Timeline for Theorems with unexpected conclusions
Current License: CC BY-SA 4.0
12 events
| when toggle format | what | by | license | comment | |
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| Jul 3, 2022 at 7:59 | history | made wiki | Post Made Community Wiki by Stefan Kohl♦ | ||
| Jun 12, 2022 at 9:23 | history | edited | Martin Sleziak | CC BY-SA 4.0 |
http -> https (the question was bumped anyway)
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| Dec 29, 2016 at 16:35 | history | edited | Joel David Hamkins | CC BY-SA 3.0 |
added 1 character in body
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| Dec 29, 2016 at 16:18 | history | edited | Joel David Hamkins | CC BY-SA 3.0 |
Fixed a typo (missing coefficient of 3) that was bugging me for six years. Also, texified...
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| Dec 29, 2016 at 16:06 | history | edited | Joel David Hamkins | CC BY-SA 3.0 |
Fixed a typo (missing coefficient of 3) that was bugging me for six years. Also, texified...
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| Sep 6, 2015 at 8:22 | comment | added | Andreas Blass | An even more impressive form of Goodstein's theorem is obtained by replacing "increase the base by $1$" at each step by "increase the base as much as you want (to any larger natural number)." The result still holds, with the same proof. I think this may have been Goodstein's original version. The "increase base by $1$" version became better known because, though weaker, it's still strong enough to be unprovable in PA (and it can be formulated in the language of PA). | |
| Apr 25, 2011 at 4:01 | comment | added | luqui | <3 Goodstein's theorem. I hope to one day understand the independence result. | |
| Jun 27, 2010 at 21:43 | comment | added | Andrés E. Caicedo | I once went through the trouble of find an explicit formula for G(n), the number of steps that the process takes to reach 0 starting with n. For example, G(3)=6, as the sequence is 3=2+1,3=(3+1)-1,3=4-1,2,1,0. We have G(0)=1,G(1)=2,G(2)=4,G(3)=6,G(4)=3x2^402653211-2, a number with 121210695 digits. The number of digits of G(5) is much larger than G(4), while (as I am fond of saying at this point) the number of elementary particles in the universe is estimated (well) below 10^90. I don't think one really understands what "fast growing" means until faced with something like this example. | |
| Mar 14, 2010 at 0:02 | comment | added | Joel David Hamkins | There is another sense, however, in which the whole theorem is more than any given case: all of the particular cases are provable in PA. What remains unprovable is the universal statement. | |
| Mar 14, 2010 at 0:01 | comment | added | Joel David Hamkins | I'm glad we agree about Goodstein's theorem, Kevin, and you're right that one can begin to understand the proof by looking at some very small numbers. The general proof proceeds with something like the same idea---you replace all the n's in a_n with the ordinal omega, and interpret the result as a countable ordinal. Now, replacing the omega's with more omega's doesn't do anything, but when you subtract one, you have to break up the ordinal, and the result is a strictly smaller ordinal. So the ordinal complexity goes down, and must eventually hit zero. | |
| Mar 13, 2010 at 22:42 | comment | added | Kevin Buzzard | My favourite example is definitely Goodstein's Theor...*doh*. Yeah, definitely a nice example---you beat me to it :-). Although when you actually write down the proof that the sequence converges to 0 starting at 4=2^2 you begin to get a very clear picture as to why it's true. | |
| Mar 13, 2010 at 21:51 | history | answered | Joel David Hamkins | CC BY-SA 2.5 |