This is a question that I fully reedited to make it more precise. Many thanks to the people that answered the previous version to get this distilled one.
Background: (from http://plato.stanford.edu/entries/independence-large-cardinals/)
1) Let T1 and T2 be recursively enumerable axiom systems. We say that T1 is interpretable in T2 (T1 ≤ T2) when, roughly speaking, there is a translation τ from the language of T1 to the language of T2 such that, for each sentence φ of the language of T1, if T1⊢φ then T2⊢τ(φ). We shall write T1 < T2 when T1≤ T2 and T2≰ T1 and we shall write T1≡ T2 when both T1≤ T2 and T2≤ T1. In the latter case, T1 and T2 are said to be mutually interpretable.
In terms of interpretability there are three possible ways in which a statement φ can be independent of a theory T, two of them are:
2)SINGLE JUMP. Only one of φ or ¬φ leads to a jump in strength, that is, T+φ>T and T+¬φ ≡ T (or likewise with φ and ¬φ interchanged)
3)DOUBLE JUMP. Both φ and ¬φ lead to a jump in strength, that is, T+φ>T and T+¬φ>T.
Argument: Let us make from definition (1): T1 be PA, φ be the Paris–Harrington theorem (known to be independent of PA), T2=T1+φ and T3=T1+¬φ . Also, let the translation τ be the identity (so τ(φ)=φ) . Thus, for every φ' in T1, if T1⊢φ' then T2⊢φ'. But there is a φ'=φ such that T2⊢φ but T1 does not prove φ. Thus T1 < T2 . A similar argument can be done using T3 and φ'= ¬φ. Thus T1 < T3 . Then φ is a case of double jump (definition 3).
Problem: Every body agrees that φ is actually a case of single jump (definition 2)!!
Question: Did I wrongly interpreted any of the definitions? Did I introduced a fallacious argument?
Please try to limit your answer to the specific definitions and argument made above. That is: Did I wrongly interpreted any of the definitions? or, introduced a fallacious argument?
Please keep it simple and try not to answer through a different path, such as by introducing unnecessary (?) concepts (such as model theory, or set theoretical arguments in general), nor self-referential statements such as Con(PA) (or ¬Con(PA) ), which cause me big trouble; and that I will not be able to grasp in the foreseeable future. Thanks!