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In summary: Undecidable means that there is no general algorithm that can determine if a given problem is decidable or not. Undecidable problems are those that have no finite or infinite solution, and this includes problems in systems of logic.

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Physics news on Phys.org

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As Turing machines can in principle simulate a human brain: Do we "understand infinity"? If yes Turing machines can do so as well.

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The problem with computing machines is that they can’t always know when to halt and thus are doomed to keep working a problem forever. It’s known as the halting problem and Turing proved that there is no general algorithm that can determine if a given program will halt or not.

https://en.m.wikipedia.org/wiki/Halting_problem

In any event, all halting problem answers are best summed up as 42, he said humorously. :-)

https://en.m.wikipedia.org/wiki/Phrases_from_The_Hitchhiker's_Guide_to_the_Galaxy#Answer_to_the_Ultimate_Question_of_Life,_the_Universe,_and_Everything_(42)

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undecidable = irresolute?? Or does the latter have other connotations?

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Gear300 said:

Well, you don't need infinite resources in order to reason about infinite objects. We can prove things about the natural numbers and the reals, etc., using finitely many axioms.

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Ah? How do our senses record infinities? When is the last time you saw, smelled, touched, tasted, or heard an inaccessible cardinal?Gear300 said:we as humans are intimate with the extant and divisible infinities mainly through our sensory-perception

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jedishrfu said:In my view to understand infinity means to know when to stop because looping forever would serve no purpose.

The problem with computing machines is that they can’t always know when to halt and thus are doomed to keep working a problem forever. It’s known as the halting problem and Turing proved that there is no general algorithm that can determine if a given program will halt or not.

But humans aren't any better. For problems like the Riemann hypothesis or Goldbach's conjecture, they may never know whether eventually they will prove them or refute them, or neither. They never know when to give up. Except when they get hungry or bored.

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nomadreid said:Ah? How do our senses record infinities? When is the last time you saw, smelled, touched, tasted, or heard an inaccessible cardinal?

We had a few in my yard. We could see them feeding but we couldn't touch as they were too skittish and flew away at the slightest sound.

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stevendaryl said:But humans aren't any better. For problems like the Riemann hypothesis or Goldbach's conjecture, they may never know whether eventually they will prove them or refute them, or neither. They never know when to give up. Except when they get hungry or bored.

This is all true but sometimes we get lucky and discover Godel who's shown us that there are always undecidable statements in any system of logic and knowing that will give us pause. Also we can get tired or die from exhaustion and infinity becomes more infinite while others laugh at our folly.

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First, far from "any" system of logic, but that is nit-picking. The more important point is that Gödel (Turing, Rosser, etc.) pointed out how to generate a few undecidables, and a few specific examples were pointed out (Cohen and Gödel, Matiyasevich et al, etc.), but unless a given problem fits into this list (modulo isomorphism or as a problem requiring the solution of an undecidable), or has been proven or disproven, or is part of a system in which all problems are decidable, there is no general procedure to tell if a given problem is undecidable. So we are back to square one, with machines and humans theoretically having the same problems.jedishrfu said:This is all true but sometimes we get lucky and discover Godel who's shown us that there are always undecidable statements in any system of logic.

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nomadreid said:First, far from "any" system of logic, but that is nit-picking. The more important point is that Gödel (Turing, Rosser, etc.) pointed out how to generate a few undecidables, and a few specific examples were pointed out (Cohen and Gödel, Matiyasevich et al, etc.), but unless a given problem fits into this list (modulo isomorphism or as a problem requiring the solution of an undecidable), or has been proven or disproven, or is part of a system in which all problems are decidable, there is no general procedure to tell if a given problem is undecidable. So we are back to square one, with machines and humans theoretically having the same problems.

You are back to square one, I am perfectly fine running in my hamster cage at work. :-)

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Philip Dick wrote a short story "The Infinities" in which hamsters evolve into pure energy and zap a meanie human. Maybe you are on your wayjedishrfu said:I am perfectly fine running in my hamster cage at work.

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nomadreid said:Philip Dick wrote a short story "The Infinities" in which hamsters evolve into pure energy and zap a meanie human. Maybe you are on your way

No, more likely to be attacked by the hamsters for maligning them and their wheel here.

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nomadreid said:Ah? How do our senses record infinities? When is the last time you saw, smelled, touched, tasted, or heard an inaccessible cardinal?

"I saw no God, nor heard any, in a finite organical perception; but my senses discover'd the infinite in every thing ..."

William Blake

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To see a World in a Grain of Sand

And a Heaven in a Wild Flower,

Hold Infinity in the palm of your hand

And Eternity in an hour.

...more stanzas here...

https://www.poetryloverspage.com/poets/blake/to_see_world.html

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