Let us consider the following two theorems by Gödel:

1) Any consistent formal system $F$ within which a certain amount of elementary arithmetic can be carried out is incomplete; i.e., there are statements of the language of $F$ which can neither be proved nor disproved in $F$;

2) For any consistent system F within which a certain amount of elementary arithmetic can be carried out, the consistency of F cannot be proved in $F$ itself.

Like others, probability theory is also an axiom-based system with 3 axioms and supports minimum arithmetic.

From the above information, it looks like that probability theory also suffers from Gödel's incompleteness theorem. Can't it? If not, then where I am ging wrong?

Thus is there a possibility that there may exist two statements in probability theory that contradict each other and existence of unprovable facts?

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    $\begingroup$ 1) ... provided that the system is consistent and is "sufficently strong" to include some relevant parts of arithmetic. $\endgroup$ – Mauro ALLEGRANZA Feb 3 at 18:07
  • $\begingroup$ @MauroALLEGRANZA Yep, instead of that I kept some, since my question is on the possibility of point 2 over probability, it may not effect. $\endgroup$ – hanugm Feb 3 at 18:09
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    $\begingroup$ Item 2) is formulated incorrectly. The point is that the proof of consistency cannot be carried out within the system itself. $\endgroup$ – Andrés E. Caicedo Feb 3 at 18:46
  • $\begingroup$ @AndrésE.Caicedo Edited... $\endgroup$ – hanugm Feb 3 at 18:50
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    $\begingroup$ In Reverse Mathematics there has been the identification of an axiom known as WWKL with (Advanced) Probability theory consequences (like existence of fairness) which is independent of the base RCA system, in which basic probability can be formulated. Whether this relates to Godel's Theorems I don't currently know. $\endgroup$ – Roy Simpson Feb 5 at 20:52

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