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Q & A: Gödel's proof

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Most recent answer: 02/06/2013
"A Brief History of Time," Bantam: 1988, 1996; pp. 159 - 160; states, "Godel was a Mathematicianwho was famous for PROVING (emphasis added) that it is impossible to prove all true statements, even if you limit yourself to trying to prove all the true statements in a subject apparently as cut and dried as arithmetic." Question 1: Is not this statement, itself, dichotomous? Question 2. May I therefore assume that one plus one might NOT equal two?
- John (age 71)
Fort Myers, FL, USA
It's not self-contradicting. Gödel showed that any logical system sufficient to generate ordinary arithmetic, including multiplication, would either be:

1. inconsistent


2. incomplete, in the the sense of having statements that could neither be  proved nor disproved.

The ordinary logical systems we use are incomplete, not inconsistent. Incomplete systems still have lots of statements that can be proved, including Gödel's theorem and 1+1=2. They just have some other statements that are "true" but unprovable. In other words, you can prove some true statements but not all true statements.

Here's an example of a statement that can neither be proved nor disproved within the ordinary rules of arithmetic. "There is no set with more elements than the integers but fewer than the reals." That's called "the continuum hypothesis".

Mike W.

(published on 02/05/2013)

Follow-Up #1: Godel again

Mike, Thank you for your prompt response. Unfortunately, I do not seem to have ability to understand the answer. In any event I'm appreciative for your effort. John R.
- John R. (age 71)
Fort Myers, LA, USA
I've tweaked the answer a bit to clarify the key point. Some but not all true statements are provable. Maybe that will help.

Mike W.

(published on 02/06/2013)

Follow-up on this answer.