Timeline for Is there formal definition of universal quantification?
Current License: CC BY-SA 2.5
11 events
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| Feb 25, 2010 at 10:29 | comment | added | kakaz | In my opinion ( but I have to learn much more to be sure) the universal quantifier ties theory and meta theory. In fact You may have serious problems to understand what means "all things" within theory if You do not have simple formula ( in level of the language of the theory, that if for example in first order level for first order theory) stating definition of Your universe. This is the most interesting do not You think? If You use "can be any formula in the meta-language" statement then easily You get Russell paradox, so probably it is not so simple though. | |
| Feb 23, 2010 at 17:10 | history | edited | Gabriel Ebner | CC BY-SA 2.5 |
Answered additional questions.
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| Feb 23, 2010 at 7:16 | comment | added | kakaz | @Gabriel please take a look into question: I have added some remark. | |
| Feb 23, 2010 at 4:11 | comment | added | François G. Dorais | You can often avoid large cardinals by using Montague's Reflection Theorem - en.wikipedia.org/wiki/Reflection_principle | |
| Feb 23, 2010 at 1:13 | comment | added | Gabriel Ebner | Of course you can extend the definition of domain of discourse to include proper classes as well, but this does not buy you anything. Say you've got a domain of discourse that is a proper class in ZFC. Then that domain of discourse is just a set in ZFC + some large cardinal axiom. This is the same approach that is usually used in category theory to handle large categories like the category of groups or the category of sets. You simply postulate the existence of a universe (and that universe is a set), and then only work with groups or sets contained in that universe. | |
| Feb 22, 2010 at 23:57 | comment | added | kakaz | No, domain of discourse do not has to be a set. For example in sentence: "for every group, ....", domain of discourse is not a set. | |
| Feb 22, 2010 at 23:50 | comment | added | Gabriel Ebner | The domain of discourse is just a set, i.e. it doesn't make sense to talk of a "first-order" domain of discourse. So yes, the above definition will work for any domain of discourse. | |
| Feb 22, 2010 at 23:32 | comment | added | kakaz | There are different domains, so is the meaning of what You say is that: for every definition of domain of discourse, we may define quantifications as follows? Or domain of discourse must belong to some class of definition ( first order formula, second order, finite order etc?) | |
| Feb 22, 2010 at 23:07 | history | edited | Gabriel Ebner | CC BY-SA 2.5 |
Fix typo.
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| Feb 22, 2010 at 23:07 | comment | added | François G. Dorais | "To be is to be the value of a variable." (Quine) | |
| Feb 22, 2010 at 22:56 | history | answered | Gabriel Ebner | CC BY-SA 2.5 |