Timeline for Can classical logic have deduction with infinite steps
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
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| Jun 3, 2019 at 5:54 | comment | added | Noah Schweber | The same goes for finitary vs. infinitary formulas, etc., and the fact that rank is the right notion of length can be seen in the way that induction-on-length arguments transform into induction-on-rank arguments (or even are better viewed that way to begin with!), as well as in various other ways. | |
| Jun 3, 2019 at 5:53 | comment | added | Noah Schweber | @JishinNoben It depends on how narrowly one interprets "infinitely long." Perhaps "infinitely wide" would be a better way to phrase it: in both classical and infinitary (first-order) logics a proof is a well-founded tree of formulas with certain properties, but in the classical case that tree has to be finite while in the infinitary case it doesn't. The right notion of "length" for an infinitary proof is then the rank of that tree, which can indeed be infinite; the rank of a well-founded tree is defined recursively: the rank of a tree is the supremum of {the ranks of its child-trees +1}. | |
| Jun 1, 2019 at 16:18 | vote | accept | Pastafarian Priest | ||
| Jun 1, 2019 at 16:18 | |||||
| May 28, 2019 at 18:26 | comment | added | Pastafarian Priest | I understand what you are expressing when you break down (*). However, I am confused as to how we can go from discussing the finitude of the rules of inference and the set we are deducing the conclusion from to discussing the steps. As I understand it, the steps would be the application of these rules. For instance, the proof of the law of excluded middle is 10 steps, since there are ten different applications of the rules established in the article. How can I gather that this must be finite from the fact that the rules and premises are finite (or can be placed in a finite number of sets)? | |
| May 28, 2019 at 15:59 | history | edited | Mauro ALLEGRANZA | CC BY-SA 4.0 |
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| May 28, 2019 at 14:56 | history | edited | Mauro ALLEGRANZA | CC BY-SA 4.0 |
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| May 26, 2019 at 17:37 | comment | added | Jishin Noben | I know what countable means. If you read carefully, I am asking something different. It is formulated as a question, since I am assuming that @Mauro_Allegranza meant the last sentence not literally, but intended something else. | |
| May 26, 2019 at 17:35 | comment | added | H Walters | @JishinNoben The word countable has a meaning that can include infinite sets: "A countable set is either a finite set or a countably infinite set." en.wikipedia.org/wiki/Countable_set | |
| May 26, 2019 at 17:32 | comment | added | Jishin Noben | How do infinitary logics allow for deductions of countable length? That makes no sense, unless one has some notion of "convergence". "Long" formulas in a derivation don't make the derivation infinite. Neither do infinitely many premises. You even said it yourself? | |
| May 26, 2019 at 15:38 | history | answered | Mauro ALLEGRANZA | CC BY-SA 4.0 |