Timeline for Interesting meta-meta-mathematical theorems?
Current License: CC BY-SA 3.0
17 events
| when toggle format | what | by | license | comment | |
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| Nov 15, 2016 at 19:28 | comment | added | user99916 | that in PM one can express statements like "PM is incomplete"? | |
| Nov 15, 2016 at 19:27 | comment | added | user99916 | @FrançoisG.Dorais: You write "note that Gödel's Incompleteness Theorems were originally meta-meta-theorems: Gödel proved that the formal system of Principia Mathematica (PM) was incomplete and PM was intended by Russell and Whitehead as the foundation of all mathematics, i.e., the ultimate meta-theory." Do you really think that Russell and Whitehead intended PM to be a meta theory. I guess Gödel was the first one who realized that theories of arithmetic (and thus also PM) can speak about logical systems (including themselves) through Gödel numbering. Did Russell and White realize | |
| Sep 29, 2016 at 20:51 | comment | added | Vincent | This answer is the best thing I read all day today! | |
| Mar 9, 2014 at 14:48 | history | edited | François G. Dorais | CC BY-SA 3.0 |
grammar
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| Mar 9, 2014 at 14:44 | comment | added | François G. Dorais | @SébastienPalcoux: That's a fair assessment. | |
| Mar 9, 2014 at 14:42 | history | edited | François G. Dorais | CC BY-SA 3.0 |
furthermore...
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| Mar 9, 2014 at 14:26 | comment | added | Sebastien Palcoux | @FrançoisG.Dorais: I agree. Now, are you willing to go further, by saying that the non mathematically conceptualizable part of philosophy of mathematics, is meta-meta-mathematics. Next, as "Theories do not have associated meta-theories" (you said), meta-theories do not have associated meta-meta-theories, and so, meta-meta-mathematics would be non mathematically conceptualizable theories about mathematics, meta-mathematics, but also meta-physics, meta-biology... and so meta-meta-mathematics would be exactly the non mathematically conceptualizable part of Philosophy in general. | |
| Mar 9, 2014 at 14:09 | comment | added | François G. Dorais | @Qfwfq: True arithmetic is the theory of the standard model of arithmetic. This is a complete theory since every arithmetic statement is either true or false in the standard model. Unfortunately, this theory is not computably axiomatizable which makes it difficult to understand. | |
| Mar 9, 2014 at 14:02 | comment | added | François G. Dorais | @SébastienPalcoux: I don't think meta-meta-theorems make sense since the theorems are mathematical ideas so talking about them is always simply meta-mathematics. As I tried to explain in my answer, this doesn't apply to all concepts. You can have meta-meta-statements, some of which are meta-meta-facts, but you can't prove such things unless you collapse them to meta-theorems. | |
| Mar 9, 2014 at 13:56 | comment | added | François G. Dorais | @Qfwfq: A "meta-theorem" is a theorem about a theory as opposed to a theorem in the theory. A "meta-meta-theorem" would be a theorem about the meta-theory. That's a fine idea but if you want to prove something about the meta-theory you first need to formalize it so that it is a theory you can prove something about. But then your meta-meta-theorem is just a meta-theorem, isn't it? This collapse of "metaness" systematically happens if you sit down to formalize and prove a meta-meta-statement. So every meta-meta-theorem is just a meta-theorem... | |
| Mar 9, 2014 at 13:36 | comment | added | Qfwfq | Also, the phrase " the theory of true arithmetic is complete and perfectly usable as a meta-theory, but the drawback is that we don't understand what the axioms of this theory actually are" doesn't make any sense to me. Perhaps I'm just missing something (after all, I'm absolutely no expert in Logic!), but my impression is that you understand the "meta-" word as indicating something philosophical and not entirely within the reach of formal mathematics, which is not the intended meaning in my question. | |
| Mar 9, 2014 at 13:32 | comment | added | Qfwfq | This answer seems to have nothing to do with my question... In particular, I really can't make sense of the phrase "There is an intrinsic problem with the idea of meta-meta-theorems because theorems are mathematical ideas and therefore talking about them belongs in the meta-theory and therefore cannot be properly meta-meta-theoretical" and of the entire second paragraph... | |
| Mar 9, 2014 at 12:44 | comment | added | Carl Mummert | One difficulty seems to be that, as some point, the $\text{(meta-)}^n\text{theory}$ has a strong chance of being inconsistent. So that may be the obstacle that prevents meta-meta-theorems from being just meta-theorems, if the meta-meta-theory is inconsistent. This seems to be the heart of the "understandable numbers" example. | |
| Mar 9, 2014 at 12:36 | comment | added | Sebastien Palcoux | I'm not sure to understand what meta-meta-theorem means for you, but compared to the content of your answer, it's normal because if there is a perfectly understandable definition, then it's a mathematical concept and then it's no more meta-meta but meta. So meta-meta theorems are statements that can't (yet?) make sense mathematically, but still make sense for the human mind, right? Then "the Riemann hypothesis is interesting" or "A. Grothendieck is a good mathematician", or ..., are meta-meta theorems: the non mathematically conceptualizable part of philosophy and history of mathematics. | |
| Mar 9, 2014 at 4:04 | comment | added | Bjørn Kjos-Hanssen | So meta - meta - mathematical in intent is a little bit like physical in intent or financial in intent... that makes sense. | |
| Mar 9, 2014 at 3:12 | history | edited | François G. Dorais | CC BY-SA 3.0 |
deleted 6 characters in body
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| Mar 9, 2014 at 2:59 | history | answered | François G. Dorais | CC BY-SA 3.0 |