Questions tagged [axioms]
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156 questions
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Why Zermelo postulated the existence of a set with no finite limit to the ranks of its elements?
The original Zermelo set theory explicitly allowed for urelements.
What was the reason that led Zermelo to formulate the Axiom of Infinity in terms of the existence of a set of the kind that has an ...
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Strength of the incompleteness theorem as an axiom
Has anyone explored an axiomatic system with the incompleteness theorem taken as an axiom?
That is, we would take as an axiom that there is some statement which is unprovable from our axioms.
This ...
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Is this theory of bottomless hierarchy, consistent?
Language: mono-sorted ${\sf FOL}(=,\in,S)$, where $S$ is a unary predicate standing for ".. is a stage".
Axioms:
Extensionality: $\forall z \, (z \in x \leftrightarrow z \in y) \to x=y$
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Addition as the "simplest" operation having certain properties
I once went to a talk by John Conway in which presented his theory of surreal numbers in a different way than the approaches taken in "Surreal Numbers", "On Numbers and Games", or &...
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A special name for Hilbert's Axiom I.7?
The famous Hilbert's Axioms of Geometry include the
Axiom I.7: If two planes have a common point, then they have another common point.
Question 1. Was David Hilbert the first mathematician who ...
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Can this salvage Cyclic Stratified Comprehension?
This is an endeavor to salvage the approach presented at earlier posting.
Is there a clear inconsistency with this axiom schema?
Cyclic Stratified Comprehension: if $\varphi$ is a stratified formula ...
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Is Cyclic Stratified Comprehension inconsistent?
Is there a clear inconsistency with this axiom schema?
Cyclic Stratified Comprehension: if $\varphi$ is a stratified formula in which $``y"; ``x"$ occur free, and only free, and where they cannot be ...
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Can Shannon's Information Theory be axiomatized?
recently I'm rethinking Shannon's Information Theory, which is not perfect enough for many applications. So I want to know has some person try to do some works on axiomatizing Shannon's theory. If not,...
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What's the proof of inconsistency of this almost naive infinite construction schema?
Define: $x= \mathcal P^n(\emptyset) \iff \\\exists x_0, \cdots, \exists x_n: \\x_0= \emptyset\land \cdots x_{i+1}=\mathcal P(x_i)
\cdots \land x=x_n$
Where $\mathcal P(x)=\{y\mid y \subseteq x\}$
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Correspondence for distribution?
Take the distribution axiom, DA, of a modal logic to be:
$\Box(A\to B)\to (\Box A \to \Box B).$
As there are distribution-free modal logics which do not have DA:
What does DA correspond to in frames ...
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Can this extension of ZC evade having distinct bi-interpretable extensions?
What property should an extension of $\sf ZC$ have in order for it to evade having distinct yet bi-interpretable extensions. Which might be seen as a merit by some, foundationally speaking.
Is ...
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What happens if we restrict Replacement to ordinals, but have successor cardinals?
Suppose we weakened Replacement in $\sf ZFC$ to the ordinals only, that is the formula $\phi(x,y)$ in Replacement scheme must be from ordinals to ordinal, so to the usual formulation of replacement ...
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At which stage of the cumulative hierarchy this iterative theory stops?
The theory axiomatized by the following.
Specification: $\forall a \exists! x \forall y \, (y \in x \leftrightarrow y \in a \land \phi)$, as long as "$x$" doesn't occur free in formula $\...
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Can this iterative set theory capture ZFC?
Define: $\square(s) \iff \exists \kappa: s=V_\kappa \land |V_\kappa|=\kappa $
$\square(s)$ to be read as $s$ is a square stage of the cumulative hierarchy; meaning that its width (i.e. cardinality) ...
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Is it possible to add a new axiom schema to classical propositional logic?
Let IPL mean the intuitionist propositional calculus. One can add a great diversity of axiom schemas to obtain intermediate logics between IPL and CPL, where CPL is the classical propositional ...
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