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Is there some finitely presented group $G$ generated by $g_1,...,g_n$ such that there is an element $g\in G$ expressed as a finite word in $g_i$'s so that it is impossible to prove neither $g=1$ nor $g\ne 1$?

Is such a group $G$ exists, what would be a relatively simple example?

Added. As Mark explains the answer to my question is rather trivially "no". So I would like to modify the question as follows:

Adjusted question. Is there $G,g$ so that $g\ne 1$ in $G$ but is it impossible to prove this in finite time?

Edited (this question contains two versions of a similar question)

Is there some finitely presented group $G$ generated by $g_1,...,g_n$ such that there is an element $g\in G$ expressed as a finite word in $g_i$'s so that it is impossible to prove neither $g=1$ nor $g\ne 1$?

Is such a group $G$ exists, what would be a relatively simple example?

Adjusted question. Is there $G,g$ so that $g\ne 1$ in $G$ but is it impossible to prove this in finite time?

Is there some finitely presented group $G$ generated by $g_1,...,g_n$ such that there is an element $g\in G$ expressed as a finite word in $g_i$'s so that it is impossible to prove neither $g=1$ nor $g\ne 1$?

Is such a group $G$ exists, what would be a relatively simple example?

Added. As Mark explains the answer to my question is rather trivially "no". So I would like to modify the question as follows:

Adjusted question. Is there $G,g$ so that is it impossible to prove in finite time that $g\ne 1$?

Is there some finitely presented group $G$ generated by $g_1,...,g_n$ such that there is an element $g\in G$ expressed as a finite word in $g_i$'s so that it is impossible to prove neither $g=1$ nor $g\ne 1$?

Is such a group $G$ exists, what would be a relatively simple example?

Added. As Mark explains the answer to my question is rather trivially "no". So I would like to modify the question as follows:

Adjusted question. Is there $G,g$ so that $g\ne 1$ in $G$ but is it impossible to prove this in finite time?