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After I read the Gödel's incompleteness theorems, I am confused about the following:

Could you tell me if the conclusions of mathematical statistics are reliable? Gödel's incompleteness theorems indicate to me that some basic concepts are wrong - Any effectively generated theory capable of expressing elementary arithmetic cannot be both consistent and complete - for this reason, especially that high-lighted.

I know some examples of undecidable statements which is excerpted from Wikipedia as below, I think there are differences between these problems and the conclusions of mathematical statistics. Right? Could you give more details about this?

The combined work of Gödel and Paul Cohen has given two concrete examples of undecidable statements (in the first sense of the term): The continuum hypothesis can neither be proved nor refuted in ZFC (the standard axiomatization of set theory), and the axiom of choice can neither be proved nor refuted in ZF (which is all the ZFC axioms except the axiom of choice). These results do not require the incompleteness theorem. Gödel proved in 1940 that neither of these statements could be disproved in ZF or ZFC set theory. In the 1960s, Cohen proved that neither is provable from ZF, and the continuum hypothesis cannot be proven from ZFC.

In 1973, the Whitehead problem in group theory was shown to be undecidable, in the first sense of the term, in standard set theory.

Gregory Chaitin produced undecidable statements in algorithmic information theory and proved another incompleteness theorem in that setting. Chaitin's incompleteness theorem states that for any theory that can represent enough arithmetic, there is an upper bound c such that no specific number can be proven in that theory to have Kolmogorov complexity greater than c. While Gödel's theorem is related to the liar paradox, Chaitin's result is related to Berry's paradox.

Thank you in advance!

@TrevorWilson What I worrried about is , for example , is the combined results of a meta-analysis of 38 medical studies investigating whether aspirin helps reduce heart attacks reliable? The combined results of a meta-analysis of 38 medical studies investigating whether aspirin helps reduce heart attacks - this bold part as I said is an example of the conclusions of statistics .

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You can relax. "Elementary arithmetic" is a technical term that has essentially nothing to do with elementary arithmetic. –  André Nicolas Oct 7 '13 at 2:35
    
Could you clarify what you are worried about? Are you worried that incompleteness might show up in statistics (some mathematical statement of interest to statisticians might turn out not to be provable or refutable from the current axioms of mathematics) or are you worried that the current axiom system for mathematics might turn out to be inconsistent and that this would invalidate some results in the field of statistics? –  Trevor Wilson Oct 8 '13 at 16:20
    
What "conclusions" of mathematical statistics are you talking about? The incompleteness and undecidability results you are concerned about essentially say "certain mathematical methods of reasoning are either inconsistent or are not capable of settling every question to which they might be applied". This is important in mathematical logic and in the foundations of mathematics, but completely irrelevant to the applications of mathematics. –  Rob Arthan Oct 9 '13 at 21:47
    
@TrevorWilson What I worrried about is , for example , is the combined results of a meta-analysis of 38 medical studies investigating whether aspirin helps reduce heart attacks reliable? The combined results of a meta-analysis of 38 medical studies investigating whether aspirin helps reduce heart attacks - this bold part as I said is an example of the conclusions of statistics . –  Lincoln Oct 10 '13 at 3:05
    
@RobArthan The "conclusions" of mathematical statistics that I am talking about is expressed above for Trevor Wilson's comments. Thank you for your comments too. –  Lincoln Oct 10 '13 at 3:14
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The problem of reliability of statistical methods is not so much related to Goedel's theorem (or any similar theorem) but the fact that statictical studies and hypothesis testing should be based on a "representative sample" in order to produce reliable statistics.

Goedel's theorem is a result of the fact that a formal theory which axiomatizes an efficient part of arithmetic is not complete (there can be statements which are valid syntacticaly in the theory but neither the "truthiness" nor "falsity" can be proved in the theory, they are not provable)

For an alternative inquiry on Goedel's 1st theorem (although unconventional) check this question of mine (of course many more questions clarifying Goedel's results on math.stackexhange)

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