
There is a scientific paper with a proof based on its own existence - msuvakov
https://zenodo.org/record/1183891#.WpSME3WYW00
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alehul
This may be a very broad question, but has anyone else thought about whether
our basic fundamentals of logic are flawed, and if so, how?

Wouldn't a logically sound system be devoid of any paradox?

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tensor_rank_0
there was a guy once who proved that any system is either capable of paradox
or reliant on external support. look up Kurt Godel.

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mikejb
Gödel's incompleteness theorems are really something. Ultimately (iirc) it
boils down to this:

If an axiomatic system is complex enough, it cannot be both consistent and
complete.

I.e. either you have statements that you can't prove despite them being true,
or your system is inconsistent (i.e. you can prove false statements).

That is a simple description of the first incompleteness theorem. The 2nd
incompleteness theorem goes along similar lines:

If an axiomatic system is complex enough and consistent, it can't prove it's
own consistency.

I.e.: If a system can prove it's own consistency, it isn't consistent.

Regarding how complex an axiomatic system has to be: Simple arithmetic is
enough.

(FYI: it has been a while since I've spent time with the incompleteness
theorems. No guarantees on accuracy, and correcting comments are welcome :) )

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tensor_rank_0
thanks for the breakdown. the example of a Godel statement is usually given in
terms of set theory. ("S is the set of all sets that do not contain
themselves. Does S contain itself?") or binary logic. ("this statement is
false"). Do you know of an example in arithmetic?

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cgmg
The examples you mentioned are not Gödel sentences, but rather Russell’s
paradox and the liar paradox, respectively. A Gödel sentence for a theory T
says “This sentence is not _provable_ by T.” Truth is very different from
provability and, unlike provability, cannot be represented as an arithmetic
formula.

See the section titled “Relationship with the liar paradox” in the Wikipedia
article on the incompleteness theorems.

------
rspeer
"Hey, why do YOU get to be the president of Tautology Club-- wait, I can
guess."

-xkcd

