The two notion (completeness and incompleteness) are strictly connected (not only by Godel's name in the name of the two theorems).

Please, take in account that Godel's Completeness Th of First-Order Logic is :
if a sentence is true in all the models of the axioms (i.e. it is a logical consequence of the axioms) then it is also formally deducible (in FOL) by the axioms.

Godel's Incompleteness Th is relative to formal systems containing "a certain amount" of arithmetic (for example : Robinson Arithmetic, that is weaker than Peano's) and says that we can find in an effective way a statement expressible in those formal systems that is "true" in the intended model (i.e. the model with domain the standard numbers and operation the standard addition and multiplication) but not deducible from the axioms.

This does not contradict the Completeness Th : the aforasaid statement is true in the standard model but is NOT true in some other "strange" model (there are many): this is the reason way it is not deducible from the said axioms.

The arithmetical statement constructed by Godel in his proof is quite "strange", but starting form a result of Paris & Harrington (1977) has been possible, in mathematical logic, to find statements that are true but not provable in Peano arithmetic and are more "natural". This was the first "natural" example of a true statement about the integers that could be stated in the language of arithmetic, but not proved in Peano Arithmetic.

The two notions (completeness and incompleteness) are not opposites but very much connected (not only by Godel's name in the name of the two theorems).

Do take into account that Godel's Completeness Th of First-Order Logic is :
if a sentence is true in all the models of the axioms (i.e. it is a logical consequence of the axioms) then it is also formally deducible (in FOL) by the axioms.

Godel's Incompleteness Th is relative to formal systems containing "a certain amount" of arithmetic (for example : Robinson Arithmetic, that is weaker than Peano's) and says that we can find in an effective way a statement expressible in those formal systems that is "true" in the intended model (i.e. the model with domain the standard numbers and operation the standard addition and multiplication) but not deducible from the axioms.

This does not contradict the Completeness Th : the aforesaid statement is true in the standard model but is NOT true in some other "strange" model (there are many): this is the reason it is not deducible from the said axioms.

The arithmetical statement constructed by Godel in his proof is quite "strange", but starting from a result of Paris & Harrington (1977) has been possible, in mathematical logic, to find statements that are true (in the standard model) but not provable in Peano arithmetic and are more "natural". This was the first "natural" example of a true statement about the integers that could be stated in the language of arithmetic, but not proved in Peano Arithmetic.

The two notion (completeness and incompleteness) are strictly connected (not only by Godel's name in the name of the two theorems).

Please, take in account that Godel's Completeness Th of First-Order Logic is :
if a sentence is true in all the models of the axioms (i.e. it is a logical consequence of the axioms) then it is also formally deducible (in FOL) by the axioms.

Godel's Incompleteness Th is relative to formal systems containing "a certain amount" of arithmetic (for example : Robinson Arithmetic, that is weaker than Peano's) and says that we can find in an effective way a statement expressible in those formal systems that is "true" in the intended model (i.e. the model with domain the standard numbers and operation the standard addition and multiplication) but not deducible from the axioms.

This does not contradict the Completeness Th : the aforasaid statement is NOT true in the standard model but is true in some other "strange" model (there are many).

The arithmetical statement constructed by Godel in his proof is quite "strange", but starting form a result of Paris & Harrington (1977) has been possible, in mathematical logic, to find statements that are true but not provable in Peano arithmetic and are more "natural". This was the first "natural" example of a true statement about the integers that could be stated in the language of arithmetic, but not proved in Peano Arithmetic.

The two notion (completeness and incompleteness) are strictly connected (not only by Godel's name in the name of the two theorems).

Please, take in account that Godel's Completeness Th of First-Order Logic is :
if a sentence is true in all the models of the axioms (i.e. it is a logical consequence of the axioms) then it is also formally deducible (in FOL) by the axioms.

Godel's Incompleteness Th is relative to formal systems containing "a certain amount" of arithmetic (for example : Robinson Arithmetic, that is weaker than Peano's) and says that we can find in an effective way a statement expressible in those formal systems that is "true" in the intended model (i.e. the model with domain the standard numbers and operation the standard addition and multiplication) but not deducible from the axioms.

This does not contradict the Completeness Th : the aforasaid statement is true in the standard model but is NOT true in some other "strange" model (there are many): this is the reason way it is not deducible from the said axioms.

The arithmetical statement constructed by Godel in his proof is quite "strange", but starting form a result of Paris & Harrington (1977) has been possible, in mathematical logic, to find statements that are true but not provable in Peano arithmetic and are more "natural". This was the first "natural" example of a true statement about the integers that could be stated in the language of arithmetic, but not proved in Peano Arithmetic.