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First, a remark. You formulate both Gödel's theorem and your question in a subjective way "we cannot prove", etc. However, this theorem is a mathematical one, therefore it is not about our ability to do something, but about the nonexistence of a mathematical object, namely a formal proof within the system concerned. You can draw some e.g. philosophical conclusions from this theorem, but this is a completely another matter.

Now, as far as your question concerned, it is almost certain that nothing analogous to Gödel's theorem can even be stated for the pure first order logic itself. The reason is simple. The analogous theorem would claim the unprovability of the formula expressing the the fact that a contradiction is unprovable within the pure first order logic. But, in the absence of the formal provability predicate, this theorem cannot even be stated. Actually, what we would like to show is that there is a formula $Pr(x)$ such that, on the one hand, it can be considered a provability predicate (that is, for any formula $\varphi$, $Pr(\ulcorner \varphi \urcorner)$ is true just in case $\vdash \varphi$ (here, of course, $\ulcorner \varphi \urcorner$ is the Gödel number of $\varphi$), on the other hand, the formula expressing the fact that a contradiction is unprovable is itself unprovable: $\not\vdash \lnot Pr(\ulcorner 0=1\urcorner)$. Now, the proof of the existence of a provability predicate seems to require much more than pure logic.

2 added 6 characters in body

First, a remark. You formulate both Gödel's theorem and your question in a subjective way "we cannot prove", etc. However, this theorem is a mathematical one, therefore it is not about our ability to do something, but about the nonexistence of a mathematical object, namely a formal proof within the system concerned. You can draw some e.g. philosophical conclusions from this theorem, but this is a completely another matter.

Now, as far as your question concerned, it is almost certain that nothing analogous to Gödel's theorem can even be stated for the pure first order logic itself. The reason is simple. The analogous theorem would claim the unprovability of the formula expressing the the fact that a contradiction is unprovable within the pure first order logic. But, in the absence of the formal provability predicate, this theorem cannot even be stated. Actually, what we would like to show is that there is a formula $Pr(x)$ such that, on the one hand, it can be considered a provability predicate (that is, for any formula $\varphi$, $Pr(\ulcorner \varphi \urcorner)$ is true just in case $\vdash \varphi$ (here, of course, $\ulcorner \varphi \urcorner$ is the Gödel number of $\varphi$), on the other hand, the formula expressing the fact that a contradiction is unprovable is itself unprovable: $\not\vdash \lnot Pr(\ulcorner 0=1\urcorner)$. Now, the proof of the existence of a provability predicate seems to require much more than pure logic.

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First, a remark. You formulate both Gödel's theorem and your question in a subjective way "we cannot prove", etc. However, this theorem is a mathematical one, therefore it is not about our ability to do something, but about the nonexistence of a mathematical object, namely a formal proof within the system concerned. You can draw some e.g. philosophical conclusions from this theorem, but this is a completely another matter.

Now, as far as your question concerned, it is almost certain that nothing analogous to Gödel's theorem can even be stated for the pure first order logic itself. The reason is simple. The analogous theorem would claim the unprovability of the formula expressing the the fact that a contradiction is unprovable within the pure first order logic. But, in the absence of the formal provability predicate, this theorem cannot even be stated. Actually, what we would like to show is that there is a formula $Pr(x)$ such that, on the one hand, it can be considered a provability predicate (that is, for any formula $\varphi$, $Pr(\ulcorner \varphi \urcorner)$ is true just in case $\vdash \varphi$ (here, of course, $\ulcorner \varphi \urcorner$ is the Gödel number of $\varphi$), on the other hand, the formula expressing the fact that a contradiction is unprovable is itself unprovable: $\not\vdash Pr(\ulcorner 0=1\urcorner)$. Now, the proof of the existence of a provability predicate seems to require much more than pure logic.