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Me and my friend came up with a cool game - we take turns in taking some mathematical theorem stated in English and turn it into a symbolic statement. The rules are this: you are only allowed to use symbols from formal logic, and the statement has to be as pedantic and succinct as possible.

Now, the statement he gave me is this:

Any foundation of mathematics has a statement that is true, yet unprovable, and the foundation cannot prove its own consistency.

Now, after thinking about it for a few minutes, I came up with this:

$$\exists \varphi \leftrightarrow \top \left (\mathrm{F}\vdash\varphi \downarrow \mathrm{F}\vdash\ulcorner\neg\varphi\urcorner \right) \wedge \mathrm{F} \nvdash \mathrm{Con}\: \mathrm{F} $$

Here, $\mathrm{F}$ is a foundation of mathematics, '$\varphi$' is a wff of $\mathrm{F}$, '$\top $' a tautology, '$\downarrow$' Pierce's arrow and the quotation marks are not a function returning the Gödel number of $\varphi$, but just quasi-quotation.

What I think is missing:

  1. Quantification over every possible foundation.
  2. Stating that $\varphi$ is a wff of $\mathrm{F}$.

The only way to way to fix this, it seems to me, is to write the statement like this:

$$\forall \mathrm{F} \exists \varphi \in \mathrm{F} \left (\left (\varphi \leftrightarrow \top \right )\wedge\left (\mathrm{F}\vdash\varphi \downarrow \mathrm{F}\vdash\ulcorner\neg\varphi\urcorner \right ) \right) \wedge \mathrm{F} \nvdash \mathrm{Con}\: \mathrm{F} $$

But I'm not sure if the notation is right and the universal quantification is legitimate in this case.

Constantine
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  • What you wrote seems to be saying that F doesn't prove $\varphi$ nor does it prove the number corresponding to $\neg\varphi$ under some encoding. It doesn't really make sense to talk about proving a number. Also, I doubt your use of $\top$ commonly means whatever you're trying to represent with it. Finally, you may want to encode the "any" in "any foundation" somehow. If you want to cheat and see how logicians actually write the first part of the statement formally (though the meaning of "true" is slightly warped), you can see http://math.stackexchange.com/a/980983/26369 . – Mark S. Apr 14 '16 at 17:54
  • @MarkS those quotation marks are quasi-quotation marks (we are being pedantic, remember?), not a function returning the Gödel number of $\varphi$. The rest of your comment I don't quite understand. It would be awesome if you could post an answer with concrete examples. – Constantine Apr 14 '16 at 18:01

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