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> The propositions themselves are not inaccessible Yes, I know. I meant that their truth value is inaccessible. What I'm saying is that claiming 'some things a

> The propositions themselves are not inaccessible Yes, I know. I meant that their truth value is inaccessible. What I'm saying is that claiming 'some things are indeed undecidable (even if they must be true or false)' suggests the existence of a Platonic realm with an existing truth value. If a proposition is undecidable, it makes no sense to talk about whether 'it actually is true/false', because truth/falsehood is attached to a proposition by virtue of proving/disproving the proposition.

zozbot234 7y ago
> truth/falsehood is attached to a proposition by virtue of proving/disproving the proposition. The thing is, this isn't necessarily the case. "Proving/disproving" a proposition can only be done starting from a set of axioms that are independently thought of as "true". But if you think of, e.g. ZFC as true, you'll also think of Con(ZFC) as true, even though the statement of Con(ZFC) can't be proven from ZFC. Similarly for Con(Con(ZFC)) and many sorts of increasingly-complex "consistency" statements.
Scarblac 7y ago
The law of the excluded middle says that for all propositions, either it or its negation must be true. But Godel showed that there exist propositions for which neither can be proven.
- ProfHewitt 7y ago
  The Law of Excluded Middle says: ⊢(Ψ or ~Ψ) In inferential incompleteness says that there is a proposition Ψ such that: (⊬Ψ) and (⊬~Ψ)
lou1306 7y ago
Ok, now I see your point. It seems you're advocating for intuititionistic logic: you only consider a proposition "true" if you can build a proof for it. Classical logics (and, therefore, classical mathematics) assumes that it makes perfect sense to talk about the truth value of a proposition without computing/proving it. Even undecidable ones are either true or false.
morelisp 7y ago
> the existence of a Platonic realm with an existing truth value. This is a standard assumption by mathematicians.

zozbot234 7y ago

> truth/falsehood is attached to a proposition by virtue of proving/disproving the proposition. The thing is, this isn't necessarily the case. "Proving/disproving" a proposition can only be done starting from a set of axioms that are independently thought of as "true". But if you think of, e.g. ZFC as true, you'll also think of Con(ZFC) as true, even though the statement of Con(ZFC) can't be proven from ZFC. Similarly for Con(Con(ZFC)) and many sorts of increasingly-complex "consistency" statements.

Scarblac 7y ago

The law of the excluded middle says that for all propositions, either it or its negation must be true. But Godel showed that there exist propositions for which neither can be proven.

ProfHewitt 7y ago
The Law of Excluded Middle says: ⊢(Ψ or ~Ψ) In inferential incompleteness says that there is a proposition Ψ such that: (⊬Ψ) and (⊬~Ψ)

lou1306 7y ago

Ok, now I see your point. It seems you're advocating for intuititionistic logic: you only consider a proposition "true" if you can build a proof for it. Classical logics (and, therefore, classical mathematics) assumes that it makes perfect sense to talk about the truth value of a proposition without computing/proving it. Even undecidable ones are either true or false.

morelisp 7y ago

> the existence of a Platonic realm with an existing truth value. This is a standard assumption by mathematicians.