What's Gödel's incompleteness theorem? And what importance does it have?
Gödel's incompleteness theorem
Caleb Foster
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Michael Hughes
Basically says for a system of axioms sufficiently powerful enough to express arithmetic there are true statements that cannot be proven in the system. It's a mathematical result on the limitation of foundational issues.
Josiah Wright
>there are true statements that cannot be proven in the system
should be
>there are statements in the language of that system which cannot be proven/disproven
Jaxon Hall
Can you provide an example?
Ethan Powell
There are statements about the natural numbers that are true, which cannot be proven within the language. That is not incorrect.
Eli Baker
The continuum hypothesis if i remember correctly
Josiah Bailey
Godel formalized the notation of the liar's paradox. Look into:
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To learn more
Owen Murphy
This statement is false.
James Bailey
What does his theorem even mean in real life?
Ethan Butler
It would be difficult to build a bridge from this kind of mathematics to areas one thinks of as "real life."
It's not at all clear to me what such a bridge is made out of, but some have attempted to apply it to philosophy of mind by giving formal descriptions to consciousness (ex: the mind is unable to prove or disprove certain statements about reality because it is, in some sense, like one of the systems Godel considers)