Re: What does Gödel's Incompleteness mean for the Working Mathematician?


george wrote:
|Timothy Murphy wrote:
|>
|> Surely it is normal usage to say
|> that theorems in a formal system
|> are true?

Only when there's an independent notion of truth
for the sentences that all the theorems satisfy.

|It's not normal usage to say "formal system"
|at all (EVEN though it says it on my transcript).
|"Formal system", like "set", is a completely INformal
|term!

Oh, I don't know about "completely" informal. There
are various ways of presenting them, but they are
usually equivalent to describing a computable
relationship "X proves Y" between strings X and Y.

|People WHO KNOW WHAT THEY ARE TALKING ABOUT
|can say that something that is a theorem of
|a formal theory is true (full stop) because THEY
|know that what has been elided (between the true and
|the stop) is "in all models". YOU, on the other hand,
|could NOT say that because you DIDN'T know that.

Sometimes there's an intended model, or otherwise
an intended interpretation of the sentences. For
example, we can define the set of truths of elementary
arithmetic, and consider whether the theorems of a
formal system are a subset of that set of truths.

Keith Ramsay

.

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