An Unprovable Theorem in Mathematics

My previous blog (Top/Ten Science Topics/Issues) was probably too general to get any comments (reactions from any readers). What about this approach - one unproven theorem in the arena of mathematics.

I started going to the local library recently to take out books on the science subjects of interest. One such book not only contained science issues, but also contained math issues. The book was
"Beyond Reason", A. K. Dewdney, Wiley Publisher, copyright 2004, On page 138 the auther was mentioning an unproven theory referred to as Goldbach's conjecture. When mathematicians encounter a statement that they think might be true, they call it a conjecture. If someone proves it to be true, the statement becomes a theorem. They can prove it untrue by showing just one counter example.

The conjecture is as follows: "Every even number greater than two is the sum of two prime numbers. Recall that a prime number is any natural number greater than one and divisible only by itself and the number one. The first 20 prime numbers are 2, 3, 5 , 7, 11, 13, 17, 19, 23, 29, 31, 37, 41 43, 47, 53, 59, 61, 67 , and 71 .

Thus, some example of this conjecture are:
(1) 4 = 2+ 2
(2) 6 = 3 + 3
(3) 8 = 3 + 5
(4) 10 = 5 + 5 = 3 + 7
(5) 12 = 5 + 7

What I find interesting is that after 250 years, mathematicians are unable to prove (to make a theorem) or find a counter example. Computer searches have shown the first 10 to the fourteenth power ( one followed by 14 zeros or 100 trillion calculations) have failed to find a counter example.

Either there exists unprovable theorems or the standard arithmetic humans have been using since they started to count, is inconsistent. This is the author's conjecture. I find it remarkable that there exists unproven theorem, given the human mind.