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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.
2
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Consistency of bounded finitely axiomatized set theories [closed]
If $T$ is a consistent first order finitely axiomatized set theory having an axiom that defines V and the rest of axioms are all of the form:
$\forall x1 ..xn \in V \exists x \in V \forall y \in V ( …
- 11.3k
1
vote
0
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365
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Ultraconsistency & Truth
Let $D$ be an effectively generated set of recursive ordinals that contains $0$ and that is closed under an effectively defined ordinal successor function "$+1$" and that have a well defined relation …
- 11.3k
5
votes
1
answer
353
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IF we weaken comprehension of second order logic to first order formulas, would the resultin...
Take second order logic, weaken the comprehension axiom schemata to using only FIRST order formulas; that is, $\phi(x_1,..,x_n)$ in the referred article is restricted to be a first order formula. Keep …
- 11.3k
-3
votes
1
answer
168
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Can there be an effectively generated consistent theory that extends PA and be consistently ...
[The question has been edited]
Can we have an effectively generated consistent theory $T$, that extends $\sf PA$, such that:
$T + \\ \forall \mathcal S \, [\exists x: \operatorname {Proof}_T(x, \ulcor …
- 11.3k
-2
votes
1
answer
183
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Can this Rosser-like trick also work as a proof of the first incompleteness theorem?
The context of this question is related to proving the first incompleteness by alternative ways related to Rosser's trick. So, for a proof by negation, we assume that $T$ is complete, and fulfills Göd …
- 11.3k
0
votes
2
answers
309
views
Can we have a theory $T$ that is complete for simple sentences in the language of $T$ that a...
Let's denote a sentence $P$ as "weak Godel sentence of theory $T$", if and only if
$$[\neg (T \vdash P) \wedge \neg (T\vdash \neg P)] \wedge [Con(T)=Con(T+P) \wedge Con(T)=Con(T+ \neg P)] $$
In Engli …
- 11.3k
6
votes
1
answer
669
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Can this provide an example of incompleteness under the assumption of mere consistency?
A try to capture the informal notion of "this sentence is not provable in less than $n$-many steps" where $n$ is a concrete natural.
Use the diagonal lemma to coin the following sentences, per each co …
- 11.3k
4
votes
1
answer
818
views
Why adhere to $\omega$-consistency with respect to Godel's proof of first incompleteness?
On MSE I've asked a question about why did Godel assume the theory in question to be $\omega$ consistent [on top of effectiveness] for his proof [actually the second part of his proof] of first incomp …
- 11.3k
8
votes
2
answers
1k
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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: $$ G_T \iff \neg \exists x: \operatorname{Proof}_T(x,\ulcorner G_T \urcorner)$ …
- 11.3k
4
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2
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334
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Is there a consistent, unsound, $\omega$-inconsistent, effective theory that doesn't prove i...
Can we have a consistent and effective (fulfilling Godel's criteria) first order theory $T$, that is both arithmetically unsound and $\omega$-inconsistent, and yet doesn't prove its own inconsistency …
- 11.3k
3
votes
1
answer
189
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Can we re-write every effective first order theory using finitely many primitives?
Let $T$ be an effectively generated (recursively enumerable) theory written in a first order language that has infinitely many extra-logical primitives.
Is it always the case that there is a theory …
- 11.3k
3
votes
0
answers
255
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Can ZF be interpreted in a theory axiomatized by a version of replacement and infinity?
Dana Scott had once proved that Zermelo's set theory $``\text{Z}"$ can be interpretable in the first order set theory whose axioms are just the axioms of:
Separation: if $\phi$ is a formula in which …
- 11.3k
5
votes
0
answers
427
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What is the shortest expression of finiteness? [closed]
What is the shortest definition of "$x$ is a finite class" that can be formulated in the class theory presented at:
Is it possible to derive the rules of set theory as transfers from the pure finite …
- 11.3k
2
votes
1
answer
641
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What is the consistency strength of definable axiomatization of ZFC?
Let $x^\phi$ be the set $x$ that is definable after the parameter free formula $\phi$, i.e. formally we have: $$\forall y (y \in x^{\phi} \leftrightarrow \phi(y))$$
Now by $\text{definable ZFC}$ it is …
- 11.3k
0
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1
answer
435
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Can cardinality be defined with essentially no practical restriction on non-well-ordered com...
Question: Can we have a model of $ZF-\text {Regularity}$ where there exist an ordinal $\kappa$ such that $H_{\kappa}$ exists and $H_{\kappa}$ is not equinumerous to any well founded set?
The moti …
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