measure sequence
I believe the argument could be saved by considering the following ``sub-extender'': For each $\beta<(2^\kappa)^+$, let $a=\bigcup \max\{\beta, \kappa^+\} \cup \bigcup\{a_\gamma: \gamma<\beta\}$ wher …
- 3,038
1
vote
A proof of $ZF \vdash AC^L$
The usual strategy is to define inductively a well-ordering on $L(\alpha)$ for each ordinal $\alpha$. Since under the assumption $V=L$, any set $x \in L (or$ $V), x\subseteq L(\beta)$ for some $\beta …
- 3,038
3
votes
1
answer
394
views
Proof of the existence of hyperimmune-free degrees
In Classical Recursion Theory Vol.I by P.Odifreddi, section V.5 on the Tree Method, the proof for the existence of hyperimmune-frees involves the construction of a series of trees.
Some definitions f …
- 3,038
8
votes
1
answer
519
views
Cohesive sets with degree below some non-high 1-generic degrees?
Terminology:
Cohesive sets: $A\subset \omega$, for each recursively enumerable set $W_e$, either $A\cap W_e$ is finite or $A\cap(\omega\setminus W_e)$ is finite.
Non-high degrees: Degree $a$ such th …
- 3,038
2
votes
Is the fusion argument on trees of uncountable height consistent?
The answer is yes. In fact, more general statements are true. See https://arxiv.org/abs/1704.06827 for more detail (in particular Theorem 3.1).
- 3,038
7
votes
1
answer
305
views
Very weak square and good points
This is probably well known but I'll appreciate pointers to references: Is there any model where for a singular cardinal $\kappa$ of cofinality $\omega$, Very Weak Square holds at $\kappa$ but every …
- 3,038
2
votes
3
answers
391
views
Indices of r.e. sets
The last part of the paper Located Sets and Reverse Mathematics [Journal of Symbolic Logic 65 (1999), 1451–1480] by Giusto and Simpson involves a proof as follows:
Given $A$ an effectively immune set …
- 3,038
8
votes
1
answer
334
views
Is the fusion argument on trees of uncountable height consistent?
In the countable context where we are given a perfect subtree $T$ of $2^{<\omega}$ and a sequence of colorings $f_i: T\to 2, i\in \omega$, it is possible to obtain a perfect subtree $T'\subset T$ and …
- 3,038
2
votes
Can there be an almost-special not-fully-special Aronszajn tree?
To supplement this with another example, it is also possible to construct a tree that is $\omega$-distributive and $S$-st-special (in Shelah's terminology) from $\Diamond^*(S^c)$ for some $S$ bi-stati …
- 3,038
3
votes
Accepted
Indecomposable ordinals and pseudointersection
I believe the claim is wrong:
If the claim is right I claim I can show $\alpha\to (\alpha)^2_2$ which is obviously wrong for countable ordinal $\alpha\geq \omega+2$.
Given a coloring $f: [\alpha]^ …
- 3,038
4
votes
1
answer
235
views
Strong partition property + DC + existence of non-principal ultrafilter on $\omega$
It was mentioned after Theorem 30.27 in Kanamori's Higher Infinite that Woodin constructed a model of $DC$ + there exists unboundedly many many $\kappa<\Theta$ such that $\kappa \to (\kappa)^\kappa_{\ …
- 3,038
6
votes
Accepted
End-extending cardinals
Suppose $\kappa$ carries an $\omega_1$-saturated $\kappa$-complete ideal $I$, given $M\prec (V_{\kappa+2},\in , <)$ ($<$ well orders $V_{\kappa+2}$) of size $<\kappa$ containing $I$, we show how to fi …
- 3,038
2
votes
1
answer
845
views
Absoluteness of Countability
Let M be a countable transitive model for ZFC, P is a partial order in M. Notions like "partial orders" and "dense" are absolute. Consider the following set
$S$={$D\in M: D$ is dense in $P$} = {$D: D$ …
- 3,038