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accepted Prove that $\lfloor\lfloor x/2 \rfloor / 2 \rfloor = \lfloor x/4 \rfloor$
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comment Are (some) axioms “unprovable truths” of Godel's Incompleteness Theorem?
Maybe I'm using the word "true" incorrectly. Here's an example in my mind. Definition of water: a molecule made of hydrogen and oxygen. This is true/real/accurate. It can be "proven" because it can be seen (or can it? What does it mean to be seen?...). Basically my understanding of Godel's Theorems is that you can keep splitting hairs like this for eternity, but eventually you just simply have to stop and accept something. So I was just using basic axioms as an example, rather than the usual "barber's paradox" and "liars paradox."
Aug
9
comment Are (some) axioms “unprovable truths” of Godel's Incompleteness Theorem?
@James But how can you have a definition if you are unable to prove that what the definition contains is true/real? Or CAN you prove what every definition contains? In which case I'm still stuck on my "0 is a natural number" axiom --- how can you prove that "0" is real, how can you prove that "natural numbers" are real?
Aug
9
comment Are (some) axioms “unprovable truths” of Godel's Incompleteness Theorem?
@PVAL If any axiom has a proof for it in any logical system, what is the proof for the axiom that "0 is a natural number" in Peano's Postulates?
Aug
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asked Are (some) axioms “unprovable truths” of Godel's Incompleteness Theorem?
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accepted Where, specifically, did Principia Mathematica fail?
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asked Where, specifically, did Principia Mathematica fail?
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accepted Help understanding $x=y\Rightarrow(x=z\Rightarrow y=z)$