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Results tagged with lo.logic
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user 95347
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.
14
votes
1
answer
678
views
Can $L$ be defined without parameters?
If we omit parameters in the definition of $L$ would the result still be $L$?
That is, we define a successor stage $L_{\alpha+1}$ in the constructible universe $L$, without including parameters; as: …
- 11.3k
12
votes
1
answer
679
views
Does synonymy seep down to the fragments of theories?
IF we have a synonymous interpretation between two theories $T$ and $H$ that uses translation $\tau$ from the language of $T$ to the language of $H$. Then I'd expect that for a sentence $\mu$ in the …
- 11.3k
12
votes
4
answers
1k
views
Is this theory synonymous with PA?
Language: Mono-sorted first order logic with equality.
Extralogical Primitives: $<, \in$
Define: $x \leq y \iff x < y \lor x=y$
$\textbf{Well ordering: }\\\textit{Transitive:} \ x < y \land y < z \t …
- 11.3k
11
votes
1
answer
528
views
Is every set being cardinal definable consistent with ZF + negation of Choice?
Recall the definition of cardinal definable, where every set being cardinal definable is proved consistent relative to ZF + V=HOD. To re-iterate it:
$Define: X \text { is cardinal definable} \iff \\\ …
- 11.3k
11
votes
1
answer
1k
views
Had this attempt to salvage naïve comprehension been studied before?
Is the following a possible way to overcome inconsistency with naive comprehension:
We add an $\in_n$ symbol for each natural $n$ to the signature of this theory, which is a first order theory with eq …
- 11.3k
11
votes
2
answers
2k
views
Can GCH fail everywhere every way?
The following question is about if it is compatible to add to $\sf ZF$ an axiom asserting the existence of a countable transitive model of $\sf ZF$ such that for every strictly increasing function $f$ …
- 11.3k
10
votes
1
answer
318
views
Is choice over definable sets equivalent to AC over axioms of ZF-Reg.?
If we add the following axiom schema to ZF-Reg., would the resulting theory prove $\sf AC$?
Definable sets Choice: if $\phi$ is a formula in which only the symbol $``y"$ occurs free, then:
$$\forall X …
- 11.3k
10
votes
2
answers
1k
views
What's the exact consistency strength of this axiom system for classes and sets?
Notation: Let $\phi$ be any formula in $\mathsf{FOL}({=},{\in}, W)$; let $\varphi$ be any formula in $\mathsf{FOL}({=},{\in})$ having $x$ free, and whose parameters are among $x_1,\dotsc,x_n$.
Note: “ …
- 11.3k
10
votes
2
answers
789
views
Does every consistent extension of ZF have a model in the minimal transitive model of ZFC?
Suppose there exists a transitive model of $\sf ZFC$. Is it the case that every consistent theory that extends $\sf ZF$ must have a model that is an element of the minimal transitive model of $\sf ZFC …
- 11.3k
9
votes
2
answers
461
views
Is it consistent to have a function that is sensitive to subset relation from the power set ...
Is it consistent with $ZF$ to have a set $S$ and a function $F: P(S) \to S$ such that:
$\forall X,Y \in P(S): X \subsetneq Y \implies F(X) \neq F(Y)$
- 11.3k
8
votes
2
answers
1k
views
Can we effectively axiomatize a theory that proves the negation of its own Gödel's sentence?
For any consistent and effective theory $T$ fulfilling Gödel's criteria, let $G_T$ be the Gödel sentence of $T$, that is: $$ G_T \iff \neg \exists x: \operatorname{Proof}_T(x,\ulcorner G_T \urcorner)$ …
- 11.3k
8
votes
1
answer
1k
views
Is there a form of choice that can elude Kunen's inconsistency theorem?
When it is said that Kunen inconsistency theorem proves that given $\sf ZFC$ no elementary embedding can exist from the universe to itself. Most references quote full choice in stating that result, an …
- 11.3k
8
votes
1
answer
351
views
Is "every infinite set of strictly subnumerous sets is supernumerous to its union" equivalen...
Is the following sentence equivalent to $\sf AC$ over the rest of axioms of $\sf ZF$?
For each infinite set $X$: if for all $y \in X$ we have $|y| < |X|$, then $| \bigcup X|\leq |X| $?
Note: The c …
- 11.3k
7
votes
2
answers
654
views
Can Category theory be founded in set theory using worldly cardinals instead of inaccessibles?
What is exactly demanded for a set theoretic foundation of Category theory? I saw generally two main approaches. One is Muller's who did the work in Ackermann's set theory, but his criteria seem to hi …
- 11.3k
7
votes
1
answer
327
views
Can we have mutual elementary embeddability between distinct transitive sets?
Is it consistent with $\sf ZFC$ to have mutual elementary embeddability between distinct transitive sets?
Formally, is the following theory consistent? $${\sf ZFC} + \exists M \exists N: M,N \text { a …
- 11.3k