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Not a logician, but it’s the first time i’ve heard Gödel incompleteness being about trying to say something about syntactically invalid formulas.. Do you have m

by bsaul 5y ago

Not a logician, but it’s the first time i’ve heard Gödel incompleteness being about trying to say something about syntactically invalid formulas.. Do you have more pointers on that ? it makes me curious.

laszlokorte 5y ago
„Valid“ does not mean syntactically valid but „true in any case“
triska 5y ago
Every statement S has a negation ¬S, read as "not S". If S is well-formed, so is its negation. For example, if we take the continuum hypothesis, i.e.: "There is no set whose cardinality is strictly between that of the integers and the real numbers." and construct S to express this statement in the language of our theory (say ZFC), then the negated statement ¬S would correspond to: "There is a set whose cardinality is strictly between that of the integers and the real numbers." Famously, the continuum hypothesis is independent of ZFC. This means that (assuming consistency of ZFC), ZFC does not prove S, and also does not prove ¬S. From a consistent and syntactically complete theory, we expect that it contains exactly one of these statements. Is it the case or not? The fact that ZFC does not "tell" us which it is means that it is incomplete in this sense. Clearly, a question that is important to us is left unanswered by this theory, since neither the statement nor its negation is provable with the given axioms. A syntactically complete theory is also called deductively complete, or negation complete. An example of a first-order theory that is consistent, complete, and also decidable is Presburger arithmetic: https://en.wikipedia.org/wiki/Presburger_arithmetic https://en.wikipedia.org/wiki/Presburger_arithmetic Every question you have about its sentences will be answered by the theory one way or the other by either proving the statement or its negation.
- ProfHewitt 5y ago
  The Continuum Hypothesis is still an open question in foundational mathematical theories that are more powerful than 1st-order ZFC such as the theory Ordinals. See the following: https://papers.ssrn.com/abstract=3457802 https://papers.ssrn.com/abstract=3457802 Someday, the Continuum Hypothesis might be proved or it might be disproved in foundational theories.
contingencies 5y ago
https://en.wikipedia.org/wiki/When_a_White_Horse_is_Not_a_Horse https://en.wikipedia.org/wiki/When_a_White_Horse_is_Not_a_Ho...

by bsaul 5y ago

laszlokorte 5y ago

„Valid“ does not mean syntactically valid but „true in any case“

triska 5y ago

Every statement S has a negation ¬S, read as "not S". If S is well-formed, so is its negation. For example, if we take the continuum hypothesis, i.e.: "There is no set whose cardinality is strictly between that of the integers and the real numbers." and construct S to express this statement in the language of our theory (say ZFC), then the negated statement ¬S would correspond to: "There is a set whose cardinality is strictly between that of the integers and the real numbers." Famously, the continuum hypothesis is independent of ZFC. This means that (assuming consistency of ZFC), ZFC does not prove S, and also does not prove ¬S. From a consistent and syntactically complete theory, we expect that it contains exactly one of these statements. Is it the case or not? The fact that ZFC does not "tell" us which it is means that it is incomplete in this sense. Clearly, a question that is important to us is left unanswered by this theory, since neither the statement nor its negation is provable with the given axioms. A syntactically complete theory is also called deductively complete, or negation complete. An example of a first-order theory that is consistent, complete, and also decidable is Presburger arithmetic: https://en.wikipedia.org/wiki/Presburger_arithmetic https://en.wikipedia.org/wiki/Presburger_arithmetic Every question you have about its sentences will be answered by the theory one way or the other by either proving the statement or its negation.

ProfHewitt 5y ago
The Continuum Hypothesis is still an open question in foundational mathematical theories that are more powerful than 1st-order ZFC such as the theory Ordinals. See the following: https://papers.ssrn.com/abstract=3457802 https://papers.ssrn.com/abstract=3457802 Someday, the Continuum Hypothesis might be proved or it might be disproved in foundational theories.

contingencies 5y ago

https://en.wikipedia.org/wiki/When_a_White_Horse_is_Not_a_Horse https://en.wikipedia.org/wiki/When_a_White_Horse_is_Not_a_Ho...