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I can't do it in the general form, but I can do it in a special case without any problems.
Take Rossel's formulation of the first incompleteness theorem: “if a formal system is consistent, then it is incomplete, that is, both formulas A and A belonging to this formal system are not deducible in it.” Now let us take the formal logic of first-order predicates in the semantics of an “ordinary” language. Let us formulate statement A as “formula A is not deducible by means of our system”. Now let's see what happens: if we take A for true, then the question is solved – A is not deducible, if we take�A for true, then it turns out that the formulaA is deducible, hence the formula A is not deducible, that is, a contradiction. Thus, A is true and thus non-deducible, and A leads to a contradiction with the deducibility of itself and thus is also non-deducible in a consistent system. You will easily recognize the so-called “liar's paradox” in the statement under consideration and you will be right. In fact, any such paradox can serve as an example and at the same time a proof of Godel's first incompleteness theorem for first-order theories in the semantics of everyday language.
The “real” proof of the first incompleteness theorem is approximately the same, only a clear algorithm (more precisely, algorithms – more than one proof) is given for constructing such a formula for a wide class of formal systems, and an idea is also given of which systems belong to this class (“any sufficiently strong recursively axiomatizable consistent first-order theory”).
If we have proved the first theorem, then the second one is already easier; in fact, it is practically contained in the first one. Formulation: “if a formal system is consistent, then a formula that meaningfully asserts the consistency of this system is not deducible in it.” Let's construct the formulaB, which expresses the impossibility of deducing any formula in our system together with its negation (that is, the condition of consistency of our system). Then the statement of the first theorem is expressed as�Where A is our non-deducible formula from the first theorem. All arguments for the proof of the first theorem can be expressed and carried out by means of our formal system (this is the essence of the “real” proof of the first theorem), that is, the formula can be derived in our systemB ⊃ A. Then if the output isB, then output and A. However, according to Godel's first theorem, if our system is consistent, then A in it is non-deducible. Therefore, if our system is consistent, then the formula is not deducible in itV.