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Questions tagged [uncountability]

45 questions
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Show that there are infinitely more problems than we will ever be able to compute

I was looking at this reading of MIT on computational complexity and on minute 15:00 Erik Demaine embarks on a demonstration to show what is stated in the title of this question. However I cannot follow his reasoning, in practice what he says is…
Yamar69
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Does there exist any work on creating a Real Number/Probability Theory Framework in COQ?

COQ is an interactive theorem prover that uses the calculus of inductive constructions, i.e. it relies heavily on inductive types. Using those, discrete structures like natural numbers, rational numbers, graphs, grammars, semantics etc. are very…
HdM
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How to determine if a set is countable or uncountable?

When I am presented with a problem of finding whether or not the given set is countable, I cannot figure out how to determine it or prove it. The general approach is to compare it with $\mathbb{N}$, but how can one prove that The set of all…
hxdshell
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Are any "standard" complexity classes uncountably infinite?

(This is a somewhat fuzzy question.) I believe that most of the "standard" complexity classes that one comes across in complexity theory are countably infinite, because they are defined in terms of decision problems that can be solved by Turing…
tparker
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Where am I wrong?: "countability" and "recursive enumerability"

I have a a few fundamental doubts in recursive enumerability and countability and below, I have written what I understand them to be with proofs. But there are contradictions at the end. What is wrong with the statements/proofs i have…
swanar
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Showing that the number of primitive-recursion programs for each function is countably-infinite

Problem Statement Prove that if a function $f$ is primitive recursive, then there are countably infinite number of primitive recursive definitions of $f$ Yes, this is a homework question. My Work I proved that there are infinite number of…
Banach Tarski
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Are all finite strings over some infinite alphabet countable?

Over some infinite alphabet $\Sigma$, can we state that the set of all possible finite strings is countable?
Joezer
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Is repetition the origin of countability?

The original question was "Do all non-regular languages have an uncountable number of strings?". How can someone prove that..? I am squeezing my head but I can't figure it out. And the other side of the coin: is a language always regular, if it…
Nick
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Is the set of all DFAs countable?

Let $\Sigma$ be a finite nonempty alphabet. Is the set of all DFAs over $\Sigma$ countable? I know the set of all regular languages is countable, however, it is impossible to build an injection from the DFAs to the regular languages since a regular…
Kantig Shoter
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What is the relation between countability and recursive enumeration?

Does recursive enumeration implies countability? Does countability implies recursive enumeration? I believe the first implication holds but not sure about the second. A good example would suffice.
Kishan Kumar
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Cantor's diagonal method in simple terms?

Could anyone please explain Cantor's diagonalization principle in simple terms?
user5507
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Decidability of Unary Languages / One-to-One Mapping

I'm trying to prove that there exists an undecidable subset of {1}* by showing a one-to-one correspondence between it and {0, 1}* (which would imply a one-to-one correspondence between their power sets), but I'm struggling with how to do the…
user60640
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Is there a non-recursive and uncountable language L?

Does there exist a non-recursive language, L, such that the cardinality of L is uncountable? I would really like an explanation as to why this question is true or false because at the moment, I have no idea. My understanding is that all recursive…
user2619645
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Can a recursive language be uncountable?

Does there exist a recursive language $L$ whose cardinality is uncountable? I would like to have an explanation whether Turing Machine can encode uncountable languages and whether we can use this to reject the initial question.
revisingcomplexity
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Undecidability and Countability

This question is prompted by Undecidable unary languages (also known as Tally languages) How does the countability of a language imply (un)decidability?
twinlakes
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